University of Southern California Dissertations and Theses

'Cost' And 'Utility' In The Prediction Of The Successful Junior College Student

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'Cost' And 'Utility' In The Prediction Of The Successful Junior College Student

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Content T h is d isserta tio n has been 61— 6281 m icro film ed ex a ctly as r eceiv ed DERIAN, A lbert Steven, 1920- "COST" AND "UTILITY" IN THE PREDICTION O F THE SUCCESSFUL JUNIOR COLLEGE STU DENT. U n iv ersity of Southern C alifornia E d .D ., 1961 Education, p sych ology University Microfilms, Inc., Ann Arbor, Michigan "COST" AND "UTILITY" IN THE PREDICTION OF THE SUCCESSFUL JUNIOR COLLEGE STUDENT A Dissertation Presented to the Faculty of the School of Education The University of Southern California In Partial Fulfillment of the Requirements for the Degree Doctor of Education by Albert Steven Derian June 1961 This dissertation, written under the direction of the Chairman of the candidate’s Guidance Committee and approved by all members of the Committee, has been presented to and accepted by the Faculty of the School of Education in partial fulfillment of the requirements for the degree of D octor of Education. D a te.................................................. ^*2^-7 ^ ....... / D ean G uidance C om m ittci \irman TABLE OF CONTENTS Page LIST OF TABLES................................. iv LIST OF GRAPHS................................. viii Chapter I. INTRODUCTION ............................. 1 The Purpose of the Study Definitions of Terms Statement of the Problem Organization of the Remaining Chapters II. REVIEW OF THE LITERATURE................. 12 Prediction of College Achievement Prediction of Categories The Discriminant Function Empirical Studies Using the Discriminant Function The Principle of Equal Likelihood Nonparametric Methods of Combining Measurements for Prediction Empirical Comparisons of Methods of Prediction Measures of Association Summary III. SOURCES OF DATA AND PROCEDURES............ 96 Description of Subjects Basic Data Criterion of Success Selection of Variables The Discriminant Equation Guilford and Michael's Formula for a Critical Score A Nonparametric Method of Determining Cutting Scores Wherry's Multiple Biserial Regression Equation Cross-validation on Group B 11 Chapter Page Croup A Compared with Group B Analysis of Sex Differences in Prediction of the Criterion Summary IV. ANALYSIS OF THE FINDINGS................. BIserlal Correlations The Discriminant Equation Principle of Equal Likelihood The Nonparametric Method The Multiple Biserlal Regression Equation Cross-validation on Group B Comparison of the Efficiency of the Different Methods of Prediction Statistical Verification of the Basic Hypothesis Group A Compared with Group B The Discriminant Equation for N = 200 Sex Differences In Prediction Cost and Utility with the Nonparametric Method for Men and Women Comparison of the Efficiency of the Discriminant Equation and the Nonparametric Method for Men and Women V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . Summary Conclusions Recommendations 119 221 BIBLIOGRAPHY APPENDIX . . 240 257 LIST OP TABLES Table Page I. Number of Successful and Unsuccessful Students In Each Criterion Classification for Groups A and B ... 121 II. Number of Successful and Unsuccessful Students on the 60 unit Criterion Resulting from Each Classification on the 24 unit criterion . ............ 122 III. Summary of BIserial Correlations for Groups A and B between Predictive Variables and Both Criteria of Success . 124 IV. Mean Scores for Successful and Unsuccessful Students in Group A on GPA, V Score, and S Score.... 131 V. Per Cent of Contribution of Each Variable in the Three Variable Discriminant Equation (Group A ) ................... 133 VI. Per Cent of Contribution of Each Variable In the Two Variable Discriminant Equation (Group A ) ................... 136 VII. Intercorrelations of Selected Predictive Variables for Group A ................ 136 VIII. Probability of Success Based on GPA and V Score and the Discriminant Function (Group A ) ................... 139 IX. Two-way Frequency Distribution of GPA and V Scores for Group A ......... 141 X. Cost and Utility for Group A Based on the Discriminant Equation............ 142 XI. Per Cent of Successful Students for Each Interval in the Distributions of GPA, V Score, and S Score (Group A ) ........ 147 iv Table Page XII. Cost and Utility for the Method of Equal Likelihood (Groups A and B) ... 153 XIII. Cost and Utility for the Nonparametric Method (Groups A and B) ........ 156 XIV. Compensating Cutting Scores (Alpha and Beta)........................... 160 XV. Cost and Utility for Two Compensating Scores............................... l6l XVI. The Cost and Utility for the Cross- validation of the Discriminant Equation............................. 166 XVII. A Comparison of the Efficiency of the Discriminant Equation and the Nonparametric Method at Minimum Cost . . 170 XVIII. A Comparison of the Efficiency of the Different Methods of Prediction at Approximately 10-12 Per Cent Cost . . . 172 XIX. A Comparison of the Efficiency of the Different Methods of Prediction at Approximately 1 5 -2 5 Per Cent Cost ... 173 XX. A Pour-cell Contingency Table of Frequencies of Unsuccessful Students Who Were Accepted and Rejected by an Optional Cutting Score and the Discriminant Equation at a Cost of 3.8 Per Cent......................... 177 XXI. A Four-cell Contingency Table of Frequencies of Successful Students Who Were Accepted and Rejected by an Optional Cutting Score and the Discriminant Equation at a Cost of 3.8 Per Cent......................... 179 XXII. A Four-cell Contingency Table of Frequencies of Unsuccessful Students Who Were Accepted and Rejected by a Single Cutting Score and the Discriminant Equation at a Cost of 10-12 Per Cent....................... 180 v Table ~ Page XXIII. A Four-cell Contingency Table of Frequencies of Successful Students Who Were Accepted and Rejected by a Single Cutting Score and the Discriminant Equation at a Cost of 10-12 Per C e n t ..................... 182 XXIV. Analysis of Variance of GPA ...... . 184 XXV. Analysis of Variance of V Score........ 184 XXVI. Analysis of Variance of S Score........ 185 XXVII. A Comparison of the Intercorrelations of the Predictive Variables for Groups A and B ........... 187 XXVIII. Chi-square Analysis of the Number of Successful Men and Women in Each Group............................... 189 XXIX. Mean Scores for the Successful and Unsuccessful Students on GPA and V Score, N = 200 191 XXX. Per Cent of Contribution of GPA and V Score in the Discriminant Equation for N « 200 ......................... 191 XXXI. Biserial Correlations for Men and Women with the 24 Unit Criterion . . . 193 XXXII. Analysis of Variance for Men and Women on G P A ........................ 197 XXXIII. Analysis of Variance for Men and Women on V S c o r e .......... 197 XXXIV. Mean Scores for Successful and Unsuccessful Men on GPA and V Score . . 199 XXXV. Per Cent of Contribution of GPA and V Score in the Discriminant Equation for Men ................. 199 XXXVI. Mean Scores for Successful and Unsuccessful Women on GPA and V Score . 201 vi Table Page XXXVII. Per Cent Contribution of GPA and V Score In the Discriminant Equation for Women ............... 201 XXXVIII. Summary of Cost and Utility for the Discriminant Equations for Men and Women............................. 204 XXXIX. Cost and Utility of Nonparametric Scores for M e n ................... 209 XL. Cost and Utility of Nonparametric Scores for Women................. 210 vii LIST OP GRAPHS Graph Page 1. Cost and Utility for Group A Based on the Discriminant Equation ................... 143 2. Cutting Score for GPA Using Method of Equal Likelihood......................... 148 3. Cutting Score for V Score Using Method of Equal Likelihood..................... 149 4. Cutting Score for S Score Using Method of Equal Likelihood ..................... 150 viii CHAPTER I INTRODUCTION The recently developed Master Plan for Higher Education in the State of California proposes that the requirements for admission from high school for both state colleges and the University of California be changed to reduce the number of high school graduates who will be eligible for admission as beginning freshmen. The Master Plan also re-affirms previous studies of higher education in California in the recommendation that one of the primary functions of the junior college is to provide the first and second years of a university or college program. The net result of these two recommendations plus an Increasing population in the state and an ever-increasing percentage of high school graduates who go on to college is to place the junior college transfer program in a position of extreme importance in the total picture of higher education in California. The Master Plan is designed to carry out the continuing policy of the State of California that free higher education be available to all of those who can succeed in such a program. The present practice, and that proposed for the future by the Master Plan, is to set 1 certain minimum standards of performance in terms of the high school program and to accept those who meet these standards for entrance as freshmen to the University or to the state colleges. Those students who do not meet the minimum standards have the opportunity to establish their eligibility by meeting certain performance criteria during their work in the junior college. The California Education Code permits all high school graduates and anyone else who is 18 years of age or over and can profit from the instruction to enter the junior college. Recent legislation has spelled out a more rigid retention policy for all junior colleges but no change has been made in admission standards. With current predictions indicating an increasing number of students in higher education and also indicating that the junior college will handle a larger percentage of the lower division students, there is the definite possibility that the continued growth of the program may well be limited by finances for higher education or availability of qualified teachers for higher education or both. In any case, it may become desirable to select those students for the junior college transfer program who have the greatest probability of achieving the minimum standards which will allow them to continue their education at the University or the state colleges. If necessary, a pre-transfer program could be established for those who were not selected in order that they may establish their eligibility for the transfer program so that the door would never really be closed to a student who desired to attempt a program leading to a four-year college degree. The procedure of selective admission to certain classes has been practiced for some time in the Junior colleges of California. An inspection of catalogs shows that selective admission has been practiced most fre quently in English, mathematics, chemistry, and physics. The determining factor is usually a certain grade in a previous course in the same subject or the passing of a critical score on a test. Long Beach City College, as of Fall i9 60, is an example of a Junior college which is attempting to classify new students as provisional, probationary, or regular at the time of entrance. The problem of clinical versus actuarial pre diction is far from settled, but if consideration is limited to actuarial prediction only, it is apparent that the methods of selection used in business and industry cannot be used by the colleges if the emphasis is to be on maximum opportunity rather than on obtaining a maximum percentage of successful students among those admitted. It is not sufficient under the present policy for higher education in California to consider as efficient a selection program which results in a relatively high percentage of successful students among those given the opportunity. This type, of prediction is sufficient for business and industry because of the limits placed on the number of workers hired and because they have no designated responsibility for those not hired. The junior college in establishing any system of selection must consider how many students are being eliminated by its selection procedure who would have been successful if they had been given the opportunity to try. Two elements must be considered simultaneously if a selection program for the junior college transfer student is to meet the two sets of conditions imposed upon it. There must be an indication of how many students, who would not be successful in the program, will be eliminated and hence reduce the pressures of finance and teaching personnel; and at the same time indicate how many potentially successful students have been deprived of the opportunity to begin their higher education. The selection of students in a transfer program need not be concerned with how well a person succeeds nor by what margin he fails to meet the requirements of the situation. Success in this Instance is based upon a minimum set of standards which must be met. The purpose of the prediction is to make a minimum discrimination, i.e., to predict the dichotomy of successful students and unsuccessful students and not a relative ranking of academic success such as their grade point average. The problem of prediction centers around three questions: 1. What variable or combination of variables will best Indicate the potentially successful transfer student? 2. What is the probability that an Individual with a given score or combination of scores will be in a designated category? 3. How many errors of prediction will be made in each predicted segment of the successful- unsuccessful classification? X. THE PURPOSE OF THE STUDY The purpose of this study was to define a two-way classification scheme, in terms of cost and utility, which would indicate the number of errors of prediction for evaluating the selection of students for the junior college transfer program and to compare several methods of predicting the dichotomy of success and non-success on the basis of this two-way classification. 6 II. DEFINITIONS OF TERMS The following terms will be used throughout the present paper according to the definitions given below. Utility.— Utility is defined as the percentage of transfer students who do not satisfy the criterion of success and who are so designated by prior information from high school grades and/or test scores. Cost.— Cost is defined as the percentage of transfer students who do meet the criterion of success but who have been incorrectly designated as unsuccessful on the basis of prior information from high school grades and/or test scores. Cutting score.— A cutting score is the minimum acceptable value on a scale of continuous measurements. Multiple cutting score.— A multiple cutting score is a combination of cutting scores which are applied according to some predetermined set of rules. Pair of cutting scores.— A pair of cutting scores is a combination of two cutting scores for which an individual must exceed the minimum value on both in order to be acceptable. Optional cutting score.— An optional cutting is a multiple cutting score for which the rule states that if the individual exceeds the minimum value in any one of the variables then he will be acceptable* Compensating cutting score.— A compensating cutting score is a predetermined series of paired cutting scores which is constructed so as to allow a high score in one variable to offset a low score in the other variable. Most efficient cutting score.— The most efficient cutting score is the selection procedure which provides the maximum utility for a given cost. III. STATEMENT OF THE PROBLEM The problem to be investigated involves the determination of which of the following four methods is most efficient in predicting potential success and non success in the Junior college transfer program when efficiency is defined in terms of cost and utility; 1. The discriminant function? 2. Multiple cutting scores based on Outtman's principle of equal likelihood? 3* A nonparametric method based on inspection of a graph of the variables and the criterion with cutting scores selected so as to yield maximum utility with a low level of cost? 4. Wherry*s multiple biserial regression equation? Questions to be answered.— In the investigation of the problem, an attempt will be made to answer the following additional questions: 1. What variables should be used for prediction of success in the Junior college transfer program? 2. If the methods do show a difference in efficiency, what are the relative magnitudes of these differences? 3. Do the four methods tend to pick the 3ame individuals as being potentially successful and unsuccessful? 4. In the nonparametric method, what are the relative efficiencies of the methods of paired cutting scores, optional cutting scores, s.nd compensating cutting scores? 5. Can a single quantitative measure or index express the relative efficiency of the results given in a four-fold contingency table as well as the two measures of cost and utility? 6. Do sex differences affect predictive efficiency? 9 Statement of the hypothesis.— The hypothesis which has been formulated is that in terms of the utility achieved for a given cost, there are no differences among the four methods of predicting the dichotomized variable of success and nonsuccess in the Junior college transfer program. The hypothesis is to be tested by identifying the total population of transfer students who begin their higher education in a given Junior college during the summer or fall of a particular year. One random sample will be drawn from the total transfer group and the data from this group will be used to establish the various prediction formulas according to the four methods listed above. After the prediction formulas have been estab lished, another random sample will be selected from the total transfer group and the four prediction formulas will be applied to the second sample and compared on the basis of their cost and utility. Two assumptions are being made. The first is that the junior college transfer student is adequately represented by all those students who attempt 24 units of college accredited course work during their first year of junior college. The second assumption is that the current regulations for admission of junior college students to the state colleges is an adequate criterion 10 of success In the transfer program. The predictive variables will be limited to the two sub-tests and the total score on the School and College Ability Teat (135), the three sub-tests and the total 3core on the Cooperative Reading Test (28), and an index of high school grades. The high school grades to be considered will be limited to those obtained in the tenth, eleventh, and twelfth grades and further restricted to those obtained in the following subject areas: English, foreign language, mathematics, natural science, and social science. IV. ORGANIZATION OF THE REMAINING CHAPTERS The remaining chapters are organized in the manner indicated below. Chapter II contains the review of the literature and includes material on all four of the methods of prediction which will be used as well as a review of the general problem of the prediction of categories and the prediction of college achievement. Chapter III contains the sources of the data and the procedures used in the study. Chapter IV contains the analysis of the findings and includes the cross- validation data and the data on sex differences. Chapter V includes a summary of the problem, the investigation, and the findings, along with the conclusions and the recommendations for the junior colleges and for additional research. CHAPTER II REVIEW OP THE LITERATURE The usual criterion for determining success In college has traditionally been the grade point average or honor point ratio as it is sometimes called. It is generally recognized that this measure leaves a lot to be desired in terms of a reliable criterion; however, no other criteria have been found to be any more effective in evaluating the general classification of "successful college student." Most of the studies concerning the prediction of college achievement which are reported in the literature are reports of the zero order correlations of various predictive variables with grade point averages or of the multiple correlations of two or more predictive variables with grade point averages. Several compre hensive reviews of the many studies made over a period of twenty-five to thirty years are available and will be summarized along with some recent studies of this type. For reasons which have been given in detail in Chapter I, the criterion of grade point average has been reduced to an artificial dichotomy in terms of a student having a 2 . 0 grade point average or better in which case he has been classified as a successful transfer student; if his grade point average is below 2. 0, then he is 12 13 classified as unsuccessful. Therefore, some of the literature which relates to the prediction of categories from measurements will be reviewed. One of the techniques for classifying an individual on the basis of n measurements into one of k classifications is called discriminant analysis. The more specific term, discriminant function, is generally used to refer to the case of classifying an individual on the basis of n measurements into one of two classifi cations. The discriminant function has been used in this study as one of the methods of predicting the dichotomy of success and nonsuccess and the literature pertaining to its development and use will be reviewed. The "principle of equal likelihood" refers to the problem of classification in terms of making decisions which will yield the maximum number of correct classifi cations with a minimum number of errors of both kinds, i.e., predicting success for one who i3 unsuccessful and predicting nonsuccess for one who is really successful on the criterion measure. One of the methods used to obtain cutting scores for classification purpose in this study is based on the principle of equal likelihood. A number of the mathematical models pertaining to the prediction problem use this principle in the derivation of the model. A few references will be cited which deal with the 14 practical application of this principle to the problem of determining cutting scores. A variety of methods and procedures for predicting categories from measurements will be reviewed in the section on nonparametric methods including approximate weighting techniques, pattern analysis, and configural scoring. The emphasis here is on methods which do not require the assumptions of normal bivariate distributions for predictors and criteria. Some evidence is available in terms of empirical comparisons between methods of prediction. In most cases it consists of a comparison of the discriminant function with multiple regression or a comparison of multiple regression with some form of nonparametric technique. These studies will be reviewed. The last section of the review of the literature will be concerned with the various types of Indices which have been suggested to convey a meaningful interpretation of the degree of association or accuracy of the predic tion of a dichotomous criterion on the basis of some predetermined rule of selection from a series of measure ments. In other words, after the selection has been made and the criterion values are known, the problem reduces itself to a 2 x 2 contingency table and an index is needed to convey the magnitude of "hits" and "misses" in the selection process. 15 I. PREDICTION OF COLLEGE ACHIEVEMENT Durflinger cites several summaries of college achievement which appeared In the literature from 1931 to 1934 and summarizes the additional studies from 1934 to 1942 (34). For the studies prior to 1934, he reports a median correlation of .45 for measures of intelligence and college grade point average; and for the studies since 1934, he reports a median correlation of . 5 2. The fact that the more recent tests used at the college level are designed for college students and that college grades are Increasingly dependent upon examinations are suggested by Durflinger as possible reasons for the rise in the median correlation. He reports a drop in the median correlation from .5 5 to .48 for the same two periods for the comparisons of achievement examinations based on high school subjects and college grade point average. There is a smaller number of such studies since 1934 and Durflinger attributes this to the acceptance of the idea that the number of units in a subject In high school has no correlation with later success in college. The median correlation reported for high school scholar ship and college scholarship Is .5 5 with the range being .50 to .60. An intelligence test, high school grade point average, and an achievement test are reported by Durflinger to be the best combination of predictors. The 16 median multiple correlation for the studies reviewed is reported as between .6 0 and . 7 0. Garrett made a review and interpretation of the various studies reporting factors related to scholastic success In colleges of arts and sciences and teachers colleges (53). This review was reported in 1949 and contains a history of college entrance requirements along with an extensive bibliography. The following is a summary of Garrett's findings: 1. The high school grade point average is consistently reported as the best predictor, with a median correlation of . 5 6. 2. General achievement tests are ranked second with a median correlation of .49; however, special achievement tests, i.e., tests in specific subject areas, do not do as well with a median correlation of .40, 3. In this summary, scholastic aptitude tests are classified along with the more general intelligence tests and rank third as predictors of college success with a median correlation of .47; however, it is to be noted that when the scholastic aptitude tests were considered separately they performed as well as the general achievement tests. Garrett reports the median correlations of the two "psychological examinations," the American Council on Education and the Ohio State, as .49. Special aptitude tests ranked fourth as predictors of academic success with a median correlation of .41. The criterion of first year grades yields higher correlations than using first quarter or semester as the criterion. Also, the first year criterion yields slightly higher correlations than a criterion of over-all grade point average for the four years. The records of graduates from one high school show higher correlations than records from several schools. Various kinds of rankings yield smaller correlations than does the grade point average. There is no relationship between the number and pattern of high school subjects and later college success. The median multiple correlation for high school grade point average and intelligence with grade point average is . 6 7. 18 10. A third variable adds very little to the multiple correlation and a fourth variable essentially nothing. In general, the findings of the more recent studies to be cited below are in agreement with the conclusions of Garrett listed above. Any findings not in agreement with these conclusions will be noted. The problem of selective admission to college.— The accuracy of predicting college success in terms of the coefficient of correlation has remained fairly stable over a period of years according to the findings reported above. Opinion remains divided as to whether it is possible to improve prediction by improving the tests, the means of combining the results of the tests, and/or by improving the criterion measures of success in college. There Is also difference of opinion as to whether any attempt should be made to make admission to college more selective because of the amount of error of prediction that cannot be eliminated even under optimum conditions. One of the attempts to answer some of these question has been to start testing for predictive purposes at an earlier age. The long-range effects of such a program in terms of the individual's self-concept and the development of vocational interest patterns are beyond the scope of this investigation. Most of the studies 19 reported in the literature are in terms of the correlation coefficient between tests taken in the ninth grade or earlier and college grades. Samenfeld investigated scores obtained in the ninth grade on the ACE* and scores obtained on the ACE in the twelfth grade by the same students (131). Correlations were obtained between these scores and the grade point average for the first year in college. The correlation for ninth grade scores was .39 and for twelfth grade, .34. The OSPE* was given to these same students in the twelfth grade and had a correlation of .5 2 with college grades. A multiple correlation of .6 8 was obtained with the OSPE and high school percentile rank. Scannell investigated 3*202 students who had taken the ITED* as high school seniors from 1948 to 1952 and who had enrolled at the State University of Iowa or Iowa State College in the fall semester following their graduation from high school (133). He formed subsamples for the various grade levels and reported that general academic success was predicted more efficiently for each grade in school with a correlation of .45 for grade four and a correlation of .6 3 for grade twelve. Scannell concluded that with restriction in range taken into ♦Because of the number of references to be made to certain tests in the literature, abbreviations will be used for simplicity. ACE will represent the American Council on Education Psychological Examination: OSPE, the Ohio State Psychological Examination; and ITED, the Iowa Tests of Educational Development. 20 account the elementary tests give good prediction. He reports a corrected correlation of .8 5 for eighth grade scores on the ITED and college freshman grade point average. In DurfUnger's summary, he reports one correlation of .5 8 for tests at the elementary level and college grades (34). Travers argues that tests are likely to have limited predictive value unless they are used on individuals living in a controlled environment which provides equal opportunities and equal distractors for all. He gives military schools and the service academies as examples of controlled environment (156:293). Travers feels that little can be done to improve present day scholastic tests as long as they are used In the type of culture which characterizes the United States. His suggestion Is to study the criterion (grades) and try to improve prediction through a better definition of the objectives of education (156:294). Hoyt studied the relationship of junior college performance to success at Kansas State University (75). The correlations between junior college grade point average and college grades varied with respect to original school, class at time of transfer, and sex. The longer the students attended junior college the higher was the correlation between junior college grades and grades at Kansas State University. By using matched samples, Hoyt compared "early" grade point average between transfer students and students originally enrolled at the university with their "later" grade point averages. The students beginning their higher education at the univer sity showed higher correlations between "early" and "later" grade point averages than those students who started at the Junior colleges. A comparison of "later" averages for both groups showed no differences in achievement. A comparison of the mean grade point averages for different Junior colleges, with the ACE scores used as a control on academic ability, suggests a different degree of predictability for students from different schools. Wagner and Strabel studied the results of the New York Regents Examinations for a period of six years at the University of Buffalo and found that the graduates of the Buffalo high schools obtained higher correlations with college grades than did the non-Buffalo high school graduates (161). The differences in correlations between boys and girls was even larger. For boys from the Buffalo high schools the correlation was .60 and for the girls, . 7 6. For boys from outside of Buffalo the correlation was .52 and for girls, . 6 6. Three other factors tended to show some improvement in prediction: 22 (1 ) age at graduation, ( 2) rank among siblings, and ( 3) amount of foreign language. Fricke takes a different viewpoint with regard to raising the predictive efficiency of tests for admission purposes (50). He indicates that low correla tions are to be desired because this should indicate that the selection process is working toward providing a more homogeneous college population. The correlations which will shrink the most are for those variables which are being used for selection. This shrinkage of correlations is one of the criteria that he proposes for evaluating admission procedures. In addition, he proposes two other criteria: (l) judge the quality of the students by the mean scores of the predictive variables, and ( 2) judge the homogeneity of the group admitted by the standard deviations of the predictive variables. 0 *Connor describes the admission procedure and its development as used by the Air Force Academy (121). This study is a good illustration of the use of a large number of predictive variables when the number of students to be selected is a small percentage of the number of applicants. It also focuses attention on a situation in which more than one criterion is necessary. The selection procedure is designed to pick out those applicants who will be good academic risks as well as good potential pilots. Chauncey and Frederiksen indicate that their preference is for rank in class instead of the grade point average because it eliminates some of the varia bility due to grading practices (99:87). In their opinion, the highest potential of tests under conditions as io sal as we can expect would be approximately a correlation of .7 5 and that it will be seldom possible to exceed .7 0 ( 9 9: 9 2). Leonard, after reviewing studies of single and multiple correlation and the eight-year study of the Progressive Education Association, reaches the conclusion that general success in high school together with general ability is a better indication of college success than any particular pattern of courses (9 7: 3 3 2). A rather unique situation was studied by Smith at the University of Kansas (140). A need arose to improve the selection process for football scholarships. High school grades had been used in the past and were found unsatisfactory for this purpose. His criterion was a dichotomy according to whether or not the scholar ship holders did or did not remain eligible for their full period of varsity competition. The variables used were the ACE, Cooperative Reading Test. Cooperative English Test, and the Kansas University mathematics test. 24 All tests except the reading test showed significant differences between the two groups; however, the ACE was the only one reported a3 worth exploring for the purpose of selection. No attempt was made to combine variables. A critical score was adopted which eliminated 75 per cent of the "drop-outs" and retained 66 per cent of the "stay-ins." Bamow discusses the median correlations for predicting college success in terms of E, the index of forecasting efficiency, and concludes that the greater part of the difference in these predictions is still unaccounted for and recommends the introduction of non intellectual influences of college achievement in the prediction equation (7)« A number of the studies cited by Durflinger (34) and Garrett (53) included personality and interest inventories and questionnaires as predictive variables without much success. Nonintellectual evidence of various sorts is used by counselors and admission officers, especially in the "borderline" cases which do not quite meet the standard admission requirements. In the absence of predictive equations containing the non- intellectual variables, they are usually applied in a "clinical" manner with weights to be determined for each individual case by the person making the admission decision. 25 Meehl has reviewed the available evidence to support the idea that clinical Judgment can be more flexible and take into account those factors which cannot be measured on tests or which must be interpreted in terns of patterns or profiles and hence yield greater accuracy of prediction than can be achieved by purely actuarial means. In the twenty studies which actually compared the two methods, the clinical procedure was more efficient in only one. The actuarial method was equal to or exceeded the clinical method in nineteen studies (111:119). One of the studies reviewed by Meehl was made by Sarbin in which academic success was the criterion to be predicted from case studies and also to be predicted from a previously derived regression equation. The correla tion coefficients demonstrate that the case study was no more efficient than a two variable regression equation. Even when the counselors were given the actuarial data their clinical Judgment could not add anything to the accuracy of prediction. Sarbin*s analysis of the predictions suggests that the clinical method takes behavior segments with known predictive weights and applies other weights to them ( 1 3 2). Angell (2) has recently written on the "clinicalM aspect of the admission problem and presents arguments 26 for the methods of multiple differential prediction and multiple absolute prediction which have been developed by Horst (72 and 73). These techniques will be discussed in a later section of this chapter. Unpromising college students who graduate.— Unpromising students who graduate is the title of a study by Munger and represents the type of evidence lacking in the usual type of report which includes only the correlation coefficients between the predictors and the grade point average (120). He reports on an experi ment in which a part of 891 students who graduated in the lower one-third of their high school class was admitted to the University of Toledo on a trial basis. The mean grade point average for this group at the end of the first semester was .82 (C = 1.0) and the mean grade point average for all other freshmen was 1.22. Of the unpromising group, 43 per cent overcame difficulties involving two or more failures in specific subjects and went on to graduate from college. Munger*s conclusion was that slightly more than one-half of the unpromising students who graduated were really rather typical students. Brown and Nemzek studied the college success of 288 recommended and 53 nonrecommended students who enrolled in five different colleges from one Detroit 27 high school (23). The mean grade point average in the freshman year for the recommended students was 1 .3 5 and for the nonrecommended, .98 (C= 1.0). In terms of a criterion of 1 . 0 or better at the end of the freshman year, 6 9 per cent of the recommended students were satisfactory and 5^ per cent of the nonrecommended students were doing satisfactory work. Munger In an earlier study used as criteria: (1) How long does a student stay in school? (2) Does he graduate? He divided the students into three groups on the basis of high school performance, low, middle, and high (119). The OSPE showed differences between all three groups but no relation to persistence In college. The first semester grade point average showed a definite linear relation to persistence for each of the three groups. Hunger's conclusion was that the best answers to the two criterion questions would be based on work in college. Kerr has studied the freshman class at the University of Arkansas in 1956 with a criterion of C average for satisfactory achievement ( 8 8). The method of analysis was to calculate the percentage of satisfac tory students for various scores on the predictive variables and for combinations of these scores. His conclusion was that no criteria have become evident for 28 selective college admission which would not do an injustice to an appreciable amount of students. Henderson and Malueg found in a study of Los Angeles City College students who had completed at least one semester and did not have a foreign ethnic background that there was no difference in ACE scores for those who had graduated from high school when compared with those who had not graduated from high school ( 6 6). Harder compares students who completed four semesters at the University of California at Davis with students who dropped out during the freshman and sophomore years (64). Comparisons were made on high school grade point average, ACE scores, and reading scores, and no differences were found between those who stayed and those who dropped out. These findings could be interpreted to support Fricke's arguments that there should be no difference if proper selection has taken place ( 5 0). Pugh reported on the policy which has been established by the West Virginia Board of Education for selective admission to their state colleges (125). The studies in that state have covered a period of at least ten years and the determination of requirements was based on two criteria to which most college admission officers could agree, but they are not all certain of the best way to achieve them. The criteria are: (l) to keep out 29 those who cannot succeed, and ( 2) to encourage desirable students to enter. The requirements which the Board of Education has adopted are of the optional multiple cutting score type. A student who is in the upper 75 per cent in terms of high school rank will be admitted auto matically. A student in the lower 25 per cent will be admitted if he scores at the 25th percentile or better on the School and College Ability Test, using West Virginia norms. Aptitude vs. achievement tests for the prediction of college achievement. Although Garrett in his review indicated that general achievement tests ranked second behind high school grade point average in terms of median correlations reported in the literature, it was pointed out that the more widely used scholastic aptitude tests ranked Just as high on the same criterion (53). Levine has given a general discussion of the differences in the two types of tests and states that they both measure the end-products of the Interaction of innate ability and motivated learning, and that the achievement tests tend to supplement the aptitude tests by measuring the more specific valid variance which is present in the criterion (9 8: 5 2 1-2 3 )0 It is possible that a Bingle criterion may require a test that is both a measure of aptitude and achievement. 30 Lennon and Schutz have made a summary of 479 unpublished correlations between common group tests of intelligence and various group achievement tests; the period covered is from 1940 to 1956 (96). The range of correlations is from .2 6 to .8 6 with a median correlation of . 6 5. It would appear difficult to make any general izations concerning the over-all relations between the two types of tests with the range of data reported in this study. 1 Travers has presented another extensive review of significant research on the prediction of academic success, including a bibliography of 272 entries. He cites evidence to support the New York Regents Examina tions as one of the best Instruments for the prediction of college success. His ranking of the relative value of the predictors is the same as Garrett's (53). Travers states that the OSPE will identify those who will not appear in the higher ranks of scholarship but will not Identify those who will fall. His findings also confirm the fact that actuarial prediction is as efficient as any other method and that there is little evidence that counseling techniques can either upset the predictions or provide better predictions. It is his conclusions that the evidence indicates that observing a sample of the student's college work Is the only satisfactory method of selection (31sl47“90). The ITED composite score was found to yield higher correlations with college grade point average than the total score on the ACE by O'Neil who reported on freshman students at the College of San Mateo, San Jose State College, Stanford University, and the University of California at Berkeley (123). The ITED also gave a higher multiple correlation when combined with high school grade point average than did the scores on the ACE. The best multiple correlations for the two variable equations were .54 .54 .83 .5 2 in the same order as the listing of the Institutions. The composite score on the ITED was a better predictor than any combinations of its sub-tests. Tracy, in predicting the academic success of junior college business students, started with five • predictor variables: ( 1) I.Q. score, ( 2) high school rank, (3) grade point average In high school English classes, (4) composite score on the ITED, and (5) grade point average In high school algebra courses (154). He found that he could reduce the predictors to (3) and (4) only and still obtain a multiple correlation of .6 0 as compared to a multiple correlation of .6 3 for all five variables. He also found that selected sub-tests of the ITED when weighted in a regression equation gave a 32 slightly higher correlation than the composite score. The selected sub-tests when combined with the high school English grade point average gave a multiple correlation of .64 with all college units attempted. When only the first semester was used as a criterion, a slightly lower correlation was obtained. In an exploratory study for the Educational Testing Service, French compared results of the Scholastic Aptitude Test of the College Entrance Examination Board with a variety of new tests and item types (49). When the tests on government information and literature Information were corrected for restricted range and test length, they proved to be the better predictors of academic success. The college grade point average as a criterion of success.— Most of the studies reported in the litera ture use the grade point average for the first year of college work as a criterion of success. The freshman year is considered to be a better criterion than the first semester or quarter because of the need for a period of adjustment to the new situation by many stu dents. Very few studies have reported on long-range predictions of grade point averages or even compared the freshman prediction to achievement at the end of the sophomore year. Munger's studies, which were reviewed 33 previously, used graduation as a dichotoraous criterion in an attempt to evaluate the degree of persistence in the college program (119 and 120). Hoyt’s study of junior college transfers indicated that in predicting "later” grade point averages from "early" averages, the number of semester on which the prediction could be based was a factor in increasing the accuracy of prediction ( 7 5). O'Neil (123) and Scannell (133) in their studies with the ITED both found that the high school grade point average was a better predictor of freshman grades than of sophomore grades. Scannell also found this to be true for the composite score of the ITED when given in the twelfth grade. O’Neil reported correlations ranging from .8 6 to .9 3 between freshman grades and sophomore grades. French in hiB exploratory study cited above reported that four-year cumulative average validities do not differ consistently from freshman validities (49). This also holds true in major subject areas. The verbal score of the Scholastic Aptitude Test and the high school grade point average yeilded the best over-all correlations with the freshman average and the four-year average. Buckton and Doppelt have investigated freshman tests as predictors of scores on graduate and professional school examinations. Their conclusion is that the freshman tests are relatively good predictors of graduate examinations (25). They report the following correla tions: (l) .6 0 to .7 0 with medical and law examinations, (2) .5 0 to .6 5 with the Miller Analogies Examination, and (3) .30 to .50 with the Graduate Record Examination. The high school grade point average tends to lower the obtained correlations when combined with the freshman test scores; also, the senior index (four-year grade point average) does not predict as well as the freshman tests. High school grade point average and high school percentile rank as predictors of college success.— Most of the studies reviewed have used the high school grade point average as a predictor rather than the high school rank. In most cases the availability of the data would be a determining factor in making a choice. Very few studies have used both and compared the two under similar conditions. Chauncey and Frederiksen have stated that they prefer to use the rank because it eliminates some of the variability due to grading practices ( 9 9: 8 7). Studies in West Virginia have resulted in using the high school rank as one of the options of admission to the state colleges (125). Of the studies reviewed, only Scannell has reported on the efficiency of both methods on the same group of students, and his findings favor the grade point average consistently (133). Garrett 35 indicates that the grade point average is constantly- reported as the best predictor but he makes no reference to the high school rank as a unique predictor ( 5 3). Work has been underway since 1953 at the University of Oregon to find a satisfactory index of high school performance. Some of the findings have been reported by Carlson and Mllstein (26). The criterion used in their studies is the first quarter grade point average. The term "prep rating" is used to designate the ratio of "A" units to all units attempted and the corre lation obtained was . 5 6, whereas the regular grade point average had a correlation of .6 2 and a correlation of .6 3 was found for the academic grade point average. The total number of academic units had a correlation of .1 1 and a correlation of .14 was found for the total number of all high school units. No attempt at ranking the students was reported. The index finally selected was a ratio of A's and B's to the total units; however, no correlation figure was reported for this index. Berdie states that the most efficient predictors of academic success at the University of Minnesota since 1923 have been a score on a college aptitude test and the percentile rank in the high school graduating class ( 9* 1 9 2). The predictive validity of the School and College Ability Test. Very few studies on the School and College Ability Test have appeared in the literature because it is a relatively new test. Kennedy reports correlations obtained between this test and grade point averages for the Fall semester of 1956 at San Fernando State College ( 8 7). Except for a small group of business majors (N = 29), the verbal score showed slightly higher corre lations than the total score. In general, the correla tions were: verbal score, .6 0; quantitative score, .2 7; and total score, . 5 5. Weeks has compared the School and College Ability Test with the ACE for 122 full-time freshmen at Eastern Michigan College ( 1 6 7). He found the total score on the School and College Ability Test to be a better predictor than the verbal score and the combination of total score and high school grade point average to yield a multiple correlation of .5 8 compared to the multiple correlation of .53 for the ACE linguistic score and high school grade point average. His conclusion was that the School and College Ability Test did as well or better than the ACE for this sample. This statement was made with some reservation because the ACE correlations were lower than usual• The School and College Ability Test was one of five academic aptitude tests which were compared by Juola at Michigan State University in 1957 (84). The other tests used were the College Qualification Test, the Scholastic Aptitude Te3t, the ACE, and the OSPE. A complete matrix of criterion correlations and inter correlations is given in his article. Several different groupings of subjects were used as criteria. In general, the total scores on all the tests are better predictors than the sub-tests and the differences between the correlations with the grade point average criteria are very small. The total score on the ACE for men was the only category to have a consistently lower correlation with the criteria. Sex differences in prediction.— Most of the early work on the prediction of college success did not investigate sex differences as factors to be studied. For each predictive variable only one correlation would be obtained for the total sample of men and women. However, there were enough studies that did separate the sexes for predictive purposes that both Durflinger and Garrett could come to some conclusions in their reviews. Durflinger summarizes these studies by indicating that the regression equation for men is likely to predict less accurately for women and vice versa (34). Also, he states that the equation for the total group will not do as well as separate equations when applied to men and women separately. Garrett's findings indicate that generally predictions can be made more accurately for women than for men, but that this holds true for intelligence tests to a much greater extent than for high school grades ( 5 3). Of the studies already discussed only four showed separate correlation figures for men and women and the obtained correlations were higher for women in every case. These studies were the ones by Hoyt (75), Jackson (77), Scannell (133)* and Wagner and Strobel (l6l). The study of Jex and Sorenson was the only one to show consistently higher correlations for men (79). They used the General Aptitude Test Battery of the United States Department of Employment Security to predict first quarter grades at the University of Utah in 1948 and found slightly higher correlations for the men on all six scales used, including the verbal scale. The Educational Testing Service conducted a study in seven different colleges and used 3*546 students to check on the reported differences between the sexes in obtained correlations and in the variance of the predictive variables. Abelson reported on the findings which indicated that the obtained differences in correlation values favored the women on the grade point average and on the multiple correlation of grade point average and 39 ACE with the college grade point average, but not when the ACE scores were considered alone (l). It was con cluded that, In general, when the high school grade point average is used for prediction, the standard error of estimate will be significantly smaller for women. This results more from the lower standard deviation of college grade point average for women than from a higher correla-? tion between the predictive variables and the criterion. Mallon combined an inventory of biographical material with English placement scores and ACE linguistic scores to predict first year college grades (102). The English placement scores and the inventory made the best two-variable equation for women, and the ACE linguistic score combined with the inventory as the best two-variable equation for men. The multiple correlation for women was .5 7 compared to .5 3 for the men. Studies by Kirk (89) and Seashore (136) with the College Qualification Test published by the Psychological Corporation both show the women to score higher on the verbal test but lower on the other parts of the test including the total score. Seashore reported some correlations with grade point average based on total 3core for Junior college students. He indicated that the magnitude of the obtained correlations was of the same order as those obtained at the four-year schools. The correlations for the women were ten to twelve points higher than those for the men. Juola in his comparison of several different scholastic aptitude tests indicated that there was some evidence to suggest that the College Qualification Test was better for men and the School and College Ability Test was better for women (84). The differences between the two tests appear to be a function of the "N" (number) test on the College Qualification Test which requires a working knowledge of algebra and geometry; the men do considerably better on this test. Holland derived separate regression equations for each sex based on the California Psychological Inventory and the Scholastic Aptitude Test for 743 merit scholars and 578 certificate of merit winners ( 6 9). The multiple correlation of .42 for women was obtained from the verbal score of the aptitude test and four variables from the inventory. The multiple correlation of .3 8 for the men was obtained from the mathematics score of the aptitude test and three different variables from the Inventory. When the equations were submitted to cross- validation, the correlation for women dropped to .2 3 and for men it dropped to .32. A grouping of science and non- science majors did not increase the zero order correla tions but when the students were grouped according to the college of attendance the zero order correlations 41 were increased but there was much variation between colleges. The range for the verbal scores for six men's colleges was .0 9 to .3 6 and the range for verbal scores for two women's colleges was .40 to .49. II. PREDICTION OP CATEGORIES The problem of sorting persons or things Into various categories on the basis of obtained measurements has been termed classification, placement, and selection. Wert, Neidt, and Ahmann make the distinction between classification and selection on the basis of the groups which are being separated, and state that classification implies different assumptions for the statistical method to be used, i.e., in classification, the groups represent a noncontinuum or nonvariable characteristic, and in selection the groups represent a continuum or variable characteristic which is only available in terms of a dichotomy ( 1 6 9: 3 6 6). This distinction is analogous to that made between the biserlal correlation and the point biserial correlation. In the next section of this chapter it will be brought out that this distinction is irrelevant if the sorting is done into only two classes but differ ences do exist if the sorting is to be done into more than two classes. Guilford states that selection presupposes a supply of applicants with the possibility of rejecting some of them and that the decision is rendered with respect to only one kind of assignment. He defines classification broadly as assigning each individual to his most appropriate category. Guilford points out that the assignment to alternative classes is a problem of differential prediction and that the emphasis here is the prediction of differences in the criterion variable (or between criterion variables). He indicates that whether there are two or more than two categories in which to assign individuals, the approximate solution lines in selection procedures, i.e., a multiple regression equation for each category to be predicted, and that the differences between composite scores will be the deciding factor in classification (63:431-32). Yule and Kendall do not discuss the process of selection but define classification in their chapter on the theory of attributes. An observer classifies objects or things when he makes the distinction between those objects or things which possess a particular attribute and those which do not, according to these authors. They Indicate that classification does not imply the existence of a natural or a clearly defined boundary but may be wholly arbitrary or be vague and uncertain. Regardless of the nature of the classification, the final judgment 43 must be made on the basis of whether or not the object or individual in question does or does not have the attribute in question and this type of decision produces two distinct classes ( 1 7 5 :1 1 * * 1 2). Guilford and Michael have discussed the general problem of the prediction of categories from measurements and also make the distinction between artificial and real dichotomies of a qualitative nature. They point out that while a two-way division represents minimum dis crimination in terms of describing human behavior, many administrative decisions concerning personnel must of necessity be of this type. They stress that the statistical techniques employed in prediction usually require explicit assumptions as to the nature of the distribution of the dichotomous dependent variable, and one of the purposes of their paper was to present some new formulas which were based on the nature of the assumptions being made. These authors state that one of the more important applications of the prediction of attributes, or categories, from measurements is that of selection; and, therefore, they equate selection and classification when the criterion is on a pass-fail or success-unsuccessful basis ( 6 2: 1-3). Cronbach and Gleser have taken a more general approach to defining classification, placement, and selection. They are primarily concerned with personnel decisions, i.e., what should be done with one or more individuals, and define the word treatment to include any type of result of a personnel decision. Classification is then defined as the most general form of the decision problem; an individual has been classified when he has been assigned to a particular treatment. If the decision between treatments is made on the basis of one variable or on the basis of a composite score of several variables, then the classification problem may be termed placement. Selection is defined as a subclass of placement and applies when one of the treatments which may be assigned to the individual is that of rejection (3 0: 1 0-1 1). Even though there is not complete agreement on the terminology to be used, there is agreement in the literature to the fact that if one wants to determine the probability of an individual being in a specified segment of a dlchotomous criterion or of determining the cutting score which will yield a certain specified probability, then certain assumptions will be necessary according to the statistical technique used in the process. Katzell and Cureton developed formulas for determining the probability of being in either segment of a normally distributed but dichotomized criterion (85)* Guilford and Michael extended the work of Katzell and Cureton by developing a formula for determining cutting scores for this problem which were based on the principle of equal likelihood (62). It is this formula which is one of the statistical methods used in the present study. In addition, Guilford and Michael developed a formula for determining the cutting score for a genuine qualitative separation in the dependent variable. Michael and Perry have more recently extended the solution to the treatment of a trichotomous dependent variable (115). Bimbaum has developed a mathematical solution to the problem of determining the fractional changes in each segment of the criterion when the cutting score is changed ( 1 3). The formulas discussed above require assumptions of a continuous dependent variable and linear regression of this variable on the independent variable along with normality and homoscedasticity in the arrays of the dependent variable. In addition, it is assumed that the errors of prediction in both segments of the criterion are of equal importance. Some attempts have been made to give differential weighting to the errors of prediction. Wald has developed a mathematical model for the solution of this problem (163). He defines risk as a function of cost and the probability of committing the error and partitions the sample space so as to equalize the risk for all classifications. Richardson has developed a formula for determining the increased efficiency which will result from using a particular test (128), This method makes no special assumptions but one of the factors in the formula is the ratio of the efficiency of the satisfactory criterion group to the efficiency of the unsatisfactory.criterion group and this ratio is difficult to determine in many situations. Thorndike has proposed a solution to the combination problems of selection and classification by weighting the importance or need of each kind of classification to be made (145). Brogden .has offered a solution to the problem of assigning values to the different kinds of errors by combining monetary values with the criterion measure (18). The theory of decision making is applicable to this problem and Edwards has presented a history and summary of this relatively new field which indicates that much of the work in this area utilizes the principle of "mlniraax loss,” a decision arrived at on the basis of determining the worst possible outcome for each strategy and then selecting the strategy which yields the least ill effects if the worst happens (37)« Berkson has proposed another approach to this same problem which will be reviewed in a later section of this chapter ( 1 0). Besides evaluating the number and kinds of errors in predicting categories from measurements, there is a fundamental problem to be solved before prediction can be initiated and that concerns the method of combining scores from several different kinds of measurements. Multiple regression and discriminant analysis are the two most widely used techniques for combining the independent measures. The use of the composite score which results from both methods is based on the assumption that the independent variables are additive and that a high score on one variable can offset, or compensate, for a low score on another variable. Guilford has shown a geometric interpretation of the comparison between the multiple regression equation method and the method of multiple cutting scores for a two-variable problem when approximately equal proportions are selected by both methods (63:427). It is apparent that there is a group of individuals whose scores exceed the minimum values of both cutting scores but whose composite score is not high enough to be selected by the regression equation. There are two other groups of individuals who have extreme scores on one variable but who are below the minimum on the other and hence would not be selected by the multiple cutting scores but who would be selected by the regression equation because of the compensating effect of adding the two variables for a composite score. Guilford states: 48 It can be argued that not enough Is known about compensatory effects in performances that serve as criteria, and that is quite true. There should be some experimental studies of this kind. A vindica tion of the regression method, however, is found in the consistency with which composite scores continue to correlate as they do in line with multiple- correlation coefficients that forecast those correlations. If compensatory effects did not occur, there would probably be much more shrinkage in correlation of sums with criteria than there is. (63:428) One of the objectives of the present study is to investi gate these compensatory effects as they relate to predicting success in college academic work. Johnson is one who takes issue with the assumption that aptitude traits can be graded and added together. According to Johnson the notion of aptitude or success in a situation is not absolute but is relative to some set of demands which are being made of the Individual and which he is trying to meet (80). The demands must be specified for "success” to have meaning and the judgment is not to what degree he meets the demands but whether he satisfies them or not. He asserts that each set of demands forms a class which is the logical product of certain other classes, and that while members of a class may be substituted for one another, one class of demands cannot be substituted for another. If Johnson's hypotheses are correct, it would seem that a multiple ‘ cutting score would provide a more accurate prediction than a multiple regression equation which would allow compensation to be a factor. Dvorak^ in writing about the advantages of the multiple cutting score as used by the Department of Employment Security, would seem to agree with Johnson (36). She cites two reasons why the multiple cutting scores are more effective than multiple regression equations: (l) In setting up norms on the basis of established workers, the Important abilities show a low standard deviation and therefore yield low zero order correlations with the criterion and hence are left out of the multiple regression equation. ( 2) Even without restricted range, the relationship of an aptitude and a job may only hold to a certain point beyond which an additional increase in this aptitude has no effect on success in the criterion. Also, on the lower end of the scale, certain minimum competencies are needed for certain kinds of jobs and the compensating effect of a regression equation does not provide a minimum level on any one skill. III. THE DISCRIMINANT FUNCTION Johnson has stated that the essential property of the discriminant function is that it will distinguish better than any other linear function between the specific groups on whom common measurements are available. He 50 adds: The principle upon which the discriminant function rests is that the linear functions of the measurements will maximize the ratio of the difference between the specific means to the standard deviations within the classes. (83:344) In terms of the specific problem of identifying success ful college students on the basis of their high school grade point average and a test score, this would mean that the discriminant function would combine the two independent variables in such a manner that the resulting composite score would provide the maximum differences between mean scores for the successful students and the unsuccessful students while at the same time making the variance for each group of students as small as possible. In other words, the overlapping of the frequency distri butions for the two criterion groups when plotted on a common baseline (the composite score) would be at a minimum. The first appearance of the discriminant function in the literature was in an article by Barnard ( 6 ) in which acknowledgement is given to R. A. Fisher as the originator of the technique. Fisher wrote his first article (43) on the use of the discriminant function in 1936 in which he used the technique to classify two species of flowers; and showed the relationship of the discriminant function to partial regression. Fisher's next article (44) demonstrated that the coefficients of the discriminant function differ only by a constant from the coefficients of a multiple regression equation when the average value of y in the dichotomy differs by unity. It was also in this article that Fisher gave consistent notation to Hotelling's T2, Mahalanobis1 generalized distance function, and the discriminant function. In his third article (45), Fisher solved the general problem of testing for the significance of a difference between an obtained discriminant function and one determined on the basis of theory or some simple form of computation. At about the same time, Welch (168) was making a contribu tion by demonstrating that Fisher's discriminant function was the optimum one for maximizing the probability of correct classification when the population distributions of the p variables were multivariate normal with equal variances and covariances. Fisher had been doing his writing in the Annals of Eugenics and was concerned with the application of the discriminant function to problems of classification in biology. Garrett (54) and Travers (155) tried to show the application of Fisher's discriminant function to psychological problems. Both writers demonstrated the relationship between the discriminant function and mul tiple regression. 52 In 19^7j Wherry (172) reviewed the articles by Garrett and Travers. Wherry stated these authors showed that the equivalence of weights obtained by the dis criminant function approach those obtained by the point biserial method, that they were implying that the dis criminant function could only be used on a discrete dichotomy rather than on a metricizable continuum, and that the weights obtained by the discriminant function are only proportional to the true weights. Wherry proves that true maximal weights can be obtained directly from correlational methods regardless of the assumption as to the nature of the criterion since the weights are not affected by the assumption. In this same article Wherry says: No method for multiple blserial or multiple point biserial, involving a dichotomized criterion, appears in the literature. Instead, we find the widespread belief that such procedures are illegitimate or impossible. The discriminant function, based upon the analysis of variance, is usually recommended as the proper approach. (172:189) He goes on to point out that the discriminant function is less informative because it only gives weights but not the degree of relationship, and more burdensome because it requires the computation of separate variances for each group. He recommends that the effectiveness of his procedure be judged by the multiple blserial correlation. 53 As was pointed out in the introduction to this chapter, the discriminant function is a special case of the more general problem of discriminant analysis in which there is no restriction on the number of independent variables or on the number of groups to be considered simultaneously except that, as Bryan (147) has indicated, a mathematical solution to the problem requires that there be no fewer variables than one less than the number of groups. Hodges ( 6 8), Tiedman, Rulon and Bryan (147), and Tiedman and Tatsuoka (151) have all presented excellent reviews of the development of discriminant analysis, including the discriminant function. Hodges cites Professor J. A. Glrshick in his address at the meeting of the Institute of Mathematical Statistics in Berkeley, California, June 16, 1949: The development of discriminant analysis reflects the same broad phases as does the general history of statistical inference. We may distinguish a Pearsonian stage, connected with the coefficient of racial likeness; followed by a Fisherian stage, connected with a linear discriminant function; also, a Neyman-Pearson stage (probability of misclassifi- cation); and a contemporary Waldian stage (of risk). ( 68: 2) Hodges points out with reference to the discriminant function, that if the two populations have the same covariance matrix, then the likelihood ratio is used and the probabilities of misclassification are equal. If the two populations do not have the same covariance matrix, 54 then the result is a quadratic discriminant function ( 6 8: 6 8-7 0). Tiedman as part of his discussion gives a history of the discriminant function (147). He makes it clear that the proportional weights demonstrated by Travers, Garrett, and Wherry apply to the special case of two groups only and that this has caused psychologists to think about classification in terms of pairs of groups when more than two groups are involved in the problem, instead of thinking in terms of which of the k groups is this Individual most like. Rulon makes the distinction between multiple regression and the discriminant function on the basis of ordered and unordered classes (147). The discriminant function applies to the unordered classes and uses group membership as the criterion and makes all comparisons between groups and none within groups. Another differ ence is that all groups take the same tests in discrim inant analysis and in regression analysis the groups frequently take different tests. Tiedman and Tatsuoka indicate that the function of discriminant analysis is to determine whether or not there is a stable difference in the p observations among the k populations of which the k sets of observations are samples (151). Hotelling's T2 is designed to determine the significance of these differences. If a difference actually exists then there are three additional problems with which to be concerned: (1) What is the distance between the pairs of k groups? Mahalanobls' distance function is designed to answer this question. (2) In what directions do the groups differ? Fisher's discriminant function provides the answer to this one. (3) How do you assign an unclassified Individual known to belong to one of the k groups on the basis of p observations? Welch has provided one solution and the likelihood ratio of Neyman-Pearson is another. All these methods required that the populations be multivariate normal with equal dispersions. Tyler has discussed these same problems of multivariate analysis and has Illustrated the methods by means of personality test data obtained from students preparing for general elementary and general secondary teaching credentials (138). Another comprehensive review of the generalized classification problem and the techniques developed for handling them is given by Pickeral (124). Some aspects of differential and comparative prediction are discussed by Mollenkopf with special attention being given to the use of tests which would be valid for more than one population (116). If the objective is to provide a good estimate of how well an individual will do in each of two populations, then the inclusion of a generally valid test has merit, according to this article. IV. EMPIRICAL STUDIES USING THE DISCRIMINANT FUNCTION Travers, in his article which was discussed in the previous section and which was written to help introduce the discriminant function to psychologists, used empirical data to aid in the explanation (155). He used six test variables to discriminate between engineering apprentices and aeroplane pilots. He demonstrated that the possibility of misclassification into an unsuitable occupation was minimized by the use of the discriminant function. Selover reported a study in 1942 of the sophomore testing program at the University of Minnesota and used the discriminant function for determining differences between students in various major fields (138). Using the four test scores with the highest t value, the discriminant function obtained a t value of 14.9 between pre-medicine majors and music majors; however, inspection of his data shows that the general science test yielded a t value of 13.8 when used alone. Using three test variables and the grade point average for the first two 57 years of college, Selover compared 39 students who were dropped for low scholarship in each subject major with the 39 students who graduated with the lowest scholastic average,, Composite scores were obtained for each student by means of the discriminant function and it was found that by eliminating one student who graduated, 17 out of the 39 who dropped out could have been eliminated. In plotting the grade point average alone, only 12 drop outs could have been eliminated at a cost of one graduate. Jackson has used the discriminant function in the selection of students for freshman chemistry at Michigan State University ( 7 6). He used the two parts of their own chemistry test and the ACE as variables. The criterion was a C grade or better in freshman chemistry and he used as a critical score the method advocated by Johnson (83:347) and by Wert, Neidt, and Ahmann ( 1 6 9: 3 6 9). Jackson determined separate equations for each sex and interpreted the different ratios of weights for each sex to support the hypothesis of sex differences in predic tion. He also found that he could eliminate the ACE score from the equation for men without any significant change in predictive efficiency. Tiedman (147) in his discussion of the development of discriminant analysis points out that J. G. Bryan, in an unpublished doctoral dissertation at the Harvard University Graduate School of Education (1950), provided a complete generalization of the discriminant analysis and developed a means of computing the k-1 set of discriminant functions from two matrices obtained from the original scores. Tiedman and Bryan have illustrated this procedure with an example of five groups and nine variables (150). The groups were students in five major fields who had made at least a C average and the test variables were scales on the Kuder Preference Record. The authors make it clear that the data were being used to illustrate a method and not as firm experimental evidence. The method presented leads to probability statements that permit inference of group membership, A concept of a "centour score" in introduced. The word itself is a combination of centile and contour and is employed to indicate the proportion of "misses" in classification, i.e., a centour of 10 on group A would mean that if everyone were classified as not-A, the classifier would be wrong 10 per cent of the time. The authors make the following statement concerning the use of the centour score: Whether applied to the test-space or the discriminant-space, the centour score concept is very useful. It provides a probability statement analogous to that from the t-test with which most psychologists are familiar. The centour score gives the per cent of errors made from rejection of the hypothesis that an individual is a member of a particular group. Like the probability statements 59 from the t-test, the centour score ignores two elements usually incorporated in the theory of decision functions. It ignores the relative size of each group and it ignores estimates of the seriousness of each type of error. For guidance purposes, though not for classification purposes, the authors feel that the relative size of each group should be ignored. Although the authors feel that the seriousness of errors of the several types should be of concern during counseling, this factor will probably continue to be incorporated clinically for some time to come. ( 1 5 0: 1 3 7) A number of the writers cited above have stressed the differences between the discriminant function and the regression equation. The basic difference is that the discriminant function does not take into account the variance within each group being classified and hence is not appropriate for determining how well a person will do in a certain subject area but it is appropriate for determining whether a person has the same character istics as other people who are known to be successful in a given subject area. Wesman and Bennett have defined differential prediction as the assessment of a person's relative chances of success in two or more fields and indicate the need for statements of both absolute and relative probability of success (170). The best example of an institution actually applying these principles is the University of Washington with a program under the direction of Paul Horst. Horst has written two monographs (72 and 7 3) explaining the development of a differential prediction battery and the development of a multiple absolute prediction battery. Horst has also written about his program in a nontechnical manner (7^). The object of the differential method is to select from a battery of predictors, a subset of specified size which will have the highest differential prediction efficiency for a given set of criterion variables. The index of differential prediction is based on the difference between the average variance of the predicted criterion measures and the average of their covariances, assuming standard measures for both predictors and criteria and that the predicted criterion measures are least square estimates. The object of the absolute prediction method is to select the subset of predictors which will have the highest efficiency for all criterion variables irrespec tive of how well it differentiates among them. Efficiency is defined as the sum of the squared multiple correlations of the predictors and the criteria. The weights used for both kinds of prediction are the conventional least-square regression weights; however, the weights are not, in general, the same for both methods. When Horst reported on his program in 1 9 5 6, they were using seven different entrance tests and six different high school subject areas as basic data for 32 equations for different subject majors plus one additional equation for the over-all college grade point average (7^). Age and sex are two other factors used to increase the accuracy of prediction and are evaluated by adding a given number of points to the composite score obtained from the equation for a certain area. It i3 interesting to note that Horst estimates a five second Interval for the data processing equipment to punch the 33 estimates in a second card after the independent variables have been punched Into a score card. Horst states that the equations yield approximately 10 per cent error in each extreme group, i.e., of those predicted to do poorly, 10 per cent would do well and vice versa. He also indicates that typical of most predictions of success, the middle group cannot be predicted so accurately. Three different sets of empirical data are used by Beall to demonstrate that the discriminant function can be used with approximate methods of finding the weighted coefficients (8). Moonan has given a summary of multivariate theory but uses the term dispersion analysis when he applies the theory to a problem of political analysis (117). Still another summary of recent work with the discriminant function Is provided by Lubin but this one is very technical In nature and is designed more for the mathematical statistician (100). 62 V. THE PRINCIPLE OF EQUAL LIKELIHOOD The theory of classification into one of two categories on the basis of a single linear measurement was worked out by Guttman. If the frequency distributions of the individuals in both categories are plotted on the same baseline., then the area which appears under both frequency curves represents the total proportion of error of misclassification, Guttman refers to this as a problem in probability density rather than probability alone. The critical score which will divide the two categories with a minimum of errors of classification of both kinds is that score which corresponds to the ordinate at the point of intersection of the two fre quency distributions, i.e., any score which is above the critical score will have a probability greater than .5 of being in the category which has the greatest area above the critical score; and any score which is below the critical score will have a probability greater than .5 of being in the category which has the greatest area below the critical score. The critical score, at the ordinate of intersection, is precisely at that point in the linear measurement where the probability of being in either category is equal to . 5. This solution is called the principle of equal likelihood ( 7 0: 2 7 1-7 2). Guilford and Michael (62) and Guilford (63:340-47) have demonstrated various methods of locating this critical score in practical situations. Analytical methods have been developed which rest on certain assumptions according to the nature of the data. These have already been discussed in the section on prediction of categories. The second approach which is described is based upon methods of approximation and requires no particular assumptions as to the nature of distributions, both interpolation and graphic methods are demonstrated which lead to approximate solutions of the critical score according to the principle of equal likelihood. The graphic method of approximation is demonstrated with the present data in Chapter IV. When the discriminant function is used in order to classify individuals into one of two groups, the principle of equal likelihood is applied to the composite scores. The usual method is to substitute the mean scores of the various predictors for one category into the discriminant equation to obtain a composite score for these values and then to do the same thing for the mean scores for the other category; the desired critical score, then, is the arithmetic mean of the two composite scores (83t3^7; 169:369). This procedure will also minimize the errors of misclassification in both categories. 64 VI. NONPARAMETRIC METHODS OP COMBINING MEASUREMENTS FOR PREDICTION Viteles, writing in 1932, listed three procedures for validating test scores ( 1 5 9). In addition to the usual correlation coefficient, he suggested a comparison of average scores of workers at varying levels of pro ficiency and the percentage comparison of individuals scoring above and below certain critical scores. The latter method was designated most useful in the selection or elimination of workers at one extreme end of the distribution. He suggested that the critical score is most frequently obtained by plotting results and by a process of trial and error, Thorndike has discussed the general problem of combining test scores into a battery and points out that there may be a problem about the concept of most effective selection. He lists three possible criteria for effective selection: (l) never miss any of the best individuals, ( 2) exceed minimum standards of competence, ( 3) select personnel who show the highest average performance on the job in question. Thorndike uses the last one as his standard working definition of effective selection. In the present investigation, the concern is more with the first two criteria if best is interpreted as meaning satisfactory (144:185-201). 65 If our objective in selective admission to college is to achieve as much utility as possible at a relatively low level of cost, it follows that the critical score or scores will be set at the lower end of the total distribution of the applicants. The usual regression equation which is based on the least squares estimate of predicted values for the whole distribution may not be the most effective way of determining these critical scores. In addition, the usual assumptions of normality in the distributions may not be too reasonable when consideration is given to the fact that the applicants to college are a select group from the general population to start with, and if the first year achievement in college is used as a criterion, then additional selection has taken place in terms of the usual mortality rate of those who do not complete the first year. The term nonparametric or distribution free is given to the statistical methods of inference which do not require any assumptions about the nature of the population distribution. The nonparametric methods which have been developed generally are concerned with tests of sig nificance for measures of differences or association. Pestinger and Katz (42), Seigal (137)» Walker and Lev (164), and Moses (118) are all good sources of information on these techniques which are recommended only when the usual assumptions of normality are not justified. Moses has cited references in the literature of mathematical statistics where the nonparametric test is clearly superior to a t-test (118). The term nonparametric has been used in the present study to designate methods of combining scores for prediction of success which do not require normal distributions or any of the other restrictions which the use of regression analysis and discriminant analysis impose on the data. The question of measuring the degree of association present in the predictive device will be postponed to the last section of this chapter. Fix has explored analytically the problem of discriminating between two populations when no parametric form can be assumed (46). She concluded that no non parametric procedure can have probabilities of error less than those of a likelihood ratio procedure based on known densities. Her approach is called "consistent with" the likelihood ratio procedure because the probabilities of error approach the same values. Fix and Hodges have compared a nonparametric method to the discriminant func tion on the same set of data and judged efficiency in terms of the percentages of misclassification (47). They conclude that the nonparametric method can be more effective if the data do not fit the assumptions of the discriminant function. They also point out that the judgment of the statistician is important in either method. In the nonparametric method, the judgment as to the relative importance of the various measurements is critical; and in using the discriminant function, the statistician must decide if the data "fit1 1 the assumptions. A number of different approaches to the problem of combining data for prediction of college success have been reported in the literature. Stalnaker has plotted eight semester college grade point averages against decile scores on the ACE for different subject majors (l4l). He found that the number of students completing the course of study in four years from the upper half of the ACE distribution was almost double that of the lower half of the distribution. Jackson found that the reading test used at Michigan State College predicted success as well as two or more variables ( 7 7)* He classified the students into high, middle, and low groups on the basis of test scores and used a criterion of a 1 .7 5 average (C = 2.0) stnd found that 98 per cent of the "highs" were successful and 50 per cent of the "lows" were unsuccess ful. Melton reports that an arithmetical average of high school percentile rank, ACE percentile, and Cooperative English percentile proved to be a practical procedure even though unjustifiable mathematically (113). He indicated that his rating could identify failure better than success. Berdie has recently reported that the University of Minnesota uses an average of the high school percentile rank and the percentile on a scholastic aptitude test, and that this has been the most efficient predictor of success at that institution since 1923 ( 9). He compares figures of 1923-27 and 195^-55 to demonstrate how many unsatisfactory students had been admitted and are now left out. Krathwohl discusses a 3 x 3 analysis table, made up of ACE scores and grades, which groups each variable into three divisions (91). Each cell of the table contains the frequency and two percentage figures each computed on different base number. Another variation of a three-way frequency table is reported by Bertrand (12). The Los Angeles-School System uses what is essentially a three-way frequency chart to locate children who need special attention or possibly a recheck on the tests (15). The third dimension is in terms of coded values and has been adapted for use with data processing equipment. The various nonparametric approaches to prediction listed above support the findings of other investigators concerning the fact that prediction is most effective at the extremes of the distribution. Eurich and Cain have stated that tables showing the number of correct placements which have been developed from scattergrams are likely to have more practical value than an index of forecasting efficiency (40:840), McCabe demonstrated the nature of the problem by preparing a table of percentile ranges of academic success for an obtained percentile on a test with a known validity coefficient (108). The fact is sometimes overlooked that in a normal distribution the probability is 2 to 1 that any individual will score between the l6th and the 84th percentile. The table suggests, according to McCabe, that with a validity coefficient of . 5 0, the scores must be below the 3 1st percentile or above the 69th percentile to offer much information for individual counseling. Ghiselli has recently demonstrated that prediction is more efficient for some Individuals than for others and the degree of prediction is a function of the difference between the standard predictor scores and the standard criterion scores, but he does not relate the difference score to the position (or location) of the individual in the original distribution of the predictive variable ( 5 7). Maxwell has outlined a statistical model for estimation and prediction which enables the investigator to check his data for nonlinear relationships and possible inter action effects (105). The model also enables one to discover optimum areas of the Independent variates for predictive purposes if they exist. In the area of Item analysis, certain methods of pattern or configural scoring have been demonstrated to have more predictive validity than the usual methods of adding scores on a test. It Is possible that these methods may be extended to the combining of the test scores themselves for prediction of a criterion variable. Meehl has made empirical comparisons of several methods of combining scale scores from the Minnesota Multiphasic Personality Inventory In order to Identify psychotic profiles (112). The least effective actuarial method was use of the discriminant function and the most efficient method was one in which the decision "rules" were essentially clinical In nature but the application of these rule3 was made on an actuarial basis. Meehl has demonstrated how two test items may each have phi coefficients of zero but when the two items are combined the phi coefficient is 1 .0 0 because one criterion group always answers the two items In the same way, i.e. both true or both false, and the other criterion group will give different answers to each question (110). Webster has developed a method of item rejection by counting frequencies above and below the 5 0th percentile on the Item and the criterion (l6 6). He has demonstrated empirically that his method comes very close to the same 71 validity coefficients as other more complicated methods. Gaier and Lee have reviewed a number of other studies which suggest that configural analysis as applied to item responses within a single test provide a higher degree of discrimination than the usual additive techniques which Ignore inter-item relationships ( 5 2). One of the methods of actuarial pattern analysis used in the study by Meehl (112) was one developed by I^ykken (101). He bias criticized the usual methods of profile analysis and argues that what is needed is not a measure of geometric distance but rather a measure of psychological distance. His method is based on dividing the test variables Into frequency cells and then computing the criterion mean for each cell and ranking the cells according to the mean values. The ranks are then analyzed for patterns. Toops has pre sented another means of organizing data in terms of sub- populations (153). It is essentially a system of code numbers which will Identify the various trait-categories which are under Investigation. Pricke has used a coded profile for predicting college achievement (51). The author sees his method as an extension of multiple cutting scores. Pricke1s argument for his approach is that different variables may have different effects at different places in the configuration. The beta weights 72 of a regression equation do not allow for any variation of this kind. At a particular combination of coded values, Frlcke demonstrated a negative relationship between the ACE and college grade point average. Multiple cutting scores is the nonparametric method of combining test scores which is most widely used and is another of the methods being investigated in the present study. The Bureau of Employment Security uses this method with the General Aptitude Test Battery (55). Dvorak has explained how the cutting scores are deter mined and the advantages of this method over multiple regression (35 and 36). Eligibility for a specific job is determined by a qualifying score on an occupational ability pattern which is usually made up of minimum scores on three of the nine aptitudes measured by the battery. If the individual is below the minimum score on any one of the specific aptitudes, then he does not qualify on that occupational ability pattern. Another multiple aptitude test which is scored on the same general principles is the Flanagan Aptitude Classification Test (48). *Supra, p. 49. 73 V. EMPIRICAL COMPARISONS OP METHODS OP PREDICTION Michael has pointed out that the development of mathematical models for differential prediction has far exceeded the empirical studies of the use of them (114). The literature in this area shows an even shorter supply of empirical studies in which different methods of differential prediction have been compared under the same set of conditions. Thi3 comparison of methods for a particular kind of problem is the main emphasis of this investigation. Some of the studies are based on simplified methods of computing a multiple correlation coefficient or of determining the multiple regression equation. Henklns has developed a technique which yields results very similar to the Wherry-Doolittle method (78). Wesman and Bennett furnish evidence from four different colleges that simple addition of the three tests of the College Qualification Test results in equivalent correlations to those obtained by calculating beta weights for the three sub-te3ts (171). The obtained correlations did not differ by more than two digits in the second decimal place. Using 14 fields of concentration and 13 variables for an experimental group of 925 and a cross-validation group of 425* Dunn compared multiple discriminant analysis with multiple regression (33). Discriminant analysis was more accurate in the cross-validation group. Separate equations were' needed for men and women and each method resulted in the use of different items and weights. Tiedman and Sternberg have done a similar study by comparing discriminant analysis to the regression equation in terms of which would better predict curricular choice of tenth grade students on the basis of scores made on the Differential Aptitude Test in the ninth grade (148). They found that when the regression lines were plotted on a two-way frequency table with grade point average and verbal reasoning as the variables, the regression line for the college preparatory group was completely above that of the business group which would lead to the conclusion that in predicting the group in which a student would make his highest grades the college preparatory group would be the choice for all. When the two variables were converted to standard scores then the reverse situation took place and all predictions for highest tenth grade average would place everyone in the business curriculum. They interpret their findings as supporting those of other investigators in demon strating that for a classification problem, discriminant 75 analysis provides "among group" data not provided by the regression analysis. Lawshe and Shucker compared the relative efficiency of four test weighting methods in multiple prediction (95). They used beta weights, simple addition of predictive test scores, the reciprocal of the standard deviation, and the exact opposite of the third one, multiplying by the standard deviation of the predictive variable. No significant differences were found between the four methods. The multiple correlations were not given but the authors reported that all methods predicted significantly better than chance. Two studies were made in which the Wherry- Doollttle method of selecting tests for a prediction battery was compared with a method based on multiple cutting scores. Wahoske compared the mean scores of four groups of naval trainees and found higher scores for those who had been selected by the method of multiple cutting scores although none of the differences were large enough to be statistically significant ( 1 6 3). Grimsley did a similar study using cross-validation with two groups of 250 accounting students (6l). Several groups were formed on the basis of their relative stand ing on the accounting grade as a criterion. No significant differences were found but the obtained 76 differences favored the multiple cutting score method in the top 13 per cent but favored the regression method at the other division points. In an empirical comparison of the per cent of satisfactory selectees, McCollum and Savard used the Taylor-Russell tables and a system of merely counting the number of successes in the distribution and found essentially no difference in the number of errors for each method (109). They raise the question: "Why correlate when you can count?" Sessions reports on a study with engineering students in which the ACE had the lowest zero order correlation with the grade point average but was the most efficient single test for determining critical scores; It was even better than the regression equation for this purpose (139). In another study with engineering students, Boyce tried various methods of predicting a dichotomous criterion and found that gross quantitative methods, such as adding raw scores, did as well as the refined techniques and showed less shrinkage on cross- validation ( 1 6). Tucker compared the relative efficiency of multiple regression and what he called "unique pattern techniques" which consisted of assigning predicted scores to various cells of different sizes in the frequency 77 tabulation (157). The pattern techniques correlated almost as well with the criterion as the composite score from the regression equation. Comparable results were found by Rosen and Rosen In making parametric and nonparametric analyses of opinion data ( 1 2 9). A unique approach was taken by Helmstadter in using artificial data to compare methods for estimating profile similarity ( 6 5). He reports that even the most simple techniques achieved considerable success in cross- validation. The discriminant function ranked second in terms of the proportion of successes even though most of the necessary assumptions were violated. VI. MEASURES OF ASSOCIATION Regardless of the nature of the predictive device used, if the criterion is a dichotomous one, then the Independent variable or variables must also be dichotomized at a critical point or combination of points If the only treatmets allowable are acceptance or rejection. The data for the college admission problem then reduce to a four-fold contingency table for all situations except where predictions would be made with 100 per cent accuracy. The problem is how to represent the degree of association between the predictors and the 78 criterion and not lose some of the essential information. If the selection is being made on the basis of a composite score then the degree of relationship is usually expressed in terms of the coefficient of multiple correlation as has been indicated in most of the studies which have been reviewed. If the selection has been made on some other basis, then the degree of association present In the contingency table may be expressed by the phi coefficient or the tetrachoric coefficient of correlation depending on the nature of the data. The phi coefficient Is widely used in item analysis and the tetrachoric correlation has been used for selection in the development of the General Aptitude Test Battery (55). Taylor and Russell were concerned with the fact that various functions of the correlation coefficient such as k, the coefficient of alienation, and E, the index of forecasting efficiency, had been widely accepted as the correct way in which to interpret a correlation coefficient and had in turn led to a certain amount of pessimism by indicating efflclencyof 2 per cent to 13 per cent for the usual validity coefficients of .20 to .5 0 (143). They demonstrated that the real efficiency was related to two other factors in addition to the magnitude of the correlation coefficient. These factors were (l) the percentage of successful persons in the criterion from the general population, and (2) the selection ratio, i.e., the percentage of persons to be selected from the total number of applicants. Guilford has indicated with a series of diagrams how the three factors are inter related ( 6 3: 3 8 3). The Taylor-Russell tables have been designed to predict the proportion of satisfactory persons among those selected when the success ratio, the selection ratio, and the validity coefficient are known. These tables are not appropriate to the present problem because they do not indicate how many "potential successes" have been eliminated and require a validity coefficient for their application; however, the general theory is applicable and indicates that there will be some limitation on our ability to predict success which will be a function of the number of persons in the general population who can succeed in the criterion. Brogden (17) has given his explanation of the meaning of a correlation coefficient under certain special conditions and Brogden (19) has extended the Taylor-Russell ideas to include the dollar cost of the actual testing procedure. Arbous and Sichel have developed a mathematical model and tables for determining "economy" and "loss" (similar to utility and cost in this paper) for a pre-screening test when selection ratios and correlations are known (3). Their objective is to weed out those with little or no chance of success. In addition to or in place of a correlation coefficient, it is possible to determine the degree of association in the four-fold contingency table by indicating the proportions in the various cells of the table. Yule and Kendall have added a word of caution concerning this approach by pointing out that a high percentage of occurrence of an attribute implies nothing in terms of association unless information is available as to the percentage of occurrence of those not possess ing the attribute in the same classification of the population (175:^5)* Yule (17^) has developed a coefficient of association for a four-fold table which has different properties than the phi coefficient or the tetrachoric correlation, Yule and Kendall (175:253). Yule's coefficient indicates perfect association (Q = 1.0) when all A's are B's but there may be B's which are not A's or vice versa. A phi coefficient reaches its maximum value when all A's are B's and vice versa; it cannot reach 1.0 unless the table is symmetrical, i.e. the A's equal the not-A's and the B's equal the not-B's. Guilford has indicated that the evaluations of predictions into categories when made from measurements can be made by the methods applicable to evaluating the 81 prediction of attributes (63:3^7)* Interest is usually centered in the percentage of correct predictions or of the errors of prediction, and evaluation is made on the basis of the gain in accuracy of prediction which results from the application of the measurements when contrasted to the accuracy of prediction without the measurements. If the percentages of correct predictions are compared with the corresponding types of cases in the entire sample, then the evaluation is similar to a chi-square test; however, the phi coefficient is a more interpretable index according to Guilford (63:3^9). Mayo has given an excellent summary of the development of various uses of chi-square and the means of computing association or correlation from a contingency table, and has demonstrated the application of some of these techniques to an educational problem (1 0 6 and 1 0 7). He also discusses higher order interaction in a 2 x 2 x 2 contingency table and in tables of higher order. The big issue, according to Mayo, is that coefficients such as phi are sensitive to all forms of departure from random order and are not sensitive to departure of a specified type. Horst has stated that the usual methods of measuring accuracy of prediction do not work well with a dichotomous criterion, and he has indicated that the 82 measures of accuracy might be more practically put in terms of the percentage of cases correctly classified (70:113). Johnson (81) has drawn from the work of Yule and Kendall (175) and developed a formula for the relative number of individuals classified by a test rather than by chance. Several writers have developed indices of correct classification. Brogden has developed an index for selection based on a dichotomous criterion by assuming that the criterion is continuous and has the same form as the distribution of the independent variable ( 2 0). Ginsberg, in working with a problem of biological classi fication, has proposed a single index of intergradation or of divergence (.58). One measure is the compliment of the other and they both are obtained from the overlap of the frequency polygons resulting from the characteristic with the greatest divergence for classification. Ohlin and Duncan, in dealing with the prediction of parole violation, suggest as an index the percentage reduction in the error of prediction] e.g., if the violation rate was 40 per cent and the new technique reduces it to 32 per cent then the percentage reduction is 8/40 or 20 per cent (122). Ward has developed a "disposition index" to measure the efficiency of the assignment of n people to n jobs ( 1 6 5). 83 No single index can adequately describe the prediction of a dichotomous criterion if errors of prediction in both segments of the criterion are to be considered and if each type of error has a different degree of importance. This was recognized by Thurstone who wrote in 1919i "We select as our standard method that which predicts the largest proportion of failures without excluding any students who make good" (146:135). He indicated that this method stressed the responsibility of the psychologist with regard to the individual. The definitions of cost and utility which are used in the present study were taken from Berkson who used them as the measure of the efficiency of a test by comparing the costs at a given utility (10). In illus trating the problems of using a single index as a measure of the effectiveness of a test, Berkson cites the example of a medical serologic test. The sensitivity of a test is the fraction of positives (those who have a disease) and the specificity of a test is the fraction of negatives (those who are free of the disease). The author points out that it is not only these two values which must be considered in determining the best serologic test, but that the importance of each kind of error must be con sidered along with the number of persons with and without the disease in the community. 84 Berkson plotted separate curves of cost and utility with test scores on the horizontal axis and per cent on the vertical axis (10). Duncan, et al. have used Berkson's ideas but have modified them to the extent of plotting cost as a function of utility, using parts per hundred on both axes (32). For each i (number of applicants accepted), there is a unique pair of values based on cost and utility. The diagonal line on which the values of cost and utility are equal represents zero efficiency. With this model, the authors are able to give geometric solutions to six different kinds of selection decisions. One of these decisions is to maximize "expense-weighted efficiency" and is accom plished by Introducing a ratio of the expense of misclassifying a success to that of misclasslfying an unsuccessful applicant into the efficiency formula. Goodman in dealing with the accuracy of prediction of parole violators has proposed a two-way test ( 6 0). "Accuracy" is defined as the percentage of correct pre dictions for violators and nonviolators of parole and is similar to Berkson's utility. "Efficiency" is a ratio which includes the proportion of parolees who violate the provisions of their parole and hence is similar to Berkson's cost. Goodman also discusses the idea of assigning values for "social cost." He suggests as an 85 example that if potential parole violators who are not paroled are assigned a value of one, then nonviolators who are not paroled would have a value of two and parolees who do violate parole would be assigned a social cost of 100. The objective of selection for parole would be to work for a minimum social cost. Reiss has also suggested two indexes for the purposes of testing the precision of prognostic instru ments (127) and they are very similar to those of Goodman mentioned above. Johnson has also discussed the amount of discrimination for a given cost and indicates that judgment depends on the supply of candidates and on the demand for candidates ( 8 2). Cronbach and Glesser discuss Berkson's paper (10) among others in their section on the interpretation of validity coefficients in selection and conclude: Our thinking is most consistent with the plan which assigns particular values to "hits" and "misses," and adjusts the cutting score to maximize expected utility. In institutional selection with fixed quota where men rejected leave the institution, it is not meaningful to consider misses, however. Rejection of men of good quality does not decrease the output of the institution; their frequency bears only on "what might have been.1 1 Two-sided evalua tions are appropriate only when the screening test divides men into two groups who remain within the institution but are treated differently. (30:46) Utility as used in the above quotation is the result of the evaluation of a given strategy by means of the average "payoff" for a large number of decisions. It is necessary 86 to assign a value to each criterion score (30:20-21). It should be pointed out that the problem of selection in the junior college does not necessarily presuppose a fixed quota and hence the number of students who are potential successes and who are rejected could affect the output of the institution. The idea of different treatments within the same institution also has applica tion. As was mentioned in Chapter I, all of the students in the junior college do not have to be classified as transfer students and in actual practice they do have a variety of classifications, VII. SUMMARY The literature which is pertinent to the problem under investigation has been reviewed under the following seven subtitles: (l) the prediction of college achieve ment, (2) prediction of categories, (3) the discriminant function, (4) the principle of equal likelihood, (5) nonparametric methods of combining measurements for prediction, (6) empirical comparisons of methods of prediction, and (7) measures of association. The informa tion contained in the literature which was reviewed will be summarized below in accordance with the same classifi cation of topics. The prediction of college achievement.— Byfar the largest percentage of studies of the prediction of college achievement has been made by means of obtaining a correlation coefficient between one or more predictive variables and the first semester or first year grade point average. The studies which have reviewed and summarized this type of investigation over a period of years report median correlations of .55 for high school grade point average and .5 0 for achievement tests and scholastic aptitude tests such as the ACE and the OSPE. The multiple correlation for high school grades and a scholastic aptitude test will be of the order of .6 5 to .7 0 and the addition of a third variable will add very little to the multiple correlation. First year grades have been established as a better criterion of success than the first quarter or first semester grades and the grade point average will usually yield a higher correla tion than the high school rank. Homogeneous grouping with regard to as many factors which can be controlled will usually result in a higher correlation with the grade point criterion. Pre dictions made on the students from a single high school, for example, will be more efficient than if several different high schoolB are represented. Prediction is more accurate for women than for men although there is some evidence that these findings may be the result of 88 less variation in the criterion measure for the women. It is likely that the prediction equations for each sex will differ in content as well as in weights for the variables. The extent of the original selection of students will influence the results of later studies on the same group; hence, the gains from homogeneous grouping will only accrue when it is done on the basis of factors not used for predictive purposes. With achievement tests such as the ITED, it is possible to obtain correlations of about .40 as early as the fourth grade; however, general academic success can be predicted more efficiently for each grade in school. High school grade point average is a better predictor of freshman grades than sophomore grades although the grades in the first two years of college will correlate-about .90. Persistence in college work is more closely related to first year performance than to test scores, but freshman tests will predict scores on graduate level examinations better than the four-year college grade point average and the addition of the high school grade point average for this purpose tends to lower the obtained correlation. The nonintellectual factors related to college achievement, in their present state of assessment, are not very efficient a3 predictive variables. The clinical 89 application of the intellectual factors and/or the nonintellectual factors of achievement is no more efficient than the actuarial method. Regardless of how the prediction is made, a relatively large number of "unpromising" or , , nonrecoInmended', students can do successful work and graduate with a "typical, f record, after a below average start, if given the chance to try. The correlations obtained with the School and College Ability Test and first year college grade point average are of the same magnitude as those usually found for the ACE, OSPE, and other similar tests. There is conflicting evidence as to whether the verbal score or the total score is the better predictor. Prediction of categories.— The terms selection and classification are used by different authors in different ways; however, for a dichotomous criterion, the definitions make little difference if the final decision is merely to accept or reject. In order to predict or determine the probability of an individual being in a particular segment of the dichotomy, certain assumptions will be necessary according to the nature of the method used. Most of the methods usually employed make the assumption that the errors of both kinds, i.e., mlsclassificatlon in either category, are of equal value or equally disadvantageous. Methods have been developed 90 which will permit differential weighting of the types of error but the difficulty lies in assigning the "weight” of the error. Multiple regression and discriminant analysis both are based on the fact that the predictive variables are additive and both allow for compensation among the predictors, i.e., a high score on one variable can offset a low score on another variable. Multiple cutting scores would not always select the same individuals as a regression equation because of the need for certain minimum skills and the fact that the relationship between the predictive variables and the criterion is not always linear throughout the entire range of the criterion. It would appear to be necessary to establish the effects of compensation for the type of criterion being used. The discriminant function.— When the population distributions of p variables are multivariate normal with equal variances and covariances, then the discriminant function, as developed by R. A. Fisher, is the optimum procedure for maximizing the probability of correct classification if equal weights are assumed for the errors of misclassification. It has been demonstrated that the true weights for the classification problem with a dichotomous criterion can be obtained from correlational methods 91 (using biserial correlations) regardless of the nature of the criterion, i.e., whether it is a qualitative or a quantitative classification. When the number of groups in which individuals are to be classified exceeds two, then discriminant analysis is more efficient if comparisons are limited to differences between groups only. If there are three or more groups and the comparison Is to be made on differences within the groups as well as between groups, then regression analysis Is to be preferred. The principle of equal likelihood.— Analytical methods, methods of interpolation, and graphic methods are available to locate a critical score which will yield the minimum amount of misclassification when a single linear measurement is used to predict a dichotomous criterion. This critical score will be located at that point In the linear measurement for which the probability of being in either segment of the dichotomy is . 5 0. The principle of equal likelihood is used In connection with the discriminant function after the predictive variables have been combined to form a composite score which then functions as the single linear measurement, and a critical composite score is determined for the purpose of classification. 92 Nonparametric methods of combining measurements for prediction,— Several criteria exist for the purposes of defining what Is to be considered effective selection. The nature of the criterion chosen may alter the effectiveness of a particular technique used for the purpose of selection. Various nonparametric methods have been used successfully to predict academic achieve ment in college including such methods as the adding and averaging of percentile ranks which cannot be justified mathematically. In general, the nonparametric techniques are most efficient at the extremes of the criterion distributions, and the selection ratio and the success ratio are important limiting factors as is the case with regression and discriminant analysis. Multiple cutting scores are being used successfully with vocational aptitude test batteries for the purpose of determining minimum qualifying scores. A nonparametric method of predicting a dichotomous criterion can be more effective then the discriminant function if the data used do not conform to the necessary assumptions of the discriminant function; however, theoretically, the nonparametric technique cannot exceed the efficiency of the discriminant function of the data conform in all aspects to the required assumptions. Evidence exists to support the hypothesis that 93 actuarial pattern analysis of items on a single test, or as a means of combining several scales or tests, may provide a higher degree of discrimination them the usual additive techniques. Empirical comparisons of methods of prediction.— Very few studies have been made which have tested the practical application of the mathematical models which have been developed for the purposes of differential prediction; an even smaller number of studies has com pared two or more methods of differential prediction on the same set of data. The largest group of studies has been concerned with a variety of methods of determining test weights which would be simpler and as effective as the method of least squares in forming a composite score for predictive purposes. Multiple cutting scores have been demonstrated to be as effective as multiple regression in cross- validation studies. Also, methods-such as counting the number of successes in a criterion distribution for various levels of achievement have been found to compare favorably with the Taylor-Russell tables in the selection of satisfactory employees. In contrast to other findings, one study demon strated the discriminant function to be an effective predictor even when the assumptions were deliberately 94 violated by means of selected artificial data. Measures of association.— If the effect of a selection device for the prediction of college success as a qualitative classification is to be evaluated on a relatively unselected population, all of whom have had the opportunity to test themselves against the success criterion, then the net result will be to obtain data in a four-way contingency table which will indicate how many successful students would be selected by this device and how many potentially successful students would be rejected along with similar information for the unsuccessful students. There are two basic approaches to evaluating the efficiency of this type of prediction. One method involves the use of a single index number to express the amount of association present in the table, and the other method is to use a two-way index which will indicate the number of correct classifications and the number of misclassifications. In either method, the amount of improvement in prediction resulting from the method used must be determined in relation to the magnitude of success in the original unselected population. An additional refinement can be made which will yield a more efficient forecast if it is possible to assign relative values to the different types of errors 95 of misclassificatlon and hence permit the maximization of the expected utility which will be a function of the stated objectives of the selection process. v CHAPTER III SOURCES OF DATA AND PROCEDURES I. DESCRIPTION OF SUBJECTS This study was based on the records of students attending El Camino College in Lawndale* Califomia. One of the minor objectives of the study was to investigate the predictive validity of the relatively new School and College Ability Test (135) when used with junior college students. This test was published in 1955 by the Educational Testing Service. El Camino was one of the first junior colleges to adopt the SCAT* as part of their regular battery of entrance tests and administered the tests for the first time In May of 1956. The basic data for this study were assembled during the school year 1958-59; therefore, El Camino College represented one of the few opportunities to obtain a full two year follow-up of course grades to use as a criterion for prediction. The students in the study took all of their tests in a period from May to August 1956, and all were making their first attempt at education beyond the high school. ♦SCAT will be used throughout the rest of the paper to-indicate the School and College Ability Test (Form 1A). 96 97 Some of the students started in the summer session of 1956 and the largest number started school in the fall semester of 1956. No distinction was made between these two groups. All these students were considered as entering freshmen on the basis of having no record of previous college work. Complete programs were on file for all students showing the amount and kinds of courses attempted from Summer 1956 to June 1958. By means of the permanent student number it was possible to identify the programs for all students who were new to El Camino College in Summer or Fall 1956. Each program was personally checked by the investigator and a list of IBM alpha numbers was prepared to include all those students who had attempted twenty-four or more units of course work, which would be acceptable to the state colleges as accredited college work, during the period from summer session 1956 to June 1957. This group consisted of 587 students and will be referred to as the total population of transfer students. By means of a table of random numbers (4:142-45), the first sample, Group A, was selected from the total population of transfer students. Each sample in the study was designed to consist of 100 students. In selecting Group A, it was necessary to select a total of 156 students at random In order to obtain the desired sample size of 100. The records of 56 students were Investigated and discarded because of the lack of test data, because the student had attempted previous college work, or in two cases because the high school diploma had been granted on the basis of the tests of General Educational Development. Later in the study a second sample, Group B, was selected from the total population of transfer students (less 156 already drawn) in the same manner as Group A. It was necessary to examine the records of 166 students to obtain the desired sample size of 100 for Group B. Table A in the appendix shows the major subjects and the desired four-year institutions as reported by the students of both Groups A and B at the time they filed their application for admission. Four students in each group indicated on their application that they were nontransfer students. They were retained in the study, however, because they met the requirements of one of the basic assumptions, i.e., these four students attempted twenty-four transfer units during the first year. Several students left this section of the applica tion blank or filled in only one part of it so the totals will not equal 100 for either sample. 99 IX. BASIC BATA A complete folder of permanent records including test scores and transcripts of high school and college work was obtained for each student selected by means of the random numbers. No one was discarded if only the high school transcript was lacking. There were fourteen such cases and the transcript was requested by the col lege from the student1s high school in order to complete the data. In all Instances the transcript was received and used in the study. The basic data for the study were obtained from each student's permanent folder and included: 1. The application for admission 2. The high school transcript 3. The converted scores on the SCAT 4. The standard scores on the Cooperative Reading Test [Form CgZ] (28) 5. The junior college transcript The high school grade point average was computed on the basis of four points for an A grade and includes only the grades received in the following subject areas: English, foreign language, mathematics, natural science, and social science. Table B in the appendix reports all the data for each student used in the study; It also includes each student's classification on the criterion 100 as described below. III. CRITERION OF SUCCESS Different admission standards are required of the junior college transfer student depending on whether he applies for admission to the state colleges or to the University of California. There are additional sub divisions of requirements according to whether or not the student was originally eligible to enter the institu tion of choice directly as a freshman from high school. The emphasis of this study is to identify the successful transfer student according to a minimum set of demands which he must meet regardless of his performance in high school. One of the basic assumptions of the study is that the requirements for the junior college transfer student, who was not eligible to enter a state college from high school, constitute an adequate criterion by which to evaluate his academic success. These requirements, which are stated in the California Administrative Code. Title 5, Education, provide uniform admission regulations for all the state colleges in California. They read as follows: 930. Students Who Transfer From Junior Colleges. An applicant who has earned credit in a junior college may be admitted to a state college if he meets the standard previously listed in this article, except that in case the applicant was Ineligible for admission to a state college on the basis of his 101 high school record he must, as a condition of admission to a state college, have completed 24 or more semester units of college work with a grade point average of 2.0 (grade of C on a five-point scale) or better in the total program attempted. Because of the above requirement it was decided to consider everyone who attempted at least 24 units of college work during the first year as a potential transfer student. Those students who completed at least 24 units* of college work with a 2.0 average by June 1957 were considered as successful transfer students. Those students who were identified as potential transfer students but who were not classified as "successful” were designated as unsuccessful transfer students. Except when specifically labeled otherwise, the terms successful and unsuccessful will refer to the above criterion. An additional criterion of success was established for those potential transfer students who were selected for study and who continued to attempt college work until June 1958. For these students, the successful group was defined as those who completed 60 or more college units with a 2.0 average and hence would *The number of "activity” units was governed by the regulation set forth in the Long Beach State College Bulletin. Activity courses are defined as those which provide practice in such areas as journalism, music, speech and drama, and physical education. A student may not earn more than eight units in one area, nor more than 20 units in all areas, to be counted toward the 124 units for graduation. 102 be eligible, in most cases, for upper division standing at a state college. The unsuccessful group was all those students who continued college work: but did not meet the 60 unit criterion of success. In Group A, 67 students were checked against the 60 unit criterion and in Group B, 74 students were classified on this basis. IV. SELECTION OF VARIABLES The first step in the selection of the variables to be used for prediction was to inspect visually the frequency distributions for all variables, i.e., test scores and high school grade point average, when plotted against the two sets of criteria. A tentative efficiency rating was given to each variable in terms of cost and utility. These data are presented in Tables C through H in the appendix. The actual selection of variables to be used was based upon the above ratings and the biserial correlations of each variable with both criteria as well as the inter correlations of those that appeared to be the best indicators of success. The high school grade point average and the verbal score of the SCAT were chosen as the two best predictive variables. There was not much difference in predictive validity between the scores on the speed of comprehension 103 scale and the total score of the Cooperative Reading Test. The speed of comprehension score was chosen because of greater simplicity in scoring and some advantage in terms of a lower Intercorrelation with the other test variable being used. The three variables selected for investigation and the abbreviations to be used to represent them are listed below: V score = the verbal score on the SCAT S score = the speed of comprehension score on the Cooperative Reading Test GPA = the high school grand point average (grades 10-12) V. THE DISCRIMINANT EQUATION The discriminant equation as formulated by Wert, Neidt, and Ahmann was used to obtain the linear function which best represents the maximum separation of the two groups (i.e. success and nonsuccess) when the significant variables used for prediction have been given their optimum weights ( 1 6 9: 2 6 3-7 1)• In essence, the procedure Is much like a regression analysis except that the criterion measure has only two categories instead of being a continuous series of measurements. The distinction between methods of selection and 104 those of classification has been pointed out in Chapter II. Wert, Neldt, and Ahmann present methods for both functions; however, the normal equations differ only by a mathematical constant throughout and the discriminant equation being used has the advantage of yielding the probabilities for each individual being In either segment of the dichotomy ( 1 6 9: 3 6 6). The necessary assumption for this method of selection by discriminant analysis is that the distribu tion in which the dichotomy exists is a single normally distributed variable. This is the same assumption that is necessary for the biserial correlation and appears to be justified by the nature of the criterion variable. The method of classification used by Wert, Neldt, and Ahmann implies that the two groups in the dichotomy represent a noncontinuum or nonvariable characteristic. This is similar to the assumption for the point biserial correlation (169:259)* and the data in the present investigation do not satisfy this assumption. Details of the discriminant analysis are given in Chapter IV. As In multiple regression techniques, it is possible to calculate the amount of contribution of each variable and to determine whether or not each variable is making a significant contribution to the composite score. It was found that the reading variable 105 did. not contribute significantly to the variation accounted for by the other two variables, the V score and the GPA, and so the S score was eliminated from the discriminant equation and all calculations of cost and utility were based on the two-variable equation. An expectancy table was prepared from the discriminant equation showing the chances per hundred of success for each possible combination of scores.* On the same table, the actual frequencies of obtained GPA and V scores were plotted with each tally indicating success or nonsuccess in the criterion, as in a scatter diagram, so that the actual cost and utility for Group A could be calculated for any given probability of the composite score (see Table IX to be presented in Chapter IV). Another table was prepared containing the actual cost and utility for each cutting score from a probability of success of 32 per hundred to that of 52 per hundred (Table X, Chapter IV). These limits appeared to be outside the functional use of the lower range of utility and the upper range of cost. After inspection of the various combinations of cost and utility, five different cutting scores were *The V score was grouped into intervals of three on the scale of converted scores and the GPA was grouped Into intervals of two-tenths of a grade point. See Table VIII, Chapter IV. selected for cross-validation on Group B and for compari sons with the other three methods of prediction to be investigated. These cutting scores were selected because they represent maximum utility for different levels of cost. The cutting scores selected were 32, 33, 40, 46, and 52 chances per hundred of success in the criterion. VX. GUILFORD AND MICHAEL'S FORMULA FOR A CRITICAL SCORE Guilford describes two graphic methods and a formula for locating the optimum cutting score in an artificial dichotomy (63:340-47). One of the graphic methods was used to compare results with those obtained by the formula. No assumptions are necessary for using the graphic solution but the results obtained are not very precise for all variables. The following formula, Guilford and Michael's, was used to calculate cutting scores on the GPA, V score, S score, and T(Read)* *T(Read) = Total score on the Cooperative Reading Test. Form C2Z. 107 where Mx - mean of the entire distribution of both categories combined p = proportion of the total population in the category having the higher mean score on X q = 1 “ p y = ordinate in the unit normal distribution at the point of division of the area under the normal curve with p proportion above it z - standard measure of the point at which y occurs (62:28) The above formula was used because of the amount of precision involved compared to fitting a curve as In the graphic method and because the data do not appear to violate the necessary assumptions that both distributions, the Independent variable and the dependent variable, are actually continuous and normal. In order to facilitate the application of all cutting scores in different combinations, it was necessary to put all the data for each individual on a 3 x 5 card. Each individual was assigned a number; this number and his sex, total grade points, GPA, all three SCAT scores, S score, T(Read), and both criterion classifications were placed on a single card. The predictive variables were coded by digits assigned to each Interval in their distributions. The data for each variable were grouped 108 in order to limit each variable to a range of 25 intervals which would accommodate both Groups A and B, and to reduce the number of possible combinations of scores for the expectancy table developed from the discriminant equation and for the nonparametric method to be described in the next section of this chapter. Results of the application of various combinations of the critical scores to the data for Group A are pre sented in tabular form in Chapter IV. All combinations listed were later cross-validated against data for Group B. VII. A NONPARAMETRIC METHOD OP DETERMINING CUTTING SCORES The method to be described was an outgrowth of giving consideration to test efficiency in Berkson's terms of cost and utility (10:246-55). If the concept was sound for a 3ingle test as he proposed, then It should be equally sound for a combination of tests. The only assumption necessary is that whatever form the distribution should take in any of the variables, the sample upon which the cutting scores are based must adequately represent the distribution of the total population from which it was drawn for all variables used in prediction. The use of random samples Is an 109 attempt to satisfy this assumption in the present study. This method is based upon a simple counting of subjects who would be eliminated by any given cutting score and noting their classification on the criterion under investigation. The cutting score may be established to yield a minimum possible cost or the cutting score can be located at the point where the rate of increase of cost becomes greater than that of utility. Other cutting scores can be established on a trial and error basis. As described in the preceding section, it was necessary to group raw scores into intervals so that there would not be an unworkable number of possible combinations. With only two variables, the simplest method is to plot scores for one predictive variable against the other, as in a scatter diagram. Each tally is identified as a successful student or as an unsuccess ful one, and it is a simple matter to count the cost and utility for any given combination of cutting scores. If the number of subjects becomes too large for the diagram procedure, or if three or more variables are Involved in the prediction, ordinary data processing equipment can be used to do the counting. It is also possible to calculate cost and utility for what will be called "compensating cutting scores." By this procedure, a score below the cutting point on one 110 variable is accepted if it is offset by a score sufficiently above the cutting point on the other variable. In other words, each interval score on one predictive variable has its own separate cutting scores on the other variable. Results of the application of various cutting scores in terras of cost and utility are presented in tabular form in Chapter IV. VIII. WHERRY'S MULTIPLE BISERIAL REGRESSION EQUATION Wherry has established a multiple biserial regression equation which is obtained directly from the intercorrelations of the predictive variables and holds for both real and artificial dichotomies because the regression weights obtained are not affected by the nature of the assumption in either case (172:189-95)• Wherry's arguments for the advantages of his equation have been presented in Chapter II. Wherry derives several forms (identities) of his normal equations and recommends the following set of equations for the purposes of test selection. This set of equations has been used in the present study and can be obtained by multiplying equations (l) Ill below by^-^p- V^' + W2‘r12 + ...... Wn'rm = ri(bis) Wl,p12 + W2* + .......... wn'r2n = r2(bis) wl* rin + w2lr2n + .......... wn* = ^(bis) where W^' = W^p^K * n = ^ k Di “ Mip “ Miq The regression equation is in deviation* or standard, scores and is as follows: xc = WlXl + Wgx2 ........ Wnxn where x is the composite score based on the maximal weights of each predictive variable. Wherry's formula for calculating the multiple correlation coefficient is: cRb or p * kV W jDj. + w2d2 + ........WnI^ The above formulas were compared algebraically to the discriminant equation formulated by Wert, Neidt, and Ahmann (169:263-71) and used in the present study. ♦General form of the biserial correlation regardless of the assumption as to the nature of the dichotomized variable, i.e. K = E& or y/pq7* 112 When deviation score formulas were substituted for the variance and covariance terms, Wherry's formulas resulted in the same set of normal equations used by Wert, Neidt, and Ahmann except for a constant in the regression weights. The comparison is demonstrated below for the normal equations for three variables only, in order to facilitate the explanation. Wherry starts with the following basic equations: W1<312 + W2ala2r>12 + w3ai<33r13 = D1 (1) W1Oi<3sr12 + W2' v J 22 + w3^2^23 = D2 wlQa< 33ri3 + w2cr2^3r23 + w^r32 = D3 2 Substitute in (l): ^-xi for Oi2 and ^ X1XJ for j N N Then Multiply (2) by N Then: W + W2£x;]X2 + W^Sx^x^ = ND^ (3) W^^x-^Xg + Wg^x2 + W^ £xgx^ — W-^x-jX^ + Wg^XgX^ + W^l.x| = ND3 If: W± = a^ and Di = = Mlp - Mlq 113 Then: Equations (3) are identical with those presented by Wert, Neidt, and Ahmann ( 1 6 9: 3 6 7), equations (4) below, for the maximum separation of two populations except for the constant factor N on the right side of each equation. allxl + a2$?lx2 + a3lxlx3 = dl (4) a1^_x1x2 + a25.x| + a^^Xrpc^ = d2 allxlx3 + a2&2x3 + a3lx§ = d3 Also: Equations (3) are Identical with those presented by Wert, Neidt, and Ahmann (169:264), equations (5) below, for selection from an artificial dichotomy which exists as a normally distributed variable, except for the constant factor z* on the right side of each equation. allxl + a2lxlx2 + a3^xlx3 = Nzdl (5) a1£x1x2 + a2£x| + a3^?2x3 “ Nzd2 alXxlx3 + a2^x2x3 + a3l?3 “ Nzd3 Therefore: The weights obtained from all three sets of equations are proportional and yield the same predictions (63:422). In (4), Wi - Hai In (5)» Wj. = Sl z *z represents the height of the ordinate dividing the normal curve of unit area into p and q parts. 114 Because equations (5) have already been used In the discriminant analysis, there is nothing to be gained by calculating composite scores by Wherry's formula since one can be derived from the other. The multiple correla tion was computed on the basis of Wherry's formula and compared with that obtained from the discriminant equation. IX. CROSS-VALIDATION ON GROUP B After establishing various cutting scores based on the methods described above, a second sample of 100 transfer students was chosen at random from the total population of transfer students and designated as Group B. The cutting scores established on the basis of Group A data were then applied to Group B and the resulting cost and utility, in terms of Group B students, were then calculated. The efficiency of the different methods of predicting the criterion was compared at different levels of cost. Data for these comparisons are given in Chapter IV. X. GROUP A COMPARED WITH GROUP B An analysis of variance was carried out for all predictive variables used in determining cutting 115 scores to find whether there were any significant differences in performances between Group A and Group B, and as a check against the random sampling procedure to insure that the two groups actually do represent the same total population. Biserial correlations between the predictive variables and both criteria of success were computed on the data from Group B and compared with those obtained from Group A. Comparisons were also made of the inter- correlations of predictive variables for both groups. A chi-square analysis was made of the number of successful men and women in Groups A and B. XI. ANALYSIS OP SEX DIFFERENCES IN PREDICTION OF THE CRITERION Groups A and B were pooled to form a single sample of 200 transfer students, 129 men and 71 women. A discriminant equation was calculated for the total group of 200. The mean scores on GPA and V 3core for men were substituted into the discriminant equation and the resulting composite scores compared with that obtained when the mean scores for women were used. An analysis of variance was carried out for the GPA and V score among the men and the women. Biserial correlations for GPA, V score, and S 116 score were computed for both men and women with the 24 unit criterion of success. Separate discriminant equations were worked out for men and for women with the two variables of O-PA and V score. The percentage of contribution of each variable was calculated and the costs and utilities were computed for a number of cutting scores. Comparisons were made between men and women on these data. Cutting scores were derived from the data on both men and women by the nonparametric method. The cost and utility figures for the nonparametric method were compared with those obtained by the discriminant equation on the same data. XII. SUMMARY A sample of 100 students, Group A, was selected at random from all those new students at El Camino College who attempted at least 24 units of college credit work between Summer Session 1956 and June 1957- Those students who had a 2.0 average in at least 24 units by June 1957 were classified as successful transfer students, and those who did not were classified as unsuccessful transfer students. High school grade point averages (limited to academic subjects), School and College Ability Test 117 scores, and Cooperative Reading Test scores were obtained for all students and used to predict the student's classification on the criterion of success. Biserial correlations and intercorrelations were used to determine the three best predictive variables for investigation. The variables chosen were high school grade point average, verbal score on the School and College Ability Test, and the speed of comprehension score on the Cooperative Reading Test. Cutting scores were calculated on the data from Group A by means of a discriminant equation, a formula based on the principle of equal likelihood, and a nonparametric approach based on the number in each criterion classification for a given cutting score. Cutting scores obtained by a graphic solution, based on the principle of equal likelihood, were com pared with those obtained by the formula; and Wherry's multiple biserial regression equation was demonstrated to be equivalent to the discriminant equation already used, the regression weights being proportional. A second random sample of 100 transfer students, Group B, was obtained from the total population of transfer students already identified for the selection of Group A. The cutting scores determined by the three different methods, on Group A data, were then 118 cross-validated on Group B and the resulting predictions were evaluated in terms of cost and utility to determine whether or not differences in efficiency existed among predictions made by the discriminant equation, the cutting score formula, and the nonparametric method. Data from Groups A and B were compared by means of chi-square and analysis of variance to verify that both samples did in fact come from the same population. The two samples were pooled and the predictive efficiency for the men (N = 129) was compared to that of the women (N = 71) by means of separate discriminant equations for men and for women. Another comparison was made between costs and utilities as determined by the discriminant equation and the nonparametric method for the new grouping of data by sex. CHAPTER IV ANALYSIS OP THE FINDINGS The basic problem under Investigation involved a comparison of several methods of predicting success In a dichotomized criterion when the basis for comparison was a measure of efficiency defined In terms of cost and utility. Wherry's multiple biserial regression equation has been demonstrated to be equivalent to the discriminant equation used in this study because the two equations yield regression weights which are proportional;* therefore, comparisons will be limited to the data obtained by means of the discriminant equation, the formula for a cutting score, and the nonparametric method. Data from Group A for each of the three methods are presented, followed by the cross-validation data on Group B and then the actual comparisons between methods at three different levels of cost. Group A data are compared with that of Group B and the two groups combined to provide data for a single sampie of 200 students. Thi3 sample Is broken down to provide separate data for men and women. *See supra. pp. 110-14. 119 120 The first section will provide descriptive data for both Group A and Group B and the blserial correlations of all predictive variables with both criteria of success. I. BISERIAL CORRELATIONS Table I shows the number of successful and unsuccessful students in'each criterion classification and in each sample group. Both sample groups had 48 unsuccessful students and 52 successful students when classified on the 24 unit criterion. There was no attempt to control this factor on the part of the investigator. It was the result of the random sampling procedures used. Of the Group A Students, 67 continued to attempt college credit work during the second year which ended in June 1958; 25 of these students had accumulated at least 60 units with at least a 2.0 average by June 1958. In Group B, 74 students continued the second year and 19 met the requirements of the 6 0 unit criterion. No attempt was made to conduct a follow-up study of those students who did not continue their college work at El Camino College. Table II shows that of the 61 students (both groups combined) who were unsuccessful on the 24 unit criterion and finished the second year, only 2 students were classified as successful on the 60 unit criterion. This 121 TABLE X NUMBER OP SUCCESSFUL AND UNSUCCESSFUL STUDENTS IN EACH CRITERION CLASSIFICATION FOR GROUPS A AND B 24 units with 2.0 average 60 units with 2.0 average Group A Unsuccessful 48 42 Successful 52 25 Total 100 67 Group B Unsuccessful 48 55 Successful 52 19 Total 100 74 122 TABLE II NUMBER OF SUCCESSFUL AND UNSUCCESSFUL STUDENTS ON THE 60 UNIT CRITERION RESULTING FROM EACH CLASSIFICATION ON THE 24 UNIT CRITERION 24 units with 2.0 average 6 0 units with 2.0 average Group A 48 (unsuccessful) . . . 24 i 2 1 (unsuccessful) (successful) 52 (successful) . . . 18 I 23 l (unsuccessful) (successful) Total 100 67* Group B 48 (unsuccessful) . . . 35 I (unsuccessful) (successful) 52 (successful) . . . 20 < 19 < (unsuccessful) (successful) Total 100 74* *This figure includes only those students who continued at El Camlno College for the second year. means that only 3.3 per cent of the unsuccessful students were able to change their classification during their second year. Of the total unsuccessful classification on the 24 unit criterion, 35 are unaccounted for in the second year. Some of this group changed to nonacademic majors at El Camino College but most of them dropped out of school. It is possible that some of them could have transferred to another junior college. Of the 104 students who were classified as successful on the 24 unit criterion, 80 continued at El Camino College through the second year. Forty-two of the students who continued through the second year, 52.5 per cent, were also classified as successful on the 60 unit criterion. No data on second year performance are available for the 24 students who were successful for one year but did not continue at El Camino College. This group would be eligible, on the basis of their first year record, to transfer to a state college. Table III presents a summary of the biserial correlations obtained from both Groups A and B between all predictive variables and both criteria of success. The table also shows the mean score and standard deviation for each group on every variable and the mean score of the successful students in each group. The figures given for the 5 per cent and the 1 per cent TABLE III SUMMARY OF BISERIAL CORRELATIONS FOR GROUPS A AND B BETWEEN PREDICTIVE VARIABLES AND BOTH CRITERIA OF SUCCESS _______ 24 unit criterion_______ 60 unit criterion______ Numerical Mean Mean S.D« Biserial Mean Mean S.D. Biserial variable Group successful total total r successful total total r Total H.S. A 59.35 53.55 18.46 .41 65.80 56.28 18.80 .50 Gr. Pts. B 59.73 53.60 18.02 .44 58.00 53.95 18.06 .18 H.S. Gr. A 2.80 2.54 .6 5 0 .52 3.01 2 .6 1 .688 .58 Pt. Average B 2 .8 2 2.55 .579 .6 0 2.75 2.53 .575 .31 SCAT V A 298.98 294.52 13.37 .44 3 0 1 .6 8 294.54 12.38 .57 B 296.50 2 9 2 .9 8 13.51 .34 2 9 8 .8 9 294.20 12.01 .31 SCAT Q A 303.00 300.80 17.70 .1 6 3 0 5 .2 0 300.31 16.51 .29 B 303.48 300.65 16.33 .23 3 0 7 .2 1 3 0 2 .2 6 17.53 .23 SCAT T A 300.29 2 9 7 .0 0 12.55 .34 3 0 2 .6 0 297.00 11.18 .49 B 299.23 296.75 9.13 .35 301.63 297.35 11.54 .30 H IO TABLE III (continued) Numerical variable Group 24 unit criterion 60 unit criterion Mean successful Mean total S.D. Biserial total r Mean successful Mean total S.D. total Biserial r Reading V A 59-08 58.02 9.19 .15 61.20 58.07 9.34 .33 B 59.88 57.00 10.01 .3 8 59.37 57.85 10.05 .12 Reading S A 58.51 56.87 7.27 .30 59.18 57.13 7 .2 6 .28 B 57.75 55.29 8 .5 6 .38 58.34 55.96 8.71 .22 Reading L A 61.15 60.00 7.20 .21 61.42 59.11 6.08 .37 B 60.19 58.92 6.12 .27 59.08 58.64 6.43 .05 Reading T A 60.13 5 8 .6 5 . 6.88 .28 6 1 .1 0 58.22 6 .5 8 .43 B 59.62 57.16 7.76 .41 59.29 57.74 7.91 .16 bis Ms ~ Mt /V crt \y N = Group A Group B 24 units: p = .5 2 0 p = .5 2 0 rbis for p N rbis for P = . p = ■ . (N = 100) (N * 100) (N N « 60 units: p - .373 . p = .2 5 7 (N = 6 7) ( k » 74 rbis for p = 100 ) . 520) 67 ) .373) 74 ) .257) 5^ level 1% level 5# level 1% level level 1$ level .197 .259 .246 .322 .251 .331 74) ro V J 1 126 levels of confidence were calculated by the procedure suggested by Wert, Neidt, and Ahman (169:260-62). The biserial correlation is converted to a point biserial correlation and the resulting figure Is corrected for coarse grouping of a product-moment correlation and the obtained value can be read in the prepared tables of levels of significance for different degrees of freedom. All correlations with the 24 unit criterion were significant at the 5 per cent level of confidence except the SCAT quantitative score for Group A and the Cooperative Re&ding Test vocabulary score for Group A. All of the obtained correlations for Group A with the 60 unit criterion were significant at the 5 per cent level of confidence. For Group B and the 60 unit criterion, only three variables (high school GPA, SCAT V score, and the SCAT total score) were significant at the 5 per cent level of confidence. None of the Group B correlations with the 60 unit criterion reaches the 1 per cent level of confidence. On the basis of Group A data, the high school grade point average and the verbal score on the School and College Ability Test appeared to be the best indica tors of the ability of a student to maintain a 2.0 average in college credit work. The data from Group B were consistent with the findings on Group A. 127 The first entry In Table III is for total high school grade points. This figure is the sum of the grade points used in computing the grade point average. This index was Investigated because of the fact that it was limited to academic units only and might reflect not only how well the student did in his work but also the extent to which he recognized himself as a pre-college student in high school. The obtained numerical difference between the biserial correlation for total grade points and grade point average favored the latter in all four cases. In comparing the 60 unit criterion correlations with each other and with those obtained on the 24 unit criterion, the nature of the biserial correlation itself mu3t be kept in mind. It is well known that the biserial correlation gives the best estimate of the product-moment correlation for the same data when the point of division in the dichotomy is at the midpoint, i.e. p 3 .5 0 ( 6 3: 3 0 1; 164:270; and 1 6 9: 2 5 8). The constant term in the formula used to compute the biserial correlations (see Table II) Is greater than unity for p = .39 and less than unity for p = . 3 8. The constant term gets increasingly smaller as p, the percentage of successful students, gets smaller. Therefore, if there is no change in the difference in means or in the standard deviation, the resulting 128 blserial correlation will be smaller because of the change In the constant term. For Group A, all of the correlations with the 60 unit criterion were numerically larger than those with the 24 unit criterion except for the S score which dropped from .30 to .28. In Group B, where the number of successful students on the 60 unit criterion dropped to 26 per cent, all of the correlations were smaller than those obtained with the 24 unit criterion. The predictive efficiency is not as low in Group B as the correlations would Indicate. This can be determined by an inspection of the mean scores for successful students or by inspection of the frequency distributions in Tables C through H In the appendix. The low correlations for Group B on the 60 unit criterion are not all a function of the nature of the blserial formula. An inspection of mean scores in Table III for Group A shows that there is very little difference In the total means for each criterion but that In each case the mean of the successful group on the 60 unit criterion is larger than the mean of the success ful group for the 24 unit criterion. For Group B, the mean of the 60 unit successful group is not consistently higher than that of the 24 unit successful group. The lower mean high school grade point average for the 60 unit 129 successful students in Group B suggests that fewer of the more able students in this group continued at El Camino College than was the case with Group A. II. THE DISCRIMINANT EQUATION The discriminant equation is expressed by Wert, Neidt, and Ahmann (169:264-65) as: v » a-^ + a2x2 + + + amxm where the x's are the numerical variables and the a's are the coefficients to be determined from a series of simul taneous equations. The necessary equations for three variables are shown as equations (5) in Chapter III.* The following were the basic data necessary for the solution of the simultaneous equations for the three variables selected for investigation: N - 100 ^X-l = 254.40 £x 2 = 9452. £X 3 = 1687. « 689.4622 £x 2 = 9 1 1,2 8 8. 2 X3 = 33,745. £XXX2 - 24,430.83 ^XXX3 = 441.13 £X 2X3 = 166,746. where: N = the number of students in Group A X]_ = the high school grade point average X2 = the converted V score (less 200 to facilitate computation) *See supra, p. 113. 130 X3 = the scaled score on the speed of reading com prehension (less 40 to facilitate computation) Table IV shows the differences in mean scores for the successful and the unsuccessful students, on the 24 unit criterion, for their grade point averages, V scores, and S scores. The sums of squares and cross-products were reduced to deviation scores from the general mean and substituted in the simultaneous equations and then the equations were solved for the three unknown values aj, a2, and a^. The discriminant equation then became: v = . 39917x1 + .01406x2 “ . 00479x3 where the v and x values were all in deviation score form. To change the discriminant equation to raw score form the following formula was used (Wert, Neidt, and Ahmann, 169:269-70): V - V = ai(Xl - Xl) + a2(X2 - X2) + a3(X3 - X3) where V is the normal deviate for 32 per cent in the suc cessful group (.0502) and the predicted V is in terms of sigma units in the normal curve. This raw score equation can be solved for any combination of scores and by con sulting a table of the normal curve it is possible to determine the probability of success for the given combina tion of scores. Table VIII, to be presented later, is a table of expectancy for the two variables, (JPA and V score, which has been constructed by this method. 131 TABLE IV MEAN SCORES FOR SUCCESSFUL AND UNSUCCESSFUL STUDENTS IN GROUP A ON GPA, V SCORE, AND S SCORE Mean Difference Numerical variable Successful (k = 5 2) Unsuccessful (k = 48) in Means (d) Nzd GPA 2.80384 2 .2 6 2 5 0 .54134 21.56915 V score 2 9 8 .9 8 0 7 6 289.68750 9.29326 370.28065 S score 58.51923 55.08333 3.43590. 136.89999 where z = the height of the ordinate of the unit normal curve at p = .5 2 0 132 Multiple correlation for three variables.— Wert, Neidt, and Ahmann (169:266) give the following formula for the multiple biserial correlation: When this formula was applied to the three variable discriminant equation indicated above, the multiple biserial correlation for GPA, V score, and S score with the 24 unit criterion of success was .570. This was slightly higher than the correlation of .5 2 found for the GPA alone. in prediction.— The value "delta" above corresponds to the sum of squares for regression in an analysis dealing with a numerical criterion and the effectiveness of each variable can best be determined by its contribution to the numerical value of the "delta." It is the absolute value of the sum that is used to determine the percentage contribution of each variable, i.e., the negative sign in the value for a^ was disregarded for this purpose. variable in the three variable discriminant equation. The high school grade point average contributed most to where A = a^zd-^ + a2Nzd2 + a-^Nzd^ The relative effectiveness of the variables used Table V shows the relative contribution of each 133 TABLE V PER CENT OF CONTRIBUTION OF EACH VARIABLE IN THE THREE VARIABLE DISCRIMINANT EQUATION (GROUP A) Numerical variable a-^Nzd^ Per cent contribution GPA 8,609757 59.5 SCAT V score 5.206145 3 6 .0 Reading S score .655750 4.5 Total 14.471652 100.0 134 the prediction of successful transfer students when three variables were used. Because of the relatively small contribution of the S score, it was decided to compute a discriminant equation for the two variables, GPA and V score, to determine whether there would be any significant loss of predictive efficiency as determined by the multiple biserial correlation with the criterion when only these two variables were used. The discriminant equation for GPA and V score was found to be: v = .40018x1 + *01209x2 (deviation score form) Using the formula from the preceding section, the computation of the multiple biserial correlation resulted in . 5 6 9. This is to be compared with the .570 which resulted from the three variable equation. Wert, Neidt, and Ahmann ( 1 6 9: 2 6 9-8 9) present the following formula for an F test of significance for the elimination of one variable which they explain may be considered as a test of significance of a partial biserial correlation from zero: 135 where: n = 1 degree of freedom for the one variable eliminated m = the number of variables in the original equation D = "delta" as previously defined The F-value, with 1 and 96 degrees of freedom was found to be 2.66 which was not significant at the 5 per cent level of confidence, which means that there was no loss in predictive efficiency by dropping the third variable from the discriminant equation. Thi3 also means that the partial correlation of the S score with GPA and V score held constant was not significantly different from zero. Table VI shows the relative contribution of GPA and V score to the prediction of success when only these two variables are used. A comparison of Table VI with Table V indicates that the GPA contributed to a greater degree in the two variable equation than it did in the three variable equation while the contribution of the V score was decreased slightly by the removal of the third variable. Table VII 3hows the intercorrelations for the three variables used In the discriminant analysis. In 136 TABLE VI PER CENT OF CONTRIBUTION OF EACH VARIABLE IN THE TWO VARIABLE DISCRIMINANT EQUATION (GROUP A) Numerical variable ajNzdi Per cent contribution GPA 8.6315^2 6 5 .8 V score 4.476693 34.2 Total 13.108235 1 0 0 .0 TABLE VII INTERCORRELATIONS OF SELECTED PREDICTIVE VARIABLES FOR GROUP A Numerical variable GPA V score S 3core T(Read) GPA .443 .316 V score .750 .8 2 2 137 addition, it contains the correlation between the V 3core and the total score on the Cooperative Reading Te3t. The relatively high correlations between the V score and both reading scores is the reason that the two variable discriminant equation is as efficient in predicting success as the three variable discriminant equation. As a result of these findings, all additional data for the discriminant equation were based on the two variable equation only, using the high school grade point average and the verbal score of the SCAT as the two predictive variables. Raw score formula for the two variable discriminant equation.— The discriminant equation for two variables in deviation form was found to be: v = .40018x2 + .01209x2 By means of the formula already presented,* this deviation score form of the discriminant equation can be converted to an equation in which raw scores may be substituted directly. The raw score formula for GPA and V score was as follows: v = . 40018X2 + .01209X3 - 4 .5 2 8 7 0 If the mean scores on GPA and V score for Group A ♦See supra, p. 130. 133 are inserted in this formula, V will be equal to .0502 which is the normal deviate for 52 per cent in the successful group. This is the probability of success without any selection whatsoever. This can be seen in the deviation form of the equation; when mean scores are used, the right side of the equation reduces to zero and V - V = 0; therefore, V = V. Probability of success based on GPA and V score.— The discriminant equation in raw score form, shown above, can be used to calculate the probability of success for any combination of grade point average and verbal score. Table VIII has been constructed from the above formula and shows the probability of success for various combina tions of GPA and V score. This table has been constructed from Group A data only. Inspection of the table demon strates the fact that has already been established, i.e., that GPA contributed more to the prediction than did the V score. The chances of success increase faster on the vertical scale than on the horizontal scale. The range of probabilities was restricted to 32 chances per hundred at the lower end and 52 chances per hundred of success at the upper end. On the basis of the findings to be presented in the next section, it was evident that the small amount of utility to be gained below 32 chances per hundred was of little practical value; and that the cost TABLE VIII PROBABILITY OP SUCCESS BASED ON GPA AND V SCORE AND THE DISCRIMINANT FUNCTION (GROUP A) V s c o r e x 1 ro o\ CM in 00 r-I -3 - l> - 0 m VO on CM m 00 1 —J O ■ = t r— O 00 VO ON CM m 00 m VO VO vo [ ■ — t - r- 00 00 00 00 on on ON no O 0 r H iH rH iH CM CM CM 1 1 1 1 1 1 t 1 1 1 1 1 t 1 1 1 1 1 1 1 1 1 r 1 c- 0 no VO o\ CM m CO rH ■ = * e- 0 00 vo ON CM m 00 H - 3 * f r — O no vo GPA (Xx) in VO VO VO VO h- t- t- 00 00 co on ON ON ON O 0 O 1—1 rH 1—1 CM CM CM cm C V 1 C V i C V I C V I C V J CM CM C V I CM CM CM CM CM CM no no no OO no no no no no 3.9-4.0 3.7-3.8 3.5-3.6 3.3-3.4 3.1-3.2 2.9-3.0 2.7-2.8 2.5-2.6 2.3-2.4 2. 1-2.2 1.9-2.0 1. 7-1.8 1. 5-1.6 1.3-1.4 1.1-1.2 0.9-1.0 Values In the table represent chances in one hundred of success on the 24 unit criterion based on the following discriminant equation: V = .40018X! + .01209X2 - 4.52870 41 42 44 45 47 48 50 51 52 54 38 39 41 42 43 45 46 48 49 51 52 54 35 36 38 39 40 42 43 45 46 48 49 50 52 53 32 33 35 36 37 39 40 41 43 44 46 47 49 50 52 53 32 33 34 36 37 38 40 4l 43 44 45 47 48 50 51 53 31 33 34 35 37 38 39 41 42 44 45 47 48 50 51 52 54 32 34 35 36 38 39 41 42 43 45 46 48 49 51 32 33 35 36 38 39 40 42 43 45 46 47 54 50 52 139 140 of cutting scores above 52 chances per hundred was too high for the present objectives of the junior college in California. Cost and utility with the discriminant equation.— Table IX is a two-way frequency distribution of the GPA and V scores. Each cell of the table contains two entries. The upper entry (S/) represents the number of students in that cell who were successful on the 24 unit criterion. The lower entry (/U) represents the number of students in that cell who were unsuccessful on the criterion. In a sense, the table is really a three-way frequency distribution as each cell is a dichotomy. Each cell in Table IX corresponds to a cell in Table VIII so that if cutting scores were to be estab lished for any given probability of success, then the number of students who would have been accepted and rejected by this particular cutting score could be counted in Table IX. Table X presents a summary of the cost and utility which would have resulted from the application of cutting scores to Group A based on the discriminant equation. Data are provided for cutting scores ranging from 32 chances per hundred of success to 52 chances per hundred of success. Graph 1 is a graphic representation of these same data. The base line represents the probability of TABLE IX TWO-WAY FREQUENCY DISTRIBUTION OF GPA AND V SCORES FOR GROUP A V Score Interval Totals O ' ! C V l in CO i £ ■ = r 1 o CO no I VO | ov | C V J I T l CO 1 __ o H VO Ov c v j in co GPA C * - O cn VO Ov OJ m 1 CO r- 1 4 3 * C * * ~ o ro VO * rn OvO C V J JD 1 00 rH t - = 4 * 1 [ * - O m l VO for GPA Lf\ 04 VO OJ vo 04 VO OJ VO OJ t * - OJ c- OJ CO Ol CO OJ CO OJ ON (M 0\ C V J o\ C v l o\m C V J o m O ro o no rH m H no rH ro c v i pO w CO C v l S U 3.9-4.0 1/0 1/0 1/0 1/0 1/0 5 0 3.7-3.8 1/0 1 0 3.5-3.6 1/0 1/0 1/0 3 0 3.3-3.4 1/0 0/1 1/0 2 1 3.1-3-2 1/0 2/0 1/0 2/0 1/1 7 1 2.9-3.0 0/1 1/0 1/2 1/0 1/0 3/0 7 3 2.7-2.8 1/0 1/2 0/1 1/0 2/0 0/1 0/1 1/0 6 5 2.5-2.6 0/2 0/1 1/0 0/1 1/2 2/1 1/0 1/2 2/0 8 9 2.3-2.4 0/1 0/1 1/0 1/0 1/0 1/2 0/1 4 5 2.1-2.2 0/1 0/1 1/0 1/0 0/1 0/1 0/1 1/0 3 5 1.9-2.0 0/1 0/1 0/1 1/2 0/1 0/1 0/1 0/1 1 9 1.7-1.8 0/1 1/1 0/1 0/1 1/1 0/1 0/1 2 7 1.5-1.6 0/1 1/0 1/0 0/1 2 2 1.3-1.4 0 0 1.1-1.2 0/1 0 1 .9-1.0 1/0 1 0 Interval s Totals for -- 0 0 0 0 0 1 0 0 3 4 3 5 5 5 7 3 1 5 4 1 2 3 0 0 52 V Score U 1 0 0 0 2 4 4 1 2 3 6 8 2 1 4 5 1 0 2 1 0 0 0 1 48 Each cell of the table contains two values. The value above the diagonal S/ Is the number of successful students and the value below the diagonal /U is the number of unsuccessful students occurring In that specific cell. Ttri 142 TABLE X COST AND UTILITY FOR GROUP A BASED ON THE DISCRIMINANT EQUATION* Cutting score (chances per 100) No. cost Per cent cost Per cent utility No. utility 32 2 3.8 10.4 5 33 3 5.8 10.4 5 34 4 7.7 10.4 5 35 4 7.7 14.6 7 36 4 7.7 16.7 8 37 4 7.7 20.8 10 38 4 7.7 22.9 11 39 6 11.5 2 9 .2 14 40 6 11.5 31.3 15 41 7 13.5 35.4 17 42 8 15.4 37.5 18 43 8 15.4 41.7 20 44 8 15.4 45.8 22 45 8 15.4 5 0 .0 24 46 8 15.4 52.1 25 47 9 17.3 54.2 26 48 10 19.2 54.2 26 49 10 1 9 .2 60.4 29 50 10 19.2 62.5 30 51 12 23.1 68.8 33 52 13 2 5 .0 68.8 33 *Based on the discriminant equation: v = .40018X 2 + .0 1 2 0 9 x 2 - 4 .5 2 8 7 0 where X^ = GPA and Xq = V score 143 GRAPH NO. 1 Cost and Utility for Group A Based on the Discriminant Equation V = .40018X1 + .01209X3 - 4.52870 Unsuccessful Students (Utility) - - Successful Students (Cost) 40 ■ a n 32 22 20 X X- -X x— x— x- -x - - x 12 10 ,x- -x- -X- -X- -X 32 33 34 35 36 37 38 39 40 4l 42 43 44 45 46 47 48 49 50 51 52 Cutting score based on chances per 100 of success obtained from the discriminant equation 144 success for those who would be accepted by a given cutting score. The vertical axis represents the percentage of students who would be screened out by the application of a given cutting score. Both cost and utility are plotted for each cutting score. It is apparent that within this range of cutting scores, utility rose faster than did the cost. Except for certain portions of the cost curve which show no increase in magnitude, an Increase in utility generally results in a corresponding increase in cost. The irregularities in the cost curve would tend to lessen as the sample size grows larger and more students are involved in the computation of the percentage figure. Five cutting scores were selected for comparison with the other methods of prediction. Cutting scores of 32 and 52 chances per hundred were selected because they / represented minimum and maximum costs for this range of cutting scores. Cutting scores of 40 and 46 chances per hundred were chosen for comparison because they repre sented intermediate values of cost and yielded maximum utility at two of the "flat" portions of the cost curve in Graph 1. III. PRINCIPLE OF EQUAL LIKELIHOOD The basic problem is to predict the dichotomy (success and nonsuccess) from a single set of measurements. 145 The solution requires the finding of a critical score on the set of measurements which represents a 50 per cent chance of being in either category. A person who scores higher than the critical score is more likely to be successful than he is unsuccessful, and a person who scores lower than the critical score is more likely to be unsuccessful. Guttmann has shown the critical score to be located at the point of Intersection of the two curves of the attributes being predicted when they are plotted on the same base line (70:271 ff.). In terms of the present data, if the distribution curve for the unsuc cessful group, for any one of the predictive variables, were plotted on the same base line, then the score directly below the point at which the curves intersect would be the desired critical score and represent the score on that variable which would give a 5 0 per cent probability of being either successful or unsuccessful. In addition to determining the cost and utility resulting from the selection of a critical score by the method of equal likelihood, investigation was made of the cost and utility which would result from combining several critical scores and applying them simultaneously, i.e., as multiple cutting scores. 146 Guilford’s graphic method of determining a critical score.— Guilford demonstrated a graphic method of finding the critical score (63:344-46). The procedure is to use the numerical variable as the horizontal axis and the per cent of successful students on the vertical axis. A point is plotted for the midpoint of each interval of the numerical variable. After the points are located, a continuous curve is drawn which best fits the obtained points. The critical score is determined by where this curve crosses the point on the vertical axis which represents a 50 per cent chance of success. The numerical score directly under this point of intersection is the desired critical score. Table XI shows the per cent of successful students for each interval of the numerical variables, GPA, V score, and S score. Graphs 2, 3, and 4 show the data from Table XI plotted according to Guilford's method. The code numbers representing each interval in Graphs 2, 3, and 4 cam be translated into the actual numerical values by means of Table XI. The process of fitting a curve to the plotted points could only be done with some degree of accuracy in Graph 1, based on the GPA data. Inspection of Table XI shows that only for the GPA values was there some degree of consistency in terms of an increasing percentage of success as the scores get TABLE XI PER CENT OF SUCCESSFUL STUDENTS FOR EACH INTERVAL IN THE DISTRIBUTIONS OF GPA, V SCORE, AND S SCORE (GROUP A) Interval Code No. GPA Per cent successful V score Per cent successful S score Per cent successful 24 3 2 6 -2 8 0 23 323-25 0 22 3 2 0 -2 2 100 21 317-19 100 20 314-16 50 79-80 0 19 311-13 67 77-78 0 18 308-10 100 75-76 100 17 305-07 100 73-74 100 16 3.9-4.0 100 302-04 38 71r72 100 15 3.7-3.8 100 299-301 64 69-70 100 14 3.5-3.6 100 2 9 6 -9 8 83 6 7 -6 8 50 13 3.3-3.4 67 293-95 71 6 5 -6 6 60 12 3.1-3.2 88 290-92 39 63-64 29 11 2.9-3.0 70 2 8 7 -8 9 33 6 1 -6 2 50 10 2.7-2.8 55 284-86 57 59-60 50 9 2.5-2.6 47 281-83 60 57-58 85 8 2.3-2.4 45 278-80 0 55-56 60 7 2.1-2.2 38 275-77 0 53-54 50 6 1.9-2.0 10 2 7 2 -7 4 20 51-52 40 5 1.7-1.8 22 269-71 0 49-50 46 4 1.5-1.6 50 2 6 6 -6 8 0 47-48 0 3 1.3-1.4 0 2 6 3 -6 5 0 45-46 0 2 1.1-1.2 0 2 6 0 -6 2 0 43-44 0 1 0.9-1.0 100 257-59 0 41-42 0 - t r - - 3 Per cent successful 148 100 40 20 10 6 8 5 7 9 10 11 12 Interval Code Numbers GRAPH 2 Cutting Score for GPA Using Method of Equal Likelihood* *See Table XI for numerical values of the interval code numbers. Per cent successful 149 100 90 80 70 60 50 40 30 20 10 0 14 16 10 11 12 Interval Code Numbers GRAPH 3 Cutting Score for V Score Using Method of Equal Likelihood* ♦See Table XI for numerical values of the interval code numbers. Per cent successful 150 100 90 x 80 70 60 50 x-- 40 30 20 10 0 -------------------------------------- 5 6 7 8 9 10 11 12 13 14 Interval Code Numbers GRAPH 4 Cutting Score for S Score Using Method of Equal Likelihood* *See Table XI for numerical values of the Interval code numbers. higher. The critical scores obtained by the graphic method were as folows: GPA critical score is in the interval with midpoint at 2 .5 5 V score is in the interval with midpoint at 291 S score is approximately 5 6.0 These scores are to be compared with those found by the formula in the next section of the findings. Guilford and Michael^ formula for a critical score.— The formula developed by Guilford and Michael for estimating the critical score has been presented in Chapter III.* This formula is closely related to the one for the biserial correlation and requires the same assumption that both the independent and the dependent variables be continuous and normal. Tables for H can be found in Guilford (63:544) and the rest of the variables have been used in the computation of the biserlal correlation. The critical scores obtained by the formula were as follows: GPA =2.48 V score = 292.98 S score - 57-42 In comparing these critical scores to those obtained by *See supra, p. 107. 152 the graphic method the differences did not exceed one Interval unit for any numerical variable. For GPA, the same interval of scores was accepted as successful; below this Interval the probability would be greater for nonsuccess. For V score, the graphic method divided the distribution one interval lower than the formula. For S score, the graphic method divided the distribution one interval higher than the formula. Cost and utility with critical scores derived from the formula.— The critical scores calculated from Guilford and Michael's formula were applied to the frequency distributions for GPA, V score, S score, and T(Read) (Tables C through H in the appendix) to determine the cost and utility for these cutting scores. In addition, several combinations of these scores were applied at the same time in the form of multiple cutting scores. Table XII summarizes the results of these cutting scores for both Group A and Group B. The cost ranged from 25 per cent to 31 per cent for the individual variables on Group A. Costs of this magnitude would appear to be too high for practical use in the junior college situation. When these same cutting scores were applied to Group B, the cost was even higher except for GPA where it remained constant. It was possible to obtain a lower cost by 153 TABLE XII COST AND UTILITY FOR THE METHOD OF EQUAL LIKELIHOOD (GROUPS A AND B) Cutting score Group No. cost Per cent cost Per cent utility No. utility GPA @ 225 A 13 2 5 .0 60.4 29 B 13 2 5 .0 64.6 31 V score @ 293 A 16 30.8 62.5 ^0 B 20 38.5 58.3 28 S score @ 55 A 16 30.8 54.2 26 B 23 44.2 62.5 30 T(Read) @ 57 A 14 26.9 58.3 28 B 25 48.1 60.4 29 GPA @2.5 (or) A 6 11.5 35.4 17 V score @ 293 (or ) B 8 15.4 39.6 19 T(Read) @ 57 GPA @ 2.5 (or) A 7 13.5 33.3 16 V score @ 293 (or ) B 8 15.4 35.4 17 S score @ 55 GPA @ 2.5 (or) A 8 15.4 39.6 19 V score @ 293 B 9 17.3 47.9 23 GPA @ 2.5 (and) A 18 34.6 85.4 4l V score @ 293 B 24 46.2 73.5 35 NOTE: Group B represents the cross-validation data for this method of prediction. 154 accepting anyone who exceeded the critical score on any one of three variables. In the two examples shown in Table XII, a student who was not accepted had less than 50 per cent probability of success in all three of the multiple cutting scores. If utility were the important consideration, without regard to cost, then it would be advantageous to require the student to pass two or more of the critical scores In order to be accepted. The combination of GPA and V score in Table XII yielded 85 per cent utility for Group A. IV. THE NONPARAMETRIC METHOD As described in Chapter III, the nonparametric method Is based on a simple counting of cases which fall below various combinations of cutting scores and calcula tions of the cost and utility on these data. No assumptions are necessary as to the nature of the distributions involved. The cutting scores are deter mined by the primary objective of the screening process which in this case was to keep cost at a minimum and still provide a workable level of utility for the junior college. Each individual cutting score and each multiple cutting score was set in the single or joint distributions just below the point where the rate of increase in cost 155 begins to exceed the rate of increase of utility. The data for GPA and V score were determined by inspection of Table IX, page l4l, the two-way frequency distribution for these variables. The data for the reading scores and the total score on the SCAT were obtained by sorting the 3 x 5 cards described in Chapter III.* Cost and utility with the nonparametric method.— Table XIII summarizes the results of the application of the various cutting scores to the data for both Group A and Group B. It must be kept in mind that the data for Group B are really cross-validation data which will be discussed in a later section of this chapter; that is, the determination of cutting scores on Group A and the application of these scores to Group B. For Group A, the range of cost was from zero to 17.3 per cent with a corresponding range of utility from 12.5 per cent to 60.4 per cent. The optimum cutting score for Group A appeared to be 281 on the verbal score of the SCAT which yielded a utility of 25.0 per cent with a cost of 1 .9 per cent. If one were willing to accept a cost of 11.5 per cent, then the most efficient cutting score would be a multiple one which required the *See supra, pp. IO7-IO8. 156 TABLE XIII COST AND UTILITY FOR THE NONPARAMETRIC METHOD (GROUPS A AND B) No. Per cent Per cent Cutting score Group cost cost utility utility GPA @ 2.5 (or) A 0 0.0 12.5 6 S score @ 49 B 0 0.0 20.8 10 GPA @2.1 (or) A 1 1.9 14.6 7 V score @ 284 B 1 1.9 16.7 8 GPA @2.5 (or) A 2 3.8 20.8 10 V score @ 284 B 2 3.8 2 5 .0 12 *Alpha A 3 5.8 27.1 13 Compensating B 3 5.8 2 9 .2 14 *Beta A 1 1.9 29.2 14 Compensating B 4 7.7 25.0 12 V score @ 281 A 1 1.9 2 5 .0 12 B 4 7.7 2 7 .1 13 GPA @2.1 (or) A 4 7.7 27.1 13 T(Read) @ 57 B 4 7.7 2 9 .2 14 GPA @2.1 (or) A 4 7.7 20.8 10 S score @ 53 B 5 9.6 22.9 11 GPA @2.1 A 6 11.5 39-6 19 B 5 9.6 37.5 18 GPA @ 2.1 (and) A 6 11.5 56.3 27 V score @ 281 (and) B S score @ 47 10 19.2 5 2 .1 25 GPA @2.1 (and) A 6 11.5 54.2 26 V score @ 281 B 8 15.4 5 0 .0 24 GPA @ 2.5 (or) A 6 11.5 39.6 19 T(Read) @ 57 B 11 21.2 43.8 21 TABLE XIII (continued) 157 Cutting score Group No. cost Per cent cost Per cent utility utility GPA @ 2.5 (or) A 5 9.6 29.2 14 S score @ 53 B 10 19.2 33.3 16 GPA @2.3 (and) A 9 17.3 60.4 29 V score @ 28l B 13 2 5 .0 56.3 27 NOTE: Group B represents the cross-validation data for this method of prediction. *See Table XIV, p. 160. 158 student to attain all three of the cutting scores; failure on any one of the scores would mean rejection. This was the combination which required the student to have a 2.1 high school grade point average, to make at least 281 on the V score, and to make at least 47 on the S score. The utility achieved by these scores, on Group A, was 56.3 per cent with the cost at 11.5 per cent. The same combination of GPA and V score without using the third variable, the S score, yielded the same cost with a utility of 54.2 per cent. This finding that the S score did not contribute much to the predictive efficiency was consistent with the evidence that the S score could be dropped from the discriminant equation without any loss of efficiency. The three variable multiple cutting score produced a higher cost on the cross-validation data than did the GPA and V score for the same relative utilities which again supported the finding that two variables were sufficient for prediction with these data. Compensating cutting scores.— The nonparametric method was investigated to see if it could be adapted to the compensation principle utilized by the regression equation and whether compensating cutting scores would be more efficient than the optional or paired type of multiple cutting score. The term compensation is used 159 here to designate the type of formula which will allow a high score in one numerical variable to offset a low score on another variable. Two different sets of compensating scores were established. They were designated as alpha and beta. Both were determined by inspection of Table IX, the two-way frequency distribution for GPA and V score. Both sets of scores started from the same point (2.1 on the GPA and 281 on the V score), but each one Included different cells of the distribution among its acceptable scores. The objective was to maintain as much of the utility of the fixed cutting scores as possible and at the same time to reduce the cost. The alpha set of scores required one additional Interval higher for the V score for each drop of one interval on the GPA. The beta set of scores was an attempt to fit the compensation line to the Group A data in such a way as to achieve an absolute minimum of cost and still retain adequate utility. Table XIV contains the acceptable V score which corresponds to each GPA for both the alpha and beta sets of cutting scores. Cost and utility with compensating: cutting scores.— Table XV shows the cost and utility figures for both sets of compensating cutting scores on Group A. It also contains the cross-validation data for these same 160 TABLE XIV COMPENSATING CUTTING SCORES (Alpha and Beta) GPA Alpha V score Beta V score 3.5-3.6 260-62 3.3-3.4 2 6 3 -6 5 3.1-3.2 2 6 6 -6 8 2.9-3.0 269-71 accept any V score 2.7-2.8 272-74 above this point 2.5-2.6 275-77 281-83 2.3-2.4 2 7 8 -8 0 281-83 2.1-2.2 281-83 281-83 1.9-2.0 284-86 284-86 1.7-1.8 2 8 7 -8 9 284-86 1.5-1.6 2 9 0 -9 2 284-86 1.3-1.4 293-95 2 8 7 -8 9 1.1-1.2 2 9 6 -9 8 2 8 7 -8 9 0.9-1.0 299-301 2 8 7 -8 9 161 TABLE XV COST AND UTILITY FOR TWO COMPENSATING SCORES Group No. cost Per cent cost Per cent utility No. utility Alpha A 3 5.8 27.1 13 B 3 5.8 2 9 .2 14 Beta A 1 1.9 2 9 .2 14 B 4 7.7 2 5 .0 12 NOTE: These data are Included in Table XIII for the purposes of comparison. score3 applied to Group B. The beta set of scores showed the greater efficiency for the Group A data. This was to be expected because the compensating scores were ’ 'fitted” to the Group A data; however, it also showed a slightly higher utility ( 2 9 .2 per cent to 25 per cent) than the optimum V score of 28l for the fixed nonparametric cutting score. Both cutting scores had the same cost (1.9 per cent). The data for Group B showed the alpha set of scores maintained the same cost with a slight rise in utility. The beta compensating scores which were "fit” more closely to Group A did not work as well on Group B, showing a rise in cost and a drop in utility. V. THE MULTIPLE BISERIAL REGRESSION EQUATION In Chapter III, it was demonstrated that the weights used in the discriminant equation were propor tional to those used by Wherry in his multiple biserlal regression equation. It can also be shown that the two formulas for the multiple biserial correlation are algebraic Identities. As an empirical check on these findings, Wherry's regression weights were computed from the biserial correlations and intercorrelations for GPA and V score and compared with the coefficients used in 163 the two variable discriminant equation. In addition, the multiple biserial correlation was calculated from Wherry's regression weights, using his formula; and this value was compared to that already obtained from the two variable discriminant equation. The regression weights.— The regression weights for the multiple biserial regression equation were calculated using the blserial correlations from Table III, page 124, and the intercorrelations from Table VII, page 136. The correlations were carried to five places for greater accuracy. Wherry's system of normal equations has been shown in Chapter III.* Inserting the values for GPA (Xi) and V score (X2), the normal equations become: w{ + W2(.44273) = .52157 (.44273) + W2 = .43528 These equations were solved simultaneously for the values of and W 2 and these values were found to be: w{ = .4 0 9 0 3 2 9 8 W2 = .25418882 . but Wi = where K = — oik y therefore Wx « 1.00420800 and W2 = .0 3 0 3 3 8 3 2. *See supra, p. Ill 164 According to the equations in Chapter III, zWj_ = For p = .520, z = .39844 then (.39844) (1.00420800) = .40011664 « zWx (.39844) (.03033832) = .0 1 2 0 8 8 0 0 = zW2 From the two variable discriminant equation: ax = .40018108 and a2 = .01209028 When W]_ was multiplied by z, it agreed with the a^ value to the fourth decimal place. When W2 was multiplied by z, it agreed with 3.2 to the fifth decimal place. Multiple biserlal correlation.— Wherry's formula for the multiple biserial correlation was presented in Chapter III and is shown below for the two variable equation along with the correlation obtained when using the regression weights found In the preceding section: cRbis = K v/w^Di + W2D2 where K = and = Mip - Mj_q cRbis = * 589149 The multiple biserial correlation found above was identical, to three decimal places, to that found with the two variable discriminant equation. The multiple correlation from the discriminant equation was . 5 8 9 2 8 6, 165 VI. CROSS-VALIDATION ON GROUP B The discriminant equation.— Another two-way frequency distribution was prepared for GPA and V score containing entries for the number of successful and unsuccessful students In each cell for Group B data. The probabilities for each cell were already determined on the basis of the Group A data and are shown in Table VTII, page 139. The five cutting scores selected for cross-validation were 3 2, 33> 40, 46, and 52 chances per hundred of success on the 24 unit criterion. The number and criterion classification of the students who would not be accepted by each cutting score were determined and are shown in Table XVI. Inspection of the cost and utility figures in Table XVI would seem to indicate that the discriminant equation developed on Group A was applicable to the Group B scores. The largest discrepancy in cost was 3.8 percentage units and the largest difference In utility was 4.2 percentage units. The cost and utility curves for Group B followed the trends established for Group A as shown in Graph 1, page 143, I.e., a general rise of cost and utility as the chances for success were increased with the rate of increase being greatest for the utility curve in this range of cutting scores. It was apparent from the inspection of both two-way frequency 166 TABLE XVI THE COST AND UTILITY FOR THE CROSS VALIDATION OF THE DISCRIMINANT EQUATION Cutting score (chances per hundred) Qroup No. Cost Per cent cost Per cent utility No. utility 32 A B 2 2 3.8 3.8 10.4 12.5 i 33 A 3 5.8 10.4 5 B 2 3.8 14.6 7 40 A 6 11.5 31.3 15 B 4 7.7 2 9 .2 14 46 A 8 15.4 5 2 .1 25 B 10 19.2 54.2 26 52 A 13 2 5 .0 6 8 .8 33 B 15 2 8 .8 66.7 32 167 distributions, that as the chance of success, as determined by the discriminant equation, exceeded the rate of success in the total group, the cost curve would rise faster than the utility curve. Formula for a critical score.— The cross- validation data for Group B which resulted from the application of the principle of equal likelihood, by means of Guilford and Michael's formula for a critical score, has already been shown in Table XII, page 153> along with the data for Group A. The relatively large increases in cost for the reading variables in Group B was a result of the fact that the mean scores for these variables in Group B were lower than the mean scores for Group A. This was true for the total group and for the successful group as can be seen in Table III, page 124. The cost remained the same for both groups on GPA and it can be seen in Table III that the mean GPA for both groups was about equal. The cost figures for Group B remained closer to those obtained In Group A when the multiple cutting scores were used giving the student the option of attaining any one of several critical scores as opposed to making a single critical score. This was not true for the GPA where the cost remained constant; however, the multiple cutting scores provided a lower level of 168 coat. The nonparametrlc method.— The data for the cross validation of the nonparametrlc method are contained In Table XIII, page 156, and in Table XV, page l6l, along with the data for Group A. Inspection of these tables demonstrates, again, that the greatest amount of similarity among the cost and utility figures between Groups A and B occurred when the GPA and the V score were involved in the cutting score. Multiple cutting scores of the optional type and the alpha compensating cutting 3core were the optimum forms of combination for GPA and V score in terms of lower costs and stability from one sample to another. For the purpose of comparing efficiency with the other methods of prediction in terms of minimum cost, the optional multiple cutting scores of a 2.5 on the GPA and either 284 on the V score or 49 on the S score were the most efficient. The V score combination had a cost of 3 .8 per cent in both groups and 2 0 .8 per cent utility in Group A and 25.0 per cent utility in Group B. The S score combination had zero cost in both groups and 1 2 .5 per cent utility in Group A and 20.8 per cent utility in Group B. A very close second in terms of efficiency with minimum cost was the combination of GPA at 2.1 or V score at 284 which yielded a constant cost of 1.9 per cent and 169 utilities of 14.6 per cent and 1 6 .7 per cent for Groups A and B respectively. In terms of selecting other cutting scores on the basis of the greatest Increase in utility for the smallest rise in cost the following cutting scores appeared to be most efficient: (l) the alpha compensating cutting score with a cost of 5*8 P©** cent, (2) GPA only at 2.1 for a cost of approximately 10 per cent, ( 3) the multiple cutting score of GPA at 2.1 and also a V score of 281 for a cost of approximately 12-15 P©*1 cent. VII. COMPARISON OP THE EFFICIENCY OF THE DIFFERENT METHODS OF PREDICTION Comparison at minimum cost. 0-6 per cent.— Table XVII contains data for cutting scores obtained by the discriminant equation and by the nonparametrlc method at minimum cost levels which were arbitrarily set at 0 to 6 per cent for the purposes of comparison. Cutting scores obtained by the formula were not included for comparison here because the cost figures were not this low. For these two samples of students, the nonpara- metric method was more efficient at this level of cost because it yielded a greater utility. 170 TABLE XVII A COMPARISON OF THE EFFICIENCY OF THE DISCRIMINANT EQUATION AND THE NONPARAMETRIC METHOD AT MINIMUM COST Cutting score Group No. cost Per cent cost Per cent utility No. utility Nonnarametric GPA & 2.5 (or) A 0 0 12.5 6 S score @ 49 B 0 0 2 0 .8 10 GPA @ 2.5 (or) A 2 3.8 2 0 .8 10 V score @ 284 B 2 3.8 2 5 .0 12 Alpha A 3 3.8 27.1 13 compensating B 3 5.8 29.2 14 Discriminant eauatlon 32 per hundred A 2 3.8 10.4 5 B 2 3.8 12.5 6 33 per hundred A 3 5.8 10.4 5 B 2 3.8 14.6 7 Comparison at approximately 10-12 per cent.— Table XVIII provides the data for a comparison of all three methods of prediction at a level of cost which was approximately 10 to 12 per cent. Both the nonparametrlc and the discriminant equation cutting, scores produced lower costs in Group B than they did in Group A. The formula for the critical score used to provide an optional multiple cutting score on three different numerical variables showed a rise in cost for Group B. For these two samples, the nonparametrlc method showed slightly more efficiency than did the other methods at this level of cost. Comparison at approximately 15-25 per cent cost. Table XIX shows the data for a comparison of efficiency for all three methods of prediction for a level of cost of approximately 15 to 25 per cent. Both the nonpara- metric method and the discriminant equation were more efficient than the method of equal likelihood for these two groups of students at the 15 per cent level of cost. There was not much difference in efficiency between the nonparametrlc method and the discriminant equation in this range of values. What difference did exist seemed to favor the nonparametrlc method at the 15 per cent level and to favor the discriminant equation at the 25 per cent level. 172 TABLE XVIII A COMPARISON OP THE EFFICIENCY OF THE DIFFERENT METHODS OF PREDICTION AT APPROXIMATELY 10-12 PER CENT COST Cutting score Group No. cost Per cent cost Per cent utility No. utility Nonparametrlc GPA @2.1 A 6 11.5 39.6 19 B 5 9.6 37.5 18 Discriminant equation 40 per hundred A 6 11.5 31.3 15 B 4 7.7 2 9 .2 14 Eaual likelihood GPA @ 2.5 (or) V score @ 293 (or) A 6 11.5 35.4 17 T(Read) @57 B 8 15.4 39.6 19 173 TABLE XIX A COMPARISON OF THE EFFICIENCY OF THE DIFFERENT METHODS OF PREDICTION AT APPROXIMATELY 15"25 PER CENT COST _ . . . a-Prtun No« Per cent Per cent No* Cutting score P cost cost utility utility NonDarametric GPA @ 2.1 (and) A 6 11.5 54.2 26 V score @ 281 B 8 15.4 5 0 .0 24 GPA @ 2.3 (and) A 9 17.3 60.4 29 V score @ 281 B 13 2 5.O 56.3 27 Discriminant eauation 46 per hundred A 8 15.4 5 2 .1 25 B 10 19.2 54.2 26 52 per hundred A 13 2 5 .0 6 8 .8 33 B 15 2 8 .8 66.7 32 Eaual likelihood GPA @ 2.5 (or) A 8 15.4 39.6 19 V score @ 293 B 9 17.3 47.9 23 GPA @2.5 A 13 2 5 .0 60.4 29 B 13 2 5 .0 64.6 31 174 Summary.— On the basis of the cross-validation data, some differences did exist among the three methods of prediction in terms of efficiency as defined for this study. The method of equal likelihood did not provide costs below the 10 per cent level. It was not as efficient as either of the other two methods at the 10 to 12 per cent level of cost or at 15 per cent cost, but approximated their efficiency at the 25 per cent level. The nonparametrlc method showed more efficiency than the discriminant equation at the level of minimum cost, but only slightly more efficiency at the 10 to 12 per cent level, and essentially no difference between the nonparametrlc method and the discriminant equation at the 15 to 25 per cent level of cost. The nonparametrlc compensating cutting scores were most useful at the level of minimum cost where they were more efficient than the discriminant equation but showed no advantage over the other nonparametrlc cutting scores except possibly furnishing different levels of cost and utility. No additional data have been presented; however, the investigator did try other compensating formulas but the general result was the same as found for the beta compensating score, I.e., those combinations of scores which worked well on Group A did not maintain the same 175 general level of cost when applied to Group B. VIII. STATISTICAL VERIFICATION OF THE BASIC HYPOTHESIS The hypothesis which was to be tested stated that there was no difference between four methods of predicting success in the junior college. The multiple biserial equation was demonstrated to be equivalent to the discriminant function for a dichotomous criterion such as the one used in this study and, therefore, comparisons were needed for three methods only. The method of equal likelihood did not yield a cost which was low enough to make comparisons of efficiency below the level of 15 to 25 per cent at which point the differences between the relative efficiencies of all three methods were negligible. The critical differences were those obtained between the nonparametrlc method and the discriminant function at the levels of minimum cost and 10 to 12 per cent cost. The obtained differences in efficiency were in favor of the nonparametrlc method at both levels. The problem was the determination of the significance of the obtained differences of these correlated percentages. Guilford ( 63:222-24) explains McNemar's method for obtaining a z ratio from a four-cell 176 contingency table; it is this method which has been used to evaluate the difference in utility for similar costs. Yates's correction was used whenever the expected fre quency in any cell was less than 10 (63:234). There are conflicting statements regarding the limitations of McNemar's method; however, it is based upon a chi-square analysis and Guilford (63:235) has indicated that chi- square can be used with expected frequencies of 2 or more in any cell; although Guilford (63:224) also gives McNemar's original suggestion that his method of determining differences between correlated proportions be restricted to cases in which the sum of the fre quencies in the two cells indicating the discrepancies is 10 or more. The smallest expected frequency at the level of minimum cost was 1. 5 0, and at the level of 1 0 -1 2 per cent cost, it was 5.20. In neither case is the sum of the two cells indicating the discrepancies equal to 1 0. The analysis for the total number of unsuccessful students in Groups A and B (N = 9 6) meets all suggested limitations. Differences at minimum cost. 0-6 per cent.— Table XX contains the number of unsuccessful students who were accepted and rejected by both the nonparametrlc method and the discriminant equation at a cost of 3.3 per 177 TABLE XX A POUR-CELL CONTINGENCY TABLE OP FREQUENCIES OP UNSUCCESSFUL STUDENTS WHO WERE ACCEPTED AND REJECTED BY AN OPTIONAL CUTTING SCORE AND THE DISCRIMINANT EQUATION AT A COST OP 3.8 PER CENT Nonparametrlc (GPA @2.5 or V score @ 284) Discriminant Equation (32 chances per 100) Accepted Rejected Both Rejected 6 /6 4/6 22 Accepted 37/3 6 1 /0 74 Both 85 11 (96) z = 2.04 (Group B only, N = 48) z = 2 .7 8 (Group A+B, N - 96) Note: The frequency on the top of the diagonal (A/ ) represents Group A and the figure on the bottom of the diagonal ( /B) represents the Group B frequency. The marginal totals are for the sum of both groups. 178 cent. Figures are given for Groups A and B and the marginal totals are for both groups. Inspection of the table reveals that the discriminant equation accepted six students in each group (A and B) who were rejected (i.e. utility) by the nonparametrlc method. In Group B, everyone that was rejected by the discriminant equation was also rejected by the nonparametrlc method. This fact accounts for the greater efficiency of the nonparametrlc method. The IT ratio for the obtained differences of efficiency in cross-validation was 2.04 which is significant at the 5 per cent level of confidence. When both groups were combined and the analysis was made on the total of 96 unsuccessful students, the z" ratio was 2 .7 8 which is significant at the 1 per cent level of confidence. Table XXI is a four-cell contingency table of the frequencies of successful students who were accepted and rejected by the same cutting scores used in Table XX. Only two successful students were rejected (i.e. cost) in Group A and In Group B. In each group the two methods agreed In the rejection of one student but did not agree on the rejection of the second student. Differences at the 10-12 per cent level of cost.— Table XXII Is a four-cell contingency table containing the frequencies of the unsuccessful students accepted and TABLE XXI A FOUR-CELL CONTINGENCY TABLE OF FREQUENCIES OF SUCCESSFUL STUDENTS WHO WERE ACCEPTED AND REJECTED BY AN OPTIONAL CUTTING SCORE AND THE DISCRIMINANT EQUATION AT A COST OF 3.8 PER CENT Nonparametrlc Rejected (GPA @2.5 or Accepted V score @ 284) Both Discriminant Equation (32 chances per 100) Accepted Rejected Both 1 /1 1 /1 4 49/49 1 /1 100 100 4 (104) Note: The frequency on the top of the diagonal (A/ ) represents Group A and the figure on the bottom of the diagonal ( /B) represents the Group B frequency. The marginal totals are for the sum of both groups. 180 TABLE XXXI A FOUR-CELL CONTINGENCY TABLE OF FREQUENCIES OF UNSUCCESSFUL STUDENTS WHO WERE ACCEPTED AND REJECTED BY A SINGLE CUTTING SCORE AND THE DISCRIMINANT EQUATION AT A COST OF 10-12 PER CENT Nonparametrlc (GPA & 2.1) Re .lee ted Accepted Both Discriminant Equation (40 chances per 1 0 0) Accepted 6/6 27/28 67 Rejected 13/12 2/2 29 z = 1.06 (Group B only, N « 48) z = 2.00 (Group A + B, N - 96) Both 37 59 (96) Note: The frequency on the top of the diagonal (A/ ) represents Group A and the figure on the bottom of the diagonal ( /B) represents the Group B frequency. The marginal totals are for the sum of both groups. 181 rejected by a cutting score of 2.1 on the GPA and by the discriminant equation with a cutting score set at 40 chances per 100 of success. The cost percentage figures were not equal in this case with the nonpara- metric cost being 9 .6 per cent and the discriminant equation cost being 7.7 per cent. The obtained differ ence in efficiency favored the nonparametrlc method with a higher utility; but the difference is not significant (z = 1. 0 6). When both Groups A and B were combined and the analysis was made on the total of 96 unsuccessful students, the z ratio was 2 .0 0 which is significant at the 5 per cent level of confidence. The percentages of cost were in closer agreement for N = 96 with the nonparametrlc method at 1 0 .6 per cent and the discriminant equation at 9 .6 per cent. Table XXIII Is a four-cell contingency table of the frequencies of successful students who were accepted and rejected by the same cutting scores used in Table XXII. Inspection of the table reveals perfect agreement between the two methods for Group A and agreement on three students to be rejected (i.e. cost) in Group B with the nonparametrlc method rejecting two additional students and the discriminant equation rejecting only one other student, 182 TABLE XXIII A FOUR-CELL CONTINGENCY TABLE OF FREQUENCIES OF SUCCESSFUL STUDENTS WHO WERE ACCEPTED AND REJECTED BY A SINGLE CUTTING SCORE AND THE DISCRIMINANT EQUATION AT A COST OF 10-12 PER CENT Nonparametrlc (GPA @ 2.1) Discriminant Equation (40 chances per 1 0 0) Accepted Rejected Both Rejected 0 /2 6/3 11 Accepted 46/46 0 /1 93 Both 94 10 (104) Note: The frequency on the top of the diagonal (A/ ) represents Group A and the figure on the bottom of the diagonal ( /B) represents the Group B frequency. The marginal totals are for the sum of both groups. 183 IX. GROUP A COMPARED WITH GROUP B It was decided to compute an analysis of variance on the three numerical variables used in the discriminant analysis so that if no significant differences existed between them then the two samples could be combined to provide information on a single sample of 2 0 0 students which might be of value to El Camino College. In addition, a chi-square analysis was made on the number of successful men and women in each group, and the inter correlations of the variables were compared. Analysis of variance.— Table XXIV contains the basic data for an analysis of variance of the grade point averages for Groups A and B. The F value at 1 and 198 degrees of freedom Indicated that there was no difference between the two groups on this variable. Table XXV contains the basic data for an analysis of variance of the verbal score on the SCAT for Groups A and B. The F value at 1 and 198 degrees of freedom indicated that there was no difference between the two groups on this variable. Table XXVI contains the basic data for an analysis of variance of the speed of reading comprehension score of the Cooperative Reading Test for Groups A and B. The F value at 1 and 198 degrees of freedom indicated that 184 TABLE XXIV ANALYSIS OF VARIANCE OF GPA Source of variation Degrees of freedom Sum of squares Mean squares Groups 1 .0 0 1 9 2 .0 0 1 9 2 Within 198 75.74580 .38255 Total 199 75.74772 Fl,1 9 8 = .0 0 5 0 2 TABLE XXV ANALYSIS OF VARIANCE OF V SCORE Source of Degrees of Sum of Mean variation freedom squares square Groups 1 118.48 118.48 Within 198 36,132.927 182.48953 Total 199 36,251.407 pl,198 = .64924 185 TABLE XXVI ANALYSIS OF VARIANCE OF S SCORE Source of variation Degrees of freedom . Sum of squares Mean square Groups 1 124.82 124.82 Within 198 12,611.90 63.69646 Total 199 12,736.72 Fl, 198 = 1 -9 5 9 6 0 186 there was no difference between the two groups on this variable. The results of the analysis of variance for GPA, V score, and S score justify the combining of Groups A and B and the calculation of a new two variable dis criminant equation for the total sample of 2 0 0 students. Intercorrelations between GPA. V score, and S score.— The intercorrelations for the three numerical variables originally used in the three-variable dis criminant equation for Group A are shown in Table XXVII along with the intercorrelations for these same three variables computed from the Group B data. These correlations are all product-moment correla tions and can be converted to values of Fisher's z in order to determine whether or not the obtained differences could be expected to arise from random samples or whether a real difference does exist between the two groups for a given pair of correlations. The result of this analysis showed that a differ ence of the amount obtained between the correlations of GPA and S score could occur 4 times out of 100 different pairs of samples of this size; hence, the null hypothesis that no difference existed between Groups A and B had to be rejected. The obtained differences for the other two intercorrelations were well within the limits of sampling 187 TABLE XXVII A COMPARISON OP THE INTERCORRELATIONS OP THE PREDICTIVE VARIABLES POR GROUPS A AND B Numerical variables Group A Group B GPA and V score .443 .424 GPA and S score .316 *519 V score and S score .750 .661 188 error. Because the intercorrelation of GPA and S score was higher in Group B than In Group A, It would be expected that the S score would be even less effective in terms of multiple prediction for Group B. Chl-sauare analysis.— A chi-square analysis was made of the number of successful men and women In both Group A and Group B, The objective of this analysis was to check on the random sampling procedure and to support the procedure of dividing the total sample of 2 0 0 students into separate samples of men and women. Table XXVIII is a multiple-cell contingency table containing the actual frequencies of successful men and women in each group along with the expected frequencies in each classification. The obtained chi-square value of 3.5^7* with 3 degrees of freedom, indicated that the actual frequencies would be expected to arise from sampling fluctuation. X. THE DISCRIMINANT EQUATION FOR N = 200 On the basis of the findings in the preceding section, Group A data were combined with Group B data to provide a single sample of 200 transfer students. A new discriminant analysis was made of the data for the high school grade point average and the verbal score of the 189 TABLE XXVIII CHI-SQUARE ANALYSIS OP THE NUMBER OP SUCCESSFUL MEN AND WOMEN IN EACH GROUP Sex Group Actual Expected Unsat. Sat. Total Unsat. Sat. Men A 36 30 66 31.68 34.32 Men B 32 31 63 30.24 32.76 Women A 12 22 34 1 6 .3 2 1 7 .6 8 Women B 16 21 37 17.76 19.24 Total 96 104 200 9 6 .0 0 104.00 With 3 degrees of freedom, Chi-square = 3.547 3.665 is at the 30 per cent level of confidence. 190 SCAT. The following were the basic data for this analysis: N = 200 £xx = 509.42 5x2 = 18,750 p = .520 = 1,373.29140 £x§ = 1,794,064. z « .39844 2x^2 = 48,474.05 where: N = the total number of students X^ = the high school grade point average X2 “ the converted V score (less 200 to facilitate computation) Table XXIX shows the differences in mean scores for the successful and the unsuccessful students, on the 24 unit criterion, for their grade point averages and their V scores. The raw scores above were converted to deviation scores and the normal equations were solved for the coefficients of the numerical variables in the discriminant equation. These coefficients were found to be: ax = .49841 and a2 = .0084314. The discriminant equation in raw score form was as follows: V = .49841X-L + .0084314X2 - 3 .6 9 6 0 2 Table XXX shows the relative contribution of each variable in the discriminant equation for N = 200. The multiple biserial correlation for GPA and V score for the total 191 TABLE XXIX MEAN SCORES FOR THE SUCCESSFUL AND UNSUCCESSFUL STUDENTS ON GPA AND V SCORE, N = 200 Mean Difference Numerical Successful Unsuccessful in means variable (k = 104) (k = 96) (a) Nzd GPA (Xx) 2.81086 2.26135 .54951 43.78935 V score (X2) 97.74038 89.42708 8 .3 1 3 3 0 662.47025 TABLE XXX PER CENT OF CONTRIBUTION OF GPA AND V SCORE IN THE DISCRIMINANT EQUATION FOR N = 200 Numerical variable aiNz(ii Per cent contribution GPA 21.82487837 79.6 V score 5.58552914 20.4 Total 27.41040751 100.0 192 group of 200 students was found to be: Rbls - -58201 The relative contribution of GPA at 80 per cent for N = 200 is higher than the contribution of GPA in the two-variable discriminant equation for Group A which was found to be 66 per cent. The increased contribution of GPA, which has the higher biserial correlation with the criterion, would account for the slightly higher multiple biserial correlation found for the total sample of 200 students. The difference in the "weights'1 of each variable can be observed by comparing the two raw score discriminant equations. The a^ coefficient is larger and the a2 coefficient is smaller for the discriminant equation determined above when compared to the two- variable equation for Group A. XI. SEX DIFFERENCES IN PREDICTION Table XXXI contains the mean GPA and V scores for men and for women. The obtained differences were in favor of the women for GPA and in favor of the men for the V score. It was decided to test the significance of these differences by means of an analysis of variance and to compare the blserial correlations obtained with each sex. In addition the mean scores for women and the mean scores for men were inserted In the raw score 193 TABLE XXXI BISERIAL CORRELATIONS FOR MEN AND WOMEN WITH THE 24 UNIT CRITERION Numerical variable Sex Mean successful Mean total S.D. total Biserial r GPA Men 2.65508 2.42534 .60062 .454 Women 3.03186 2.76830 .57928 .716 V score Men 297.27868 294.04651 12.96857 .2 9 6 Women 298.3953^ 293.21126 14.30248 .570 194 formula for the discriminant equation for N = 200. The V scores obtained for the mean scores of the men and women were as follows: V = .15591 (women) p = .5 6 V = -.00798 (men) p = .5 0 These data Indicated that on the basis of the discriminant equation based on the total sample of 200 students, the average woman had 56 chances per hundred of success and the average man had only 50 chances per hundred of success. This finding is a result of the fact that the discriminant equation places more emphasis on the GPA in which the women do have a higher mean score than the men. The small difference in the mean V score, which numeri cally favors the men, is not sufficient to offset the gain on GPA by the women. The above findings along with the findings of the next two sections led to the decision to make a discriminant analysis for each sex and to compare the cost and utility obtained with each discriminant equation. Biserial correlations.— Table XXXI also contains the biserial correlations for men and women with the 24 unit criterion. The obtained correlations were higher for the women on both variables. Inspection of the mean scores shows the mean V score for the successful women was slightly higher than the mean V score for the 195 successful men, although the difference was reversed for the total mean scores for each sex. Also, it can be seen that the total mean score for women on GPA was higher than the mean score for the successful men on the GPA. Walker and Lev discuss some recent findings by R. F. Tate, of the Institute of Mathematical Statistics at the University of North Carolina, In a study made on biserial coefficients ( 1 6 5: 2 6 9-7 0)• One of the outcomes of this study was the development of a logarithmic transformation for the biserial correlation into a function similar to Fisher's function of z for the product-moment correlation. Tate has called this quantity Z* (read "z star") and has shown that with the division of the dichotomous variable approximately at the center and when the biserial correlation is not close to -1 or +1, z* has a normal distribution with a standard error • The biserial correlations in Table XXXI were converted to z* values In order to test the sig nificance of the obtained differences In correlation between men and women. On the basis of the analysis with the z* values, the null hypothesis of no difference between the obtained correlations was rejected at the 5 P©** cent level of confidence for the obtained differences between 196 correlations on GPA (C.R. = 1.98). The obtained differ ences between the correlations for V score were not significant at the 5 per cent level (C.R. = 1. 7 6). Analysis of variance.— Tables XXXII and X X X III contain the basic data for an analysis of variance between men and women on GPA and V score. The F value of 1 5 .1 5 8 with 1 and 198 degrees of freedom obtained for the analysis of GPA was significant at the 1 per cent level of confidence and indicated that the difference between the men and women in terms of their grade point average was a real one and not a result of sampling fluctuation. The F value of .175 with 1 and 198 degrees of freedom did not represent a significant difference and Indicated that the difference in total mean scores for the V score which favored the men could be expected to occur as the result of sampling fluctuation. The discriminant equation for men.— The following were the basic data for the computation of the dis criminant equation for men: 197 TABLE XXXII ANALYSIS OF VARIANCE FOR MEN AND WOMEN ON GPA Source of variation Degrees of freedom Sum of squares Mean square Groups 1 5.38651 5.38651 Within 198 70.36121 .35535 Total 199 75.74772 pl,1 9 8 = * 15.158 ANALYSIS TABLE XXXIII OF VARIANCE FOR MEN AND WOMEN ON V SCORE Source of Degrees of Sum of Mean variation freedom squares square Groups 1 31.9480 31.9480 Within 198 36,219.459 1 8 2 .9 2 6 5 6 Total 199 36,251.407 *1,198 * ^T464 198 N = 129 2*1 = 312.87 2*2 = 12,132. p = .473 £ x f = 8 0 5 .3 5 4 9 £ x | « 1, 162, 668. z = .39803 ^XXX2= 29,826.10 where: N = the number.of men transfer students X i = GPA X2 = V score (less 200 to facilitate computation) Table XXXIV shows the differences in mean scores for the successful and unsuccessful men students on their GPA and V score. After solving the normal equations the coeffi cients were found to be: = .42325 and a2 = .0 0 6 6 7 3 5 The raw score discriminant equation for men was: V = .42325Xx + .0066735X2 - 3 .0 5 6 5 4 Table XXXV shows the relative contribution of each variable in the discriminant equation for men. The multiple biserial correlation for GPA and V score for men was found to be: Rbis = .471 The findings on the discriminant equation for men showed a lower value for the multiple biserial correlation than was obtained on Group A or on the total sample of N = 200. There was a slight rise in the relative per centage contribution of the grade point average over that 199 TABLE XXXIV MEAN SCORES FOR SUCCESSFUL AND UNSUCCESSFUL M EN ON GPA AND V SCORE Mean Difference Numerical Successful Unsuccessful in means variable (k = 6 1) (k = 68) w Nzd GPA (Xx) 2.65508 2.21926 .4 3 5 8 2 22.37756 V score (X2) 9 7 .2 7 8 6 8 91.14705 6 .1 3 1 6 3 314.83388 TABLE XXXV PER CENT OF CONTRIBUTION OF GPA AND V SCORE IN THE DISCRIMINANT EQUATION FOR M EN Numerical variable a^Nzdi Per cent contribution GPA 9.47130809 81.8 V score 2.10103130 18.2 Total 11.57233939 100.0 200 obtained for the total group, 8 1 .8 per cent compared to 79.6 per cent. The difference in means on both variables was smaller than that obtained for the total sample of N = 200. The discriminant equation for women.— The follow ing were the basic data for the computation of the discriminant equation for women: N = 71 2 x x = 1 9 6 .5 5 Z X 2 - 6618. p = .606 = 5 6 7 .9 3 6 5 £ x | = 6 3 1 ,3 9 6 . z = .3 8478 ^ X X X2 = 1 8 ,6 4 7 .9 5 where: N = the number of women transfer students X1 = GPA X2 = V score (less 200 to facilitate computation) Table XXXVI shows the difference in mean scores for the successful and unsuccessful women students on their GPA and V scores. After solving the normal equations the coeffi cients were found to be: a . ± = .6 1 7 9 1 and a2 = .010803 The raw score discriminant equation for women was: V = .61791X-L + .010803X2 - 4.609222 Table XXXVII shows the relative contribution of each variable in the discriminant equation for women. The 201 TABLE XXXVI MEAN SCORES FOR SUCCESSFUL AND UNSUCCESSFUL WOMEN ON GPA AND V SCORES Mean Difference Numerical Successful Unsuccessful In means variable (k = 43) (k * 28) <a) Nzd GPA (X-l ) 3.03186 2.36357 .6 6 8 2 9 18.25727 V score (Xg) 98.39534 85.25000 1 3 .1 4 5 3 4 359.12254 PER CENT TABLE XXXVII OF CONTRIBUTION OF GPA AND V SCORE IN THE DISCRIMINANT EQUATION FOR WOMEN Numerical variable a^Nzd^ Per cent contribution GPA 11.28135 74.4 V score 3.87946 2 5 .6 Total 15.16081 100.0 202 multiple biserial correlation for GPA and V score for women was found to be: Kbls " *^5 The findings on the discriminant equation for women showed a higher value for the multiple blserial correlation than was obtained on any of the other dis criminant analyses. This was a result of the fact that the difference in mean scores between the successful and the unsuccessful women was greater on both variables than any difference previously obtained. The relative percentage of contribution of GPA was lower than that obtained for men or for the total group of N « 200. Only Group A showed a lower contribu tion of the GPA variable in the two-variable discriminant equation. It was the relative size of the differences in GPA and V score which was the determining factor in each case. Cost and utility for men and women.— Separate two- way frequency tables were prepared for the men and the women in the manner described previously in this chapter. As described, the table also shows the criterion classifi cation for each student. Using the raw score discriminant equations for each sex, the chances per hundred for success on the criterion were computed for each cell in 203 the tables. Table XXXVIII contains a summary of the cost and utility data for men and for women for the range of 33 to 52 chances per hundred of success. Inspection of the cost and utility data reveals differences between men and women in terms of efficiency as defined for this study. For all cutting scores the women had a lower cost and the men had a higher utility. For women, it was possible to obtain 28.6 utility without any cost whatsoever. The minimum cost for men was 9.8 per cent, with only 1 1 .8 per cent utility at this point. At a comparable cost, 9 .3 per cent, the women showed a utility of 53.6 per cent. The general trend of the data for the men showed, except for the area of minimum cost, that the utility figure is approximately double that of the cost. Summary.— The analysis of variance showed a significant difference between the men and women on the high school grade point average but no difference on the verbal scores of the SCAT. When the mean scores for men and women on GPA and V score were inserted in the raw score discriminant equation determined on the total sample of N = 200, the results showed the women to have 56 chances per hundred of success and the men to have 50 chances per hundred of success. The actual percentage of successful women was 6 0 .6 per cent; and the percentage of 204 TABLE XXXVIII SUMMARY OP COST AND UTILITY FOR THE DISCRIMINANT EQUATIONS FOR MEN AND WOMEN Cutting No. Per cent Per cent No. score* Sex Cost cost utility utility 33 M 6 9.8 11.8 8 F 0 0.0 10.7 3 34 M 6 9.8 1 3 .2 9 F 0 0.0 10.7 3 35 M 6 9.8 13.2 9 F 0 0.0 10.7 3 36 M 8 13.1 20.6 14 F 0 0.0 14.3 4 37 M 8 13.1 26.5 18 F 0 0.0 17.9 5 38 M 9 14.8 33.8 23 F 0 0.0 21.4 6 39 M 9 14.8 38.2 26 F 0 0.0 21.4 6 40 M 10 16.4 41.2 28 F 0 0.0 21.4 6 41 M 13 21.3 44.1 30 F 0 0.0 28.6 8 42 M 14 22.9 5 0 .0 34 F 1 2.3 28.6 8 43 M 16 26.2 54.4 37 F 1 2.3 28.6 8 44 M 16 2 6 .2 55.9 38 F 1 2.3 35.7 10 45 M 16 2 6 .2 60.3 41 F 1 2.3 35.7 10 205 TABLE XXXVIII (continued) Cutting score* Sex No. Cost Per cent cost Per cent utility No. utility 46 M 18 29.5 64.7 44 F 2 4.7 35.7 10 47 M 18 29.5 64.7 44 F 2 4.7 35.7 10 48 M 19 31.1 64.7 44 F 2 4.7 39.3 11 49 M 21 34.4 67.7 46 F 3 7.0 46.4 13 50 M 24 39.3 69.1 47 F 4 9.3 53.6 15 51 M 26 42.6 6 9 .1 47 F 4 9.3 53.6 15 52 M 27 44.3 76.5 52 F 5 11.6 57.1 16 ♦Chances per 100 of success on the 24 unit criterion determined by the following equations: Men (N = 129), V * .42325XX + .0066735X2 " 3.05654 Women (N = 71), V * .6l791Xx + .010803X2 “ 4.60922 206 successful men was 47.3 pe** cent. Therefore, the single discriminant equation, without regard to sex, was under estimating the chances for women and overestimating the chances for the men. Separate biserial correlations were calculated for each sex for GPA and V score with the 24 unit criterion of success. The women had the higher correla tions on both variables: .72 compared to .45 for GPA, and .57 compared .30 for the V score. The differences in correlations were tested for significance and the null hypothesis of no difference was rejected at the 5 per cent level of confidence for GPA and accepted for the V score. Separate discriminant equations were calculated for the men and women. The multiple biserlal correlation for women was found to be .7 6 and the correlation for men was .47. The relative contribution of GPA to V score was 3 to 1 for women and 4 to 1 for the men. In terms of cost and utility, the discriminant equation for women was found to be considerably more efficient than the equation for the men. The minimum cost for women was zero and did not exceed 10 per cent cost for 50 per cent utility. The minimum cost for men wa3 approximately 10 per cent and ran to 23 per cent for 50 per cent utility. 207 XII. COST AND UTILITY WITH THE NONPARAMETRIC METHOD FOR MEN AND WOMEN Using the two-way frequency tables for men and women on GPA and V score, cost and utility data were obtained for various cutting scores determined by the nonparametric method. Several forms of the cutting scores were tried: (l) A single cutting score refers to a score on one variable only. (2) An optional multiple cutting score refers to a combination of scores on two variables which would accept the student if he reaches the minimum score on either one of them. (3) A pair of cutting scores refers to a combination of scores on two variables which would accept the student only if he exceeds the minimum score on both of them. (4) The compensating cutting scores are of the type previously described as an alpha compensating score, i.e., one higher score interval is required on the V score as each GPA score interval is lowered by one, and vice versa. The compensating cutting scores will be designated by a reference point which will be one of the two-way cells which would be accepted by that series of cutting scores, e.g., compensating score W-l (GPA 2.1, V 28l) refers to the number one compensating score for women which includes a GPA of £.1 and a V score of 281 as one of Its minimum acceptable combinations of scores. 208 Cost and utility for men.— Table XXXIX contains the cost and utility data for men obtained by the nonparametric method. The optional multiple cutting score of GPA at 2.3 and the V score at 284 yielded the minimum cost. If the GPA was raised to 2.9 on this combination, it was possible to obtain 28 per cent utility for 5 per cent cost. The lowest cost figure shown for the discriminant equation was 10 per cent cost. The pair of cutting scores of GPA - 2.1 and V score = 281 gives 56 per cent utility for 20 per cent cost. The optional multiple cutting scores, at the lower levels of cost, were more efficient for the men than were the single cutting scores or the compensating cutting scores. When the single score used was 284 on the V score, the efficiency was about equal to that of the compensating type. The single cutting score with GPA was least efficient at the lower levels of cost; at GPA = 1.7> the cost exceeded the utility. Cost and utility for women.— Table XL contains the cost and utility data for women obtained by the nonparametric method. The compensating cutting score W-l (GPA @2.1, V @ 28l) showed the greatest efficiency at the level of minimum cost. As was found for the men, the single cutting score with the V score was more efficient than the GPA at the lower levels of cost. At a 209 TABLE XXXIX COST AND UTILITY OP NONPARAMETRIC SCORES FOR MEN Cutting score No. cost Per cent cost Per cent utility No. utility GPA @2.3 (or) V score @ 284 2 3.3 20.6 14 GPA @ 2.9 (or) V score @ 284 3 4.9 27.9 19 V score @ 281 4 6.6 2 3 .6 16 GPA & 1.7 4 6.6 5.9 4 V score @ 284 5 8.2 27-9 19 Compensating M-l (GPA & 2.1, V @ 281) 6 9.8 27.9 19 GPA @ 2.1 (and) V score @281 12 19.7 55.9 . 38 210 TABLE XL COST AND UTILITY OP NONPARAMETRIC SCORES FOR WOMEN Cutting score No. cost Per cent cost Per cent utility No. utility Compensating W-l (GPA @ 2.1, V @ 281) 0 0 28.6 8 GPA @ 2.5 (or) V score @ 281 0 0 21.4 6 GPA @2.1 (or) V score @ 281 0 0 14.3 4 V score @ 281 1 2.3 32,1 9 GPA @ 2.1 (and) V score @ 28l 2 4.7 42.9 12 GPA @2.3 2 4.7 28.6 8 Compensating W-2 (GPA @ 2.5, V @ 287) 3 7.0 53.6 15 GPA @ 2.3 (and) V score @ 2 8l 3 7.0 46.4 13 211 level of 7 per cent cost, the compensating cutting score W-2 (GPA @ 2.5, V @ 287) showed slightly more utility than the pair of cutting scores, GPA @ 2.3 and V score @ 281. The pair of cutting scores, GPA @2.1 and V score @ 281 showed 43 per cent utility with a cost of 5 per cent. This was the only Instance of a pair of scores yielding a cost which would be classified as being at the minimum level, which was arbitrarily set a 0 to 6 per cent. The compensating cutting score (W-l), includ ing this same point, showed zero cost; and the optional cutting score at this same point, also yielded zero cost but with less utility than the compensating score. XIIX. COMPARISON OP THE EFFICIENCY OF THE DISCRIMINANT EQUATION AND THE NONPARAMETRIC METHOD FOR MEN AND WOMEN The comparisons to be made below will be based on the data for the discriminant equations for men and women found in Table XXXVIII, page 204, and the data for the nonparametric method to be found in Tables XXXIX and XL. Similar levels of cost for the two methods will be compared in terms of the amount of utility achieved. In addition, the efficiency of the compensating cutting 212 score will be compared to the efficiency of the optional cutting score. Men.— The nonparametric method showed greater efficiency in predicting the 24 unit criterion of success than did the discriminant equation for the sample of 129 men in this study. Table XXXVIII does not show data below 33 chances per hundred of success; however, at 31 chances per hundred of success on the discriminant equation, the cost was found to be 3.3 per cent and the utility 7.4 per cent. This finding is to be compared with the utility of 20.6 per cent at a cost of 3.3 per cent for the nonparametric method. At a level of cost of 10 per cent, the discriminant equation showed a utility of 12 per cent and the nonparametric method a utility of 28 per cent. In order to achieve 56 per cent utility, the discriminant equation equired 26 per cent and the nonparametric method, 20 per cent cost. Women.— The nonparametric method and the dis criminant equation showed very little difference In terms of efficiency as defined in this study for the sample of 71 women. At the zero level of cost, the utility was exactly the same for both methods. At 2,3 per cent cost, the discriminant equation screened out one more unsuc cessful woman than did the nonparametric method. This 213 situation was reversed at the 4.7 per cent level of cost with the nonparametric method screening out one additional woman. At the 7 per cent level of cost, the pair of cutting scores gave the same utility as the discriminant equation and the compensating cutting scores screened out two additional individuals from the sample. Compensating scores versus optional cutting scores.— The findings differed for each sex with regard to the relative efficiency of compensating cutting scores and optional cutting scores. For the sample of women, the compensating cutting scores showed greater efficiency than the use of optional cutting scores at zero cost and at 7 per cent cost. For the sample of men, the optional cutting score yielded 28 per cent utility with 5 pier cent cost; and the compensating cutting score showed 10 per cent cost for the same utility of 28 per cent. XIV. SUMMARY This chapter included the statistical analysis of the basic data obtained on the two groups of transfer students along with the statistical analysis of the total sample of N = 200 and a breakdown of the total sample of 200 students into separate samples of 129 men and 71 women for an additional analysis by sex. The findings were: 1. Fifty-two per cent of the students in both 214 Groups A and B were successful on the 24 unit criterion. In Group A, 37 per cent of those who continued for the second year were successful on the 6 0 unit criterion. Twenty-six per cent of those who continued for the second year in Group B were successful on the 60 unit criterion. Of the total number for both groups who were unsuccessful at the 24 unit criterion, only 3 per cent of those who continued were able to change their classification to successful on the 60 unit criterion. Of the total number of students who were successful on the 24 unit criterion and continued for the second year, 53 per cent retained their successful classification. 2. On the basis of the biserial correlations and inspection of the frequency distributions in terms of cost and utility, the high school grade point average and the verbal score of the SCAT were the two best indicators of success for both criteria. The discriminant analysis showed no significant loss in prediction when the reading score was dropped from the equation and only the GPA and the V score were used. Correlations were higher for the 60 unit criterion than for the 24 unit criterion on Group A but the reverse was true for Group B. 3. For Group A, the multiple correlation for GPA and V score with the 24 unit criterion was .5 7 and the relative contribution of each variable was in the ratio 215 of two to one. The intercorrelation of the predictive variables was .44. The V score was found to correlate .75 with the S score and .82 with the total reading score. 4. The cost and utility curves for the Group A discriminant equation showed the rate of change in utility to Increase faster than the rate of change in cost throughout the range of 32 to 52 chances per hundred of success. The range of cost was 4 to 25 per cent and the range of utility was 10 to 6 9 per cent. 5. The critical scores for the method of equal likelihood derived by means of Guilford and Michael's formula were: GPA * 2.48, V score = 293, and S score - 47 for Group A. The critical scores found by the graphic method were identical for GPA and differed by only several score points for the V and S scores. The cost obtained.for single variables ranged from 25 to 39 per cent. The cost was reduced by using the optional multiple cutting scores which brought the cost down to 12 per cent. 6. For the nonparametric method, the cost ranged from zero to 17 per cent and the utility from 13 to 60 per cent. A pair of cutting scores on GPA and V score provided 54 per cent utility at 12 per cent cost. The addition of the S score to this combination raised the 216 utility to 56 per cent for the same cost. The compensat ing cutting scores yielded a slightly higher utility than the fixed cutting scores at the 2 per cent level of cost on both Groups A and B. 7. The regression weights from Wherry's multiple biserial regression equation were empirically demon strated to be proportional to the coefficients of the discriminant equation. In addition, the same value (.5 6 9) was obtained, using GPA and V score, for the multiple biserial correlations by means of Wherry's weights and formulas and for the discriminant equation. 8. For the cross-validation data, it was found that the discriminant equation computed from Group A showed relatively stable levels of cost and utility when applied to Group B. The method of equal likelihood produced higher costs on Group B than either the dis criminant equation or the nonparametric method. Optional cutting scores and compensating cutting scores showed more stability on Group B than did the single or paired cutting scores. 9. At minimum cost, 0 to 6 per cent, the non parametric method was more efficient than the discriminant equation for both groups. The difference was found to be significant at the 5 P®1* cent level of confidence for the cross-validation on Group B. When Groups A and B were 217 combined, the level of significance reached 1 per cent. The method of equal likelihood did not yield costs low enough to be compared at this level. 10. At 10 to 12 per cent cost, the nonparametric method was only slightly more efficient than the dis criminant equation. The method of equal likelihood produced a rise in cost for the cross-validation compared to a drop in cost for the other two methods. The difference obtained on Group B between the nonparametric method and the discriminant equation could arise by chance 29 times out of a hundred. When Groups A and B were combined, the difference in favor of the nonpara metric method was significant at the 5 P©*1 cent level of confidence. 11. At 15 to 25 per cent cost, there was very little difference between the three methods of prediction. The method of equal likelihood was leabt efficient at the 15 per cent level but was approximately as efficient as the other methods at the 25 per cent level of cost. Because of varying levels of cost within this range between the nonparametric method and the discriminant equation, no precise comparisons of efficiency could be evaluated statistically; however, inspection of Table XIX, page 173, shows essentially no differences between these methods. 218 12. An analysis of variance between Groups A and B showed no significant differences for GPA, V score, or S score. 13. The differences between the intercorrela tions of these same three variables, obtained on each group, were tested for significance. The only signifi cant difference was for the correlation of GPA and S score which was .32 for Group A and .3 2 for Group B. 14. A chi-square analysis of the number of successful men and women in each group revealed no values to be larger than expected by random sampling. 15. With the discriminant equation for N = 200, the multiple biserial correlation was found to be .5 8 and the relative contribution to GPA and V score was 4 to 1. This equation underestimated the chances of success for women and overestimated them for the men. 1 6. The difference in blserial correlations for GPA and V score between men and women were tested for significance. Although the obtained difference favored the women on both variables (GPA, 72 to .45 and V score, 57 to .30)* only the difference for the GPA was significant at the 5 per cent level of confidence. 17. An analysis of variance showed the GPA for women to be significantly higher (at the 1 per cent level) than that of the men; however, no significant 219 difference was found for the V score although the obtained difference favored the men. 1 8. The discriminant equation for men gave a multiple correlation of .47 with the relative contribu tion of GPA and V score being slightly over 4 to 1 (82 to 18 per cent). 19. The discriminant equation for women gave a multiple correlation of .7 5 with the relative contribu tion of GPA and V score being approximately 3 bo 1 (74 to 26 per cent). 20. The discriminant equation for women was more efficient than the equation for men. The women showed 29 per cent utility with no cost. At 10 per cent cost, the women's equation gave 54 per cent utility and the men's equation only 12 per cent utility. 21. With the nonparametric method for men, the optional cutting scores were most efficient at minimum cost; compensating cutting scores showed a higher cost than the optional cutting scores for the same utility. 22. With the nonparametric method for women, the compensating cutting scores were most efficient at minimum cost and at 7 per cent cost. 23. The nonparametric method showed greater efficiency in predicting the 24 unit criterion than did the discriminant equation for the sample of 129 men. 220 24. No differences in efficiency were found between the nonparametric method and the discriminant equation for the 71 women in this study. 25. In comparing the individuals who were selected and rejected by the nonparametric method and the discriminant equation for Groups A and B, greater agreement was found between the two procedures in the rejection of successful students (i.e. cost) than in the rejection of unsuccessful students (i.e. utility). The discriminant equation accepted more unsuccessful students than did the nonparametric method. CHAPTER V SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS I. SUMMARY The purpose of this study was to establish a method of evaluating the selection of students for the junior college transfer program which would be based on the errors of prediction in forecasting the dichotomous criterion of success and nonsuccess in the program. The emphasis was placed on determining the effectiveness of selection as a function of how many potentially unsuc cessful students would be rejected by a given cutting score (defined as "utility1 ' ) accompanied by the rejection of a known frequency of potentially successful students (defined as "cost"). Cost and utility were then used to compare several methods of selecting the successful transfer students. The nature of the problem.— Efficiency was defined in terms of the amount of utility for a given cost. A criterion of success was established as a 2.0 grade point average for 24 units of college work (i.e. transfer credits). The major part of the investigation was concerned with comparing the efficiencies of the following 221 222 four methods of predicting the criterion of success: 1. The discriminant function 2. The principle of equal likelihood 3. A nonparametric method 4. Wherry's multiple blserial regression equation The second important phase of the investigation was concerned with determining the extent and nature of sex differences in predicting the same criterion of success. As a by-product of these two areas of investi gation, data were obtained on the comparison of variables which were available for prediction; on the comparison of several methods of combining scores in the nonparametric method; and some data were obtained on the extent to which the discriminant function and the nonparametric method tended to select the same individuals as being potentially successful or unsuccessful. The basic data.— Two random samples of 100 stu dents each were selected from the students at El Camino College who were making their first attempt at education beyond the high school in the summer or fall semesters of 1956, and who had attempted at least 24 units of college work by June 1957* The permanent record folders of these students contained the basic data for the study and included the following items: 1. The application for admission 2. The high school transcript 3. The converted scores on the School and College Ability Test 4. The standard scores on the Cooperative Reading Test (Form C2Z) 5. The junior college transcript The procedures.— The high school grade point average (academic subjects only) and the sub-scores and total scores of the two tests named above were used as the predictive variables. Blserial correlations were obtained for all the variables with both the 24 unit criterion and a 60 unit criterion of success. Inter correlations were obtained for those variables which demonstrated the greatest amount of predictive efficiency, and the following three variables were chosen for use in the study: 1. The verbal score on the School and College Ability Test 2. The speed of comprehension score on the Cooperative Reading Test 3. The high school grade point average The remainder of the investigation Involved only the 24 unit criterion of success. The first of the two random samples was desig nated as Group A and the data for this group were used 224 to establish various cutting scores with the discriminant function, the principle of equal likelihood, and the nonparametric method. Wherry's multiple biserial regression equation was demonstrated to be the algebraic equivalent of the discriminant function for a dichotomous criterion and therefore was not used in the investigation, as any conclusion reached concerning the one would apply to the other. The second sample of 100 students was designated as Group B and used in the cross-validation process of applying the formulas developed on Group A for a compari son of their efficiency in terms of cost and utility. An analysis was made of the data by means of chi-square and analysis of variance and the two groups were pooled in order to develop a discriminant equation for the total group of 200 students. The total group was reclassified by sex and the predictive efficiency of the men was compared to that of the women. The efficiency of the discriminant function was again compared to that of the nonparametric method on the basis of the new grouping of the data by sex. The findings.— Group A and Group B each had 52 per cent of the students successful on the 24 unit criterion. Of the total number for both groups who were unsuccessful at the 24 unit criterion, only 3 P® *1 cent 225 of those who continued in college work through the second year were able to change their classification to suc cessful on the 60 unit criterion. Of the total number of students who were successful on the 24 unit criterion and continued for the second year at El Camino College, 53 per cent retained their successful classification. The high school grade point average and the verbal score of the School and College Ability Test were found to be the best indicators of success for both the 24 unit and the 60 unit criteria of success. The discriminant analysis showed no significant loss in prediction when the reading score was dropped from the discriminant equation and only the two variables indicated above were used. The multiple correlation was .57 in both cases. It was also found that the same two variables were sufficient for use with the nonparametric method; however, by including the reading score as part of an optional cutting score with the method of equal likelihood, it was possible to obtain a cutting score with a lower cost than if only two variables were used. By plotting cost and utility curves for the Croup A data, it was found that the rate of Increase for utility was greater than that of the cost throughout the range of cutting scores used in the study. For the cross-validation data, the discriminant 226 equation was found to produce relatively stable levels of cost and utility when "applied to Group B. The method of equal likelihood showed greater increases in cost when applied to Group B than either the discriminant equation or the nonparametric method. The optional cutting scores and the compensating cutting scores of the nonparametric method were more stable when applied to Group B than were the single or paired cutting scores. At a minimum cost level which was arbitrarily defined as being from 0 to 6 per cent, the nonparametric method was found to be more efficient than the discrimi nant equation for both groups. The difference in utility for the cross-validation on Group B was significant at the 5 per cent level of confidence. When Groups A and B were combined, the significance of the difference in utility reached the 1 per cent level. The method of equal likelihood did not yield a cost low enough to be compared in this range. At levels of cost ranging from 10 to 12 per cent, the nonparametric method was found to be only slightly more efficient than the discriminant equation. The difference in utility obtained on Group B between these two methods could have arisen by chance 29 times out of one hundred and hence, was not statistically significant. However, when Groups A and B were combined, the difference 227 in favor of the nonparametric method was significant at the 5 per cent level. The method of equal likelihood produced a rise in cost for the cross-validation data in comparison to a drop in cost for the other two methods in this range. At levels of cost ranging from 15 to 25 per cent, no differences in efficiency were found between the discriminant equation and the nonparametric method. The method of equal likelihood was less efficient than the other two methods at the lower end of this range of cost and approximately as efficient as the other methods of prediction at the upper end of this range. Using the two variable discriminant equation for the combined sample of 200 students, a multiple correla tion of .5 8 was obtained with the 24 unit criterion of success. This is consistent with the value of .57 found for Group A. The relative contribution of the high school grade point average and the verbal score to the over-all prediction was in the ratio of 4 to 1 for the combined sample as compared to a ratio of 2 to 1 for Group A. When the mean grade point averages and verbal scores for men and for women were substituted into the discriminant equation based on the total sample of 200 students, it was found that the obtained values overestimated the chances of success for the men and underestimated the chances of success for the women. The blserial correlations for high school grade point average and verbal score with the 24 unit criterion were higher for women (.72 and .57) than for the men (.45 and .30). The difference in the correlations for grade point average was significant at the 5 per cent level, but the difference in the correlations for verbal score was not statistically significant. The women had obtained a significantly higher high school grade point average than the men. The obtained difference in the mean verbal score favored the men but was not statistically significant. The separate two variable discriminant equations for each sex produced a multiple correlation of .47 for the men and .75 for the women. The relative contribution of the high school grade point average to the verbal score was in a ratio of 4 to 1 for the men and 3 to 1 for the women. The discriminant equation for women was considerably more efficient than the equation for men in terms of cost and utility. The nonparametric method also demonstrated much greater efficiency in the prediction of the criterion for women than for men. In a further comparison of the relative efficiencies of the discriminant equation and the nonparametric method, it was found that the latter 229 was the more efficient in predicting success for the sample of 129 men in this study and that there was no difference between the two methods in predicting the success criterion for the J1 women. In the range of minimum cost, the optional cutting scores and the compensating cutting scores were found to be more efficient than the single or paired scores for use with the nonparametric method. In terms of the data for Group A and Group B, there were no differences found between the use of optional and compensating cutting scores. However, when the data were regrouped according to sex, then it was found that the optional cutting scores were more efficient for men and the compensating cutting scores were more efficient for the women. In comparing the nonparametric method and the discriminant equation, at cost levels ranging from 0 to 12 per cent, in terms of the extent of agreement as to which of the same students would be accepted and which rejected, there was found to be substantial agreement on the rejection of successful students (i.e. the cost). The discriminant equation accepted some of the unsuccessful students which the nonparametric method rejected, and therefore was less efficient because it did not reject any additional students who were not also 230 rejected by the nonparametric method. II. CONCLUSIONS The two major conclusions to be drawn from this investigation are that differences do exist among the four methods of prediction which were used, and that sex differences do make a difference in the level of efficiency attained. The detailed conclusions will be organized in terms of the hypothesis and the questions stated in Chapter I. The null hypothesis.— The hypothesis must be rejected that no difference in predictive efficiency exists among the four methods used for the junior college transfer students in this study. The nonparametric method is most efficient at the level of minimum cost (0 to 6 per cent). The method of equal likelihood did not produce a cost figure below 12 per cent, and there fore would place greater restrictions on educational opportunity for some potentially successful students. The discriminant equation and Wherry's multiple biserial equation are equivalent and would yield similar results for this type of prediction problem. The conclusions reached concerning the nonpara metric method and the discriminant equation must be modified in light of the Information obtained concerning 231 the differences between men and women indicated below. For the data not classified by sex, the utility of the nonparametric method is double that of the discriminant equation at 4 per cent cost. For men, at 3 pe** cent cost, the utility of the nonparametric method is three times that of the discriminant equation; for women, however, similar costs produce equal utilities for these two methods of prediction. Sex differences in prediction.— In terms of the obtained correlation figures and in terms of the cost and utility data, the junior college performance of the women students can be predicted with greater accuracy than can the performance of the men. The performance of the women Is more closely related to their performance in high school than is the case for the men. An Inspection of the scatter diagrams for men and women revealed that the women also have greater agreement between their verbal scores and their high school grade point average. The scatter diagram for men also revealed that a substantial number of those who made adequate verbal scores did not produce academically in high school and also did not produce In junior college. The Increased efficiency of the discriminant equation when used with the women would seem to indicate that the basic assumptions of this - method were being met by the data for women, whereas the 232 data for men did not adequately meet the assumptions. The magnitude of the differences in prediction between men and women can best be illustrated by citing actual figures. On the basis of the obtained data, three potentially successful men would be rejected per hundred students for every 21 potentially unsuccessful men. Twenty-nine out of a hundred potentially unsuccess ful women would be eliminated without any cost whatsoever. The best variables for prediction.— The best single indicator of success for these junior college students is their high school grade point average. This is in agreement with the usual findings reported in the literature although most of the studies reported have been done with four-year college students. The average of the grade points is a more efficient predictor than the sum of the grade points, which would seem to indicate that the amount of academic work in high school is not as critical as the quality of the work done in the academic courses which are completed. The verbal score on the School and College Ability Test is the second most efficient predictor of the success criteria. The verbal score along with the grade point average provides a sufficient number of variables for use with the discriminant equation and the nonparametric multiple cutting scores. As would be 233 expected from the zero order correlations, the grade point average contributes considerably more to the multiple correlation than does the verbal score. A 2.0 grade point average for the first year of junior college work is a necessary but not sufficient condition for the prediction of success with the 6 0 unit criterion. Only one-half of those students meeting the 24 unit criterion can be expected to have a 2 . 0 average for 60 units of work. This estimate does not include those who transfer with 24 units of work nor does it include those who drop out of school during the second year. The different types of multiple cutting scores.— The original design of the investigation called for an analysis of the various types of cutting scores in terms of three dimensional space; however, with the finding that only two variables were necessary for prediction, it was necessary to limit the analysis to a two-way scatter diagram with the tallies Indicating the classifi cation on the criterion. It is helpful to think of the scatter diagram in terms of four quadrants with the axes through the median of each distribution. The efficiency of the different methods is related to the sex of the students Involved and hence to the degree of correlation between the variables being used 234 and also to the degree of correlation between the variables and the criterion. When the correlations are relatively high (as in the case of the women), then the compensating scores are the most efficient. When the correlations are lower (as for the men), then the optional cutting scores perform best. In general, the optional and compensating cutting scores are more efficient than the single or paired cutting scores when the objective is to keep the cost at a minimum, because the former are rejecting students from the lower left quadrant in which is found the greatest percentage of unsuccessful students, i.e., those students who have low grades and low verbal scores. The single cutting score in this type of scatter diagram would be rejecting students from two quadrants and a pair of cutting scores rejects students from three quadrants. It appears logical to assume that the con clusions reached for two dimensions would also apply to a three variable problem. The single index versus the two-way index of efficiency.— In predicting a dichotomous criterion, two types of error are inevitable, and a meaningful index must indicate the degree of both types. With a given correlation, different cutting scores will yield different ratios of cost and utility and therefore the correlation 235 index does not provide sufficient information upon which to base a decision. Even if the number rejected is expressed a3 a ratio of utility to cost, such as 5 to 1, there is still no Indication of the magnitude of cost involved. Both cost and utility, or some equivalent terminology, are necessary in order to adequately evaluate any selection procedure for a program which is based on providing maximum opportunity for each student. Comparison of Individuals accepted and rejected.— One of the questions to be answered by the investigation was the extent to which the different methods of selection tended to identify the same students as being potentially successful or unsuccessful. The discussion above con cerning the quadrants from which the different types of cutting scores select students is relevant to this question also. At the lower levels of cost, the discriminant equation and the nonparametric method tended to agree on the individuals who were rejected, but the discriminant equation accepted more of the unsuccessful students. This conclusion is based on the data for Groups A and B and needs to be modified when separate equations are set up for each sex. For the men, the discriminant equation tends to reject more students with low grades and average verbal 236 scores than does the optional cutting score, and some of these students do meet the criterion of success in junior college, thereby raising the cost figures of this method. For the women, the discriminant equation and the compen sating cutting score tend to accept and reject the same individuals. The advantage of these two methods over the optional cutting scores for women is their ability to reject individuals with high grade point averages and low verbal scores. None of the men in this study were in this classification. III. RECOMMENDATIONS The recommendations to be made on the basis of the foregoing conclusions appear to group themselves into two major categories. While the following recom mendations in both categories are directed to the junior college program, the first concerns the direct applica tion of the findings and the second group is made up of suggestions for additional research in order to provide answers to questions suggested by the present study. For the junior colleges.— The following recom mendations are made on the basis of the findings of this study. 1. If selection procedures are, or become, necessary in the junior college transfer program, the 237 data on cost and utility should be otained from an adequate sample of students who have been given the opportunity to attempt the program under investigation with a minimum of initial restrictions in order that a satisfactory evaluation of the efficiency of the selection process can be made. 2. Consideration should be given to the use of some form of nonparametric selection procedure in order to be certain that there is no loss of efficiency resulting from the data not adequately meeting the necessary assumptions of the multiple regression or discriminant equations. 3. If selection is being made in order to determine who should be allowed to attempt the transfer program in the junior college, separate cutting scores should be established for each sex. 4. The sex differences in prediction have implications for the counseling program even if selection is not a factor. Separate expectancy tables for each sex should be prepared which would allow each student to evaluate his own chances of meeting some criterion of success. For additional research.— The following recom mendations are made as a result of questions suggested by this study: 238 1. In order to more adequately determine the level of cost which can be tolerated in the selection of transfer students, information must be obtained on how efficiently the potentially successful students can be identified in a pre-transfer program and on the length of time involved in the identification. Information is also needed concerning the content of such a program. 2. An investigation similar in nature to the present one needs to be conducted with a 6 0 unit criterion of success. During the period of the completion of this study, the state colleges have compiled data which are consistent with the present finding that the 24 unit criterion is not a sufficient condition upon which to predict success if the student does not do well in his high school work. As of the fall semester of 1961, junior college transfer students who desire to attend the state colleges and who were not eligible to enter as freshmen must accumulate 6 0 units of college work with a 2 .0 average. 3. Along with the previous recommendation, each junior college should be provided with a continuous feedback of Information concerning their transfer stu dents who graduate from the state colleges in order to establish a criterion of success involving completion of the four-year degree program. 239 4. The question of why the women students are more predictable than the men needs to be answered, and also why they have better high school records. One hypothesis would be that the women tend to conform to the demands of the school situation to a greater extent, and another hypothesis would be that a greater degree of selection has already taken place prior to entrance in junior college work because of the greater oppor tunities for young women to go to work immediately upon high school graduation. 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APPENDIX 258 TABLE A SEX, MAJOR SUBJECT, AND TRANSFER OBJECTIVE FOR EACH STUDENT IN GROUPS A AND B Alpha No. (Fall 1956) Sex Major subject College or university GROUP A: 1890 M Preforestry 3120 M Mathematics U.C.L.A. 4030 M U.S.C. or U.C.L.A. 4770 M Bus. Admin. U.S.C. or Pepperdine 5390 7180 M M Engineering U.C.L.A. 9190 M Business 9440 M Mathematics U.C.B. 10270 13490 M M Bus. Admin U.C.L.A. 15450 M Business U.S.C. 19270 M Bus. Admin. U.S.C. 19600 M Music B.Y.U. 21540 M Engineering U.C.L.A. 21570 F Elem. Educ. U.C.L.A* 21680 M Engineering U.C.L.A. 24140 F Elem. Educ. U.C.L.A. 25180 M Bus. Admin. U.S.C. or U.C.L.A. 26530 F Art U.C.L.A. 26780 M Chemistry U.C.L.A. 28460 M Engineering U.C.L.A. 28670 F Elem. Educ. L.B.S. 28940 F Non-transfer 29560 M Art U.C.L.A. 33990 35230 M F A.B. 35750 F Pharmacy U.S.C. 35840 M Phys. Educ. Pepperdine 35990 M Mathematics U.C.L.A. 37230 F Psychology Stanford 37270 37840 F Elem. Educ. L.B.S. M Bus. Admin. San Jo3e 39550 M Business San Diego 40960 M Bus. Admin. U.S.C. 41730 M Engineering U.C.L.A. 42880 M Phys. Educ. L.B.S. 45290 M Elem. Educ. L.B.S. 46260 F Elem. Educ. L.B.S. 53520 M Elem. Educ. L.B.S. 53690 M Biology 259 TABLE A (continued) Alpha No. (Fall 1956) Sex Major subject College or university 538 3 0 M Engineering U.C.L.A. or U.S.C 54780 M Th. Arts 55340 M Pre-med U.C.L.A. 55620 M Bus. Admin U.C.L.A. 56360 F Home Econ. Santa Barbara 56400 F Elem. Educ. L.B.S. 57820 F Music U.C.L.A. 5 81 7 0 F Business L.B.S. 58550 F Elem. Educ. U.S.C. 61230 M P.E. L.B.S. 61420 F Non-transfer 6 2 0 0 0 M Bus. Admin. U.C.L.A. 62230 M Science L.B.S. 6 29 9 0 M Engineering U.S.C. 63020 M Chemistry U.C.L.A. 65090 M Mathematics U.C.L.A. 6 85 2 0 F History U.C.L.A. 69130 M History L.B.S. - 71040 M Music U.S.C. 350 M Geology 780 M Phys. Educ. U.C.L.A. 1400 M Engineering U.C.L.A. 4640 M Engineering U.C.L.A. 7790 M Non-transfer 8770 F Bus. Admin. U.C.L.A. 11020 M Bus. Admin. U.S.C. 12010 F Phys. Educ. U.C.L.A. 14210 F Lib. Arts. 14660 M Bus. Admin. L* B. S • 14940 F Elem. Educ. L.B.S. 16180 F Art U.C.L.A. 18280 M Bus. Admin. U.C.L.A. 22450 M Law Loyola 24050 F Elem. Educ. Santa Barbara 27440 M Arch. U.S.C. 28100 M A.B. 32450 M Engineering U.C.L.A. 32550 F Art U. of Wash. 33630 F Life Science U.C.L.A. 35020 M Engineering U.C.B. 42090 F Non-transfer 43820 M Engineering U.S.C. 46360 F Elem. Educ. L.B.S. 48720 F U.S.C. TABLE A (Continued) 260 Alpha No. (Pall 1956) Sex Major subject College or university 7 4 1 0 0 P Elem. Educ. U.S.C. 5470 P English U.C.L.A. 6 5 7 0 M Bus. Admin. L.B.S• 11970 M Cinema U.S.C. 26410 M Engineering U.C.B. 2 7 6 5 0 P A.B. 28130 M Bus. Admin. U.S.C. or U.C 29570 M Bus. Admin. Loyola 34540 M Engineering U.C.L.A. 39790 P Dent. Hyg. U.S.C. 40680 M Pre.Med U.C.L.A. 45000 M Pre-Law U. of Ariz. 49140 M Engineering U.C.L.A. 51490 P Elem. Educ. 52660 M Mathematics U.C.L.A. 71020 P Psychology GROUP B: 63670 P Elem. Educ. U.C.B. 1460 M Chemistry 2370 F Educ. S.D.S. 2640 F Non-transfer 2940 M Phys. Educ. L.B.S. 4240 F A.B. 4550 M Engineering U.S.C. 4650 P Home Econ. L.B.S. 5150 M Bus. Admin. Utah U. tS S F Elem. Educ. Redlands M Engineering 6930 M Pre-Med 7640 F Elem. Educ. L.B.S. 7880 P Non-transfer 8430 P Non-transfer 9030 M Engineering U.C.L.A. 9080 P Apparel Design U.C.L.A. 9240 M Bus. Admin. U.S.C. 103 1 0 P Dental Hyg. U.S.C. 11160 M Bus. Admin. U.C.L.A. 14920 M Music 15430 F Bus. Admin. U.S.C. 18000 M Vet. U.C. at Davis 18500 M Engineering U.C.L.A. 19920 P Nursing U.C.L.A. 20040 M Pre-Dental 261 TABLE A (Continued) Alpha No. (Fall 1956) Sex Major subject College or university 2 0 1 8 0 M Pre-Dental U.S.C. 2 0 3 4 0 M Bus. Admin. 22100 F Educ. 23250 F Elem. Educ. L.B.S. 23410 M Engineering U.C.L.A. 24810 M Language U.C.L.A. 24830 M Physics U.C.B. 2 5 1 2 0 M Engineering 2 5 2 2 0 M Engineering U.C.L.A. 2 5 7 1 0 F Phys. Educ. U.S.C. 26040 F Nursing 2 7 3 8 0 M Engineering U.S.C. 29040 M History L.B.S. 29170 M Engineering 29460 M Bus. Admin. U.C.L.A. 3 0 0 2 0 M Soc. Science • U.C.L.A. 33230 M Engineering 33640 F Psychology U.C.L.A. 33910 M Engineering U.C.L.A. 3 5 2 1 0 F Psychology U.C.L.A. 35280 M Bus. Admin. 35710 F Non-transfer 36180 F Elem. Educ. L.B.S. 36220 M Engineering U.C.L.A. 36640 M Astronomy U.C.B. 37280 F A.B. 37900 M Pre-Dental U.S.C. 38700 F Dental Hyg. U.S.C. 39040 F Language Indiana U. 39060 M Geology 39670 M Engineering 40330 M A.B. 40760 M History U.C.L.A. 41130 M Theatre Arts 42220 M Engineering U.C•L.A. 42720 F Elem. Educ. 42930 M A.B. 43580 44 3 6 0 M M Speech Pre-Med. 45 4 0 0 M Engineering U.C.L.A. 46070 M Engineering 46540 M Pre-Dental U.S.C. 47250 F Elem. Educ. L.B.S. 47840 M Phys. Educ. U.C.L.A. TABLE A (Continued) 262 Alpha No. (Pall 1956) Sex Major subject College or university 50480 M Bus. Admin. U.C.L.A. 50550 M Bus. Admin. U.C.L.A. 51580 M Phys. Educ. U.S.C. 51790 F Bus. Admin. U.C.L.A. 52070 M Bus. Admin. U.C.L.A. 53030 M Forestry Montana St 54180 P Elem. Educ. 54560 M Art U.C.L.A. 55150 M Engineering U.C.L.A. 57060 M Engineering 57450 M Engineering U.C.L.A. 57570 M Engineering U.C.L.A. 57 6 1 0 F Elem. Educ. L.B.S. 60180 M Elem. Educ. U.C.L.A. 60800 M Mathematics U.C.L.A. 61560 M Pre-Med. U.S.C. 6 20 5 0 F Pre-Med. 62330 62790 F F Art 6 3 2 9 0 F U.C.L.A. 63340 M Pre “Pham. U.S.C. 6 61 5 0 F Art U.C.L.A. 66580 F Elem. Educ. L.B.S. 67420 F A.B. 72220 F Educ. 7 2 3 1 0 M Bus. Admin. U.C.L.A. 72990 M Engineering 75 2 1 0 M Pre-Law 75410 M A.B. U.C.L.A. 76810 M Engineering U.C.L.A. TABLE B TEST SCORES, HIGH SCHOOL GRADE POINT AVERAGE, AND CRITERION CLASSIFICATION FOR EACH STUDENT IN GROUPS A AND B Criterion classification 24 units 60 units ~ SCAT Coop. Reading Test Less 2.0 Less 2.0 Alpha No. fConverted score) (Scaled score) H.S. than or than or (Fall 1956) Verbal Quant. Total Vocab. Speed Level Total GPA 2.0 better 210 GROUP A: 1890 281 286 284 47 46 51 48 1.79 X X 3120 300 308 303 59 61 64 62 3.42 X X 4030 257 310 287 43 47 53 47 1.96 X 4770 273 292 284 50 49 55 51 2.48 X 5390 296 327 308 61 63 68 65 . 1.88 X X 7180 301 289 296 58 59 68 62 .85 X X 9190 269 281 276 46 49 50 48 1 .7 6 X X 9440 290 324 304 54 52 54 53 3.86 X X 10270 273 253 264 48 41 48 45 1.67 X 13490 302 291 297 61 49 50 53 1.74 X 15450 290 296 292 54 54 59 56 1.88 X X 19270 286 305 294 53 50 52 52 2 .1 8 X X 19600 290 297 293 55 56 70 61 2 .3 6 X 21540 273 292 284 48 59 58 52 1.68 X X 21570 295 313 303 52 66 66 62 3.16 X 21680 287 299 292 58 49 55 54 1.75 X 24140 302 268 289 58 59 57 58 2 .3 8 X 25180 269 299 285 50 46 50 48 2 .0 8 X X 26530 283 289 286 49 51 53 51 2.46 X X 2 6 7 8O 291 319 303 48 58 59 55 2 .5 0 X X 28460 300 299 299 61 59 60 60 2.55 X X 28670 296 274 287 58 57 62 59 2.40 X X 28940 290 289 289 51 57 58 54 3.00 X TABLE B (Continued) Criterion classification 24 units 60 units SCAT Coop. Reading Test Less 2 .0 Less 2 .0 Aloha No. (Converted score) (Scaled score) H.S. than or than or (Fall 1956) Verbal Quant, Total Vocab. Speed Level Total QPA 2, 0 .better 2 .0 better 29560 284 278 281 57 51 54 54 1.14 X X 33990 291 304 296 57 51 59 56 2 .1 2 X 35230 298 286 293 56 57 55 57 2.80 X 35750 273 297 286 50 51 59 53 2.63 X X 35840 281 311 295 61 59 57 59 1.91 X X 35990 309 311 310 61 58 72 64 3 .8 2 X X 37230 312 297 305 63 73 66 68 3.56 X X 37270 287 284 286 48 46 51 48 2 .8 0 X X 37840 312 321 316 66 74 66 69 2 .1 8 X X 39550 307 302 304 91 66 62 74 1.95 X X 40960 302 297 300 61 54 58 58 2.42 X X 41730 279 316 296 83 54 59 66 1.90 X 42880 284 308 295 55 50 53 53 2 .0 0 X 45290 286 264 276 53 52 58 54 1.87 X X 46260 319 291 306 72 61 70 68 3.39 X 53520 308 299 304 52 59 68 60 2.91 X X 53690 300 286 294 55 55 57 56 2.13 X X 53830 54780 300 330 311 58 70 66 65 2 .6 6 X 298 310 303 52 61 64 59 3.34 X 55340 308 336 319 68 62 57 63 3.04 X X 55620 303 313 307 63 66 64 65 2 .7 0 X X 56360 287 288 287 56 61 62 60 2 .5 8 X X 56400 286 304 294 55 61 62 60 2.84 X X 57820 281 288 284 51 54 55 53 2.22 X X 58170 308 317 312 68 54 57 60 4.00 X X ro 58550 295 283 290 50 55 59 55 3.14 X X o\ 4^ TABLE B (Continued) Criterion classification SCAT Coop. Reading Test Less 2.0 Less 2.0 Alpha No. (Converted score) (Scaled score H.S. than or than or (Fall 1956) Verbal Quant. Total Vocab. Speed Level Total GPA 2.0 better 2.0 better 61230 292 297 294 55 56 57 56 2.95 X X 61420 275 289 283 47 51 58 52 2.27 X 62000 302 299 301 65 63 57 62 2 .2 8 X 62230 277 305 291 50 52 51 51 2 .6 2 X X 62990 287 299 292 45 57 59 54 1.87 X X 63020 288 308 297 52 49 57 53 1.53 X X 65090 288 308 297 50 52 54 52 2.84 X X 68520 320 268 298 66 75 72 72 2.57 X X 69130 71040 303 319 310 64 66 68 67 3.13 X X 291 313 300 53 54 55 54 2.82 X X 350 312 292 303 65 62 68 66 1 .8 0 X 780 294 313 302 58 53 49 53 2.52 X X 1400 298 307 301 61 55 66 61 2 .4 5 X X 4640 320 316 319 69 58 64 64 2.57 X 7790 290 294 291 56 54 60 57 1.75 X X 8770 295 323 306 58 56 57 57 2 .6 0 X 11020 312 311 311 66 63 55 62 3.21 X X 12010 293 291 292 55 63 62 60 2 .5 6 X X 14210 302 310 305 58 57 57 58 3.21 X X 14660 290 278 284 55 53 53 54 1.84 X 14940 319 311 315 85 58 66 70 4.00 X X 16180 291 283 287 56 56 66 60 2.48 X X 18280 299 296 297 62 67 66 66 2.55 X X 22450 326 333 330 68 79 75 75 2.41 X ro 24050 296 268 285 61 56 59 59 3.00 X v 0 X V J l 27440 294 302 297 61 57 62 60 2.88 X TABLE B (Continued) Criterion classification 24 units 60 units SCAT Coop. Reading Test Less 2.0 Less 2.0 ilpha No. (Converted score) (Scaled score H.S. than or than or Pall 1956) Verbal Quant. Total Vocab. Speed Level Total GPA 2.0 better 2.0 better 28100 273 284 280 51 47 50 49 2.10 X X 32450 313 330 321 62 64 72 67 3.14 X 32550 292 2 66 281 52 54 64 57 2 .6 0 X 33630 305 313 308 68 58 64 64 3.54 X X 35020 316 319 318 56 67 72 66 2.73 X X 42090 286 286 286 47 49 57 51 2.00 X 43820 300 336 313 58 63 85 69 2.50 X 46360 287 289 288 49 53 52 51 2.75 X X 48720 275 284 280 47 46 50 47 2.87 X X 74100 301 308 304 62 50 58 57 3.22 X X 5470 313 302 308 63 59 64 63 3.90 X X 6570 290 313 299 56 68 61 2.87 X X 11970 301 321 309 61 56 58 59 2.27 X 26410 303 324 312 70 63 85 74 1.47 X 27650 287 262 276 53 50 58 54 3.23 X X 28130 275 291 284 5 0 46 49 48 1 .6 1 X X 29570 34540 284 286 285 55 5? 50 52 1.55 X X 294 299 296 95 54 51 68 2.17 X X 39790 301 288 296 61 55 62 60 2.68 X X 40680 308 327 316 61 72 62 66 2.88 X X 45000 320 323 323 73 66 72 71 2.75 X 49140 300 323 309 68 57 59 62 3.64 X X 51490 281 300 290 50 51 57 53 3.00 X 52660 314 336 324 69 74 70 72 3.95 X x M 71020 287 294 290 48 61 58 56 2.43 X ro TABLE B (Continued) SCAT Coop. Reading Test Criterion classification 24 units 60 units H.S. Less than 2.0 or Less than (Fall 1956) Verbal Quant. Total Vocab. Speed Level Total GPA 2.0 better 2.0 GROUP B: 6367O 300 291 296, 51 61 55 56 3.13 X 1460 277 297 287 52 47 53 51 2.36 X X 2370 299 289 295 57 57 58 58 2.86 X X 2640 292 281 287 57 49 55 54 2.00 X X 2940 277 294 286 42 41 48 43 2.66 X X 4240 279 307 292 46 55 51 ■ 51 2.00 X 4550 303 316 308 61 63 68 65 - 2.88 X X 4650 300 299 299 52 53 62 56 2.72 X 5150 273 258 266 54 45 50 50 1.69 X X 5870 309 300 305 70 59. 62 64 2.74 X X 6140 291 317 302 56 54 58 56 2.86 X 6930 294 296 295 57 47 53 52 1.79 X X 7640 290 308 297 49 56 66 57 2.81 X 7 8 8O 275 283 280 43 52 .49 48 1.39 X 8430 284 291 287 54 55 54 54 3.00 X 9030 292 314 301 61 59 60 60 3.25 X X 9080 263 304 ' 286 43 43 49 44 1.93 X 9240 291 284 288 52 48 54 51 1.41 X X 10310 284 281 283 50 43 49 47 2.92 X 11160 301 316 307 58 57 60 59 2 .5 0 X X 14920 287 281 284 51 52 58 54 2.23 X X 15430 296 302 299 61 49 57 56 2.44 X 18000 290 310 297 50 58 54 54 2.14 X X 18500 283 297 289 52 59 60 57 3.54 X X 19920 291 297 294 50 59 59 56 2.64 X X 2.0 or better X X X r o ( J \ —3 TABLE B (Continued) Criterion classification 24 units 50 units SCAT Coop. Reading Test Less 2 .0 Less 2 .0 llpha No. (Converted score) (Scaled score H.S. than or than or k Fall 1956) Verbal Quant. Total Vocab. Speed Level Total QPA 2 .0 better 2 .0 better 20040 308 324 315 72 64 72 70 2. 7O X X 20180 295 314 303 52 49 55 52 3.40 X X 20340 271 281 277 51 47 51 50 1.77 X X 22100 287 268 279 54 54 58 55 3.00 X X 23250 251 279 269 45 46 50 47 2.25 X 23410 271 323 296 95 49 57 68 2.41 X X 24810 316 330 323 69 83 70 75 3.46 X X 24830 317 324 322 68 59 59 63 3.37 X X 25120 296 299 297 61 54' 58 58 1.72 X 25220 306 300 303 70 72 62 69 2 .5 0 X X 25710 273 2 66 271 46 38 46 42 1.05 X •X 26040 288 296 291 51 49 50 50 2.40 X X 27380 294 308 300 59 55 58 58 2.77 X X 29040 305 279 294 66 58 54 60 2.95 X X 29170 284 294 289 53 50 48 50 2.31 X X 29460 303 327 313 58 53 57 56 2.13 X X 30020 269 297 284 51 59 52 54 2.21 X X 33230 288 319 301 52 58 62 58 2.00 X X 33640 316 305 311 83 63 70 73 3.67 X X 33910 306 321 312 72 75 67 72 2.93 X X 35210 328 304 315 89 98 73 90 3 .8 0 X X 35280 294 314 303 77 54 62 65 2.64 X X 35710 300 313 305 57 57 60 58 2 .5 2 X X 36180 281 286 283 45 54 57 52 3 .0 8 X 36220 288 316 300 51 49 55 52 1.79 X X I N 36640 317 324 322 71 77 70 74 3.21 X X c c TABLE B (Continued) Criterion classification 24 units 60 units SCAT Coop. Reading Test H.S. Less than 2.0 or Less than (Fall 1956) Verbal Quant. Total Vocab. Speed Level Total GPA 2 .0 better 2 .0 37280 313 283 300 71 63 66 67 3.36 X 37900 292 300 296 51 61 55 56 1 .8 1 X X 38700 291 289 290 58 50 58 53 2.29 X 39040 294 284 290 58 52 55 55 3 .6 0 X 39060 303 294 299 59 54 57 57 2 .0 8 X X 39670 283 333 303 44 53 62 53 2.04 X X 40330 312 300 306 61 57 57 59 2.41 X 40760 298 314 304 53 54 55 54 2.57 X 41130 294 276 286 57 46 50 51 2 .5 6 X X 42220 298 314 304 53 49 57 53 2 .7 2 X X 42720 287 264 277 50 54 54 51 2.22 X X 42930 295 300 297 45 56 53 51 2.77 X X 43580 305 305 304 61 59 57 59 2.54 X 44360 291 321 303 58 57 62 59 3.20 X X 45400 299 319 307 66 59 60 62 3 .2 6 X X 46070 291 286 289 52 59 60 57 2.29 X X 46540 291 297 294 53 45 50 49 2.68 X X 47250 294 284 290 51 58 59 56 2.42 X 47840 294 279 288 58 51 57 55 2.00 X X 50480 273 294 284 47 49 55 50 1 .8 1 X 50550 271 310 291 52 50 55 52 3.55 X 51580 291 279 286 54 47 51 51 2.09 X X 51790 257 296 280 50 45 50 48 2.88 X 52070 313 299 306 71 59 62 65 2.55 X X 53030 54180 300 316 306 62 66 66 65 3.11 X X 303 299 301 62 50 58 57 2.00 X 2.0 or better X ro C T \ X TABLE B (Continued) Criterion Classification 24 units 60 units SCAT Coop. Reading Test Less 2 .0 Less 2.0 Alpha No. (Converted score) (Scaled score ) H.S. than or than or (Fall 1956) Verbal Quant. Total Vocab. Speed Level Total GrPA 2 .0 better 2 .0 better 54560 286 297 290 52 52 60 55 1.40 X X 55150 312 310 311 68 55 64 63 3 .0 0 X X 57060 291 302 296 46 52 58 52 2.24 X X 57450 306 313 308 53 56 70 60 2 .7 6 X X 57570 298 327 309 55 58 58 57 2.67 X X 57610 291 286 281 56 54 62 58 1.86 X X 60180 292 323 305 56 53 62 57 2.80 X X 6080Q 277 296 286 45 41 48 44 1.96 X 61560 296 333 310 56 61 72 64 3.27 X X 62050 313 308 311 68 61 72 68 3.44 X 62330 299 276 289 83 55 68 69 2.50 X 62790 311 310 310 61 54 66 61 2 .6 5 X 63290 294 321 305 55 69 68 65 3.91 X 63340 300 294 297 62 63 62 63 1.96 X X 66150 292 311 300 54 55 68 59 2 .3 0 X X 66580 275 279 278 48 49 53 50 2.59 X X 67420 305 296 301 61 62 70 65 3.10 X 72220 295 299 296 52 53 60 55 2 .8 1 X 72310 295 307 300 56 51 59 55 1.89 X X 72990 306 330 315 61 68 75 69 2 .6 7 X X 75210 297 283 281. 50 51 59 53 2.00 X 75410 284 286 285 51 54 53 53 2.15 X X 76810 277 313 294 47 54 60 54 2 .3 5 X to o 271 TABLE C FREQUENCY DISTRIBUTIONS FOR HIGH SCHOOL GRADE POINT AVERAGE AND TWO CRITERIA OF SUCCESS GPA 24 unit criterion 60 unit criterion Less 2 than .0 2.0 or better Less 2 than .0 2.0 or better A B A B A B A B 3.9-4.0 0 0 5 1 1 0 4 0 3.7-3.8 0 0 1 2 0 1 1 1 3.5-3.6 0 0 3 4 1 1 2 1 3.3-3.4 1 1 2 6 1 3 0 2 3.1-3.2 1 0 7 6 2 3 4 1 2.9-3.0 3 2 7 8 3 3 4 5 2.7-2.8 5 10 6 5 4 9 4 3 2.5-2.6 9 4 8 7 10 8 2 1 2.3-2.4 5 9 4 3 l 6 1 0 2.2-2.2 5 4 3 5 5 8 2 1 1.9-2.0 9 10 1 2 6 6 0 2 1.7-1.8 7 5 2 2 4 4 0 2 1.5-1.6 2 0 2 0 2 0 1 0 1.3-1.4 0 2 0 1 0 2 0 0 1.1-1.2 1 1 0 0 1 1 0 0 0.9-1.0 0 0 1 0 1 0 0 0 272 TABLE D FREQUENCY DISTRIBUTION FOR THE VERBAL SCORE ON THE SCAT AND TWO CRITERIA OF SUCCESS 24 unit criterion 60 unit criterion V score rerted score) Less 2. than ,0 2.0 or better Less 2 than .0 2.0 or better A B A B A B A B 326-328 1- 0 0 1 0 0 0 1 323-325 0 0 0 0 0 0 0 0 3 2 0 -3 2 2 0 0 3 0 1 0 0 0 317-319 0 0 2 2 0 1 1 1 314-316 1 0 1 2 1 1 1 1 311-313 2 3 4 3 0 2 4 0 308-310 0 1 5 1 1 2 4 0 305-307 1 2 1 5 1 3 1 3 302-304 5 1 3 3 1 2 3 2 299-301 4 4 7 5 3 6 3 1 296-298 2 3 5 4 3 2 1 1 293-295 2 6 5 6 4 5 0 3 290-292 8 10 5 7 5 14 3 2 2 8 7 -2 8 9 6 3 3 3 6 4 1 2 284-286 3 1 4 4 3 3 2 1 2 8 1 -2 8 3 2 1 3 2 3 2 1 0 278-280 0 1 0 0 0 0 0 0 275-277 4 3 0 1 3 3 0 0 272-274 4 2 1 1 3 2 0 1 2 6 9 -2 7 1 2 2 0 2 2 3 0 0 2 6 6 -2 6 8 0 0 0 0 0 0 0 0 2 6 3 -2 6 5 0 1 0 0 0 0 0 0 2 6 0 -2 6 2 0 0 0 0 0 0 0 0 257-259 1 1 0 0 0 0 0 0 254-256 0 0 0 0 0 0 0 0 251-253 0 1 0 0 0 0 0 0 273 TABLE E FREQUENCY DISTRIBUTION FOR THE QUANTITATIVE SCORE ON THE SCAT AND TWO CRITERIA OF SUCCESS 24 unit criterion 6 0 unit criterion Leas than 2.0 or Less than 2.0 or Q score 2.0 better 2.0 better (Converted score) A B A B A B A B 3 3 6 -3 4 0 1 0 2 0 0 0 2 0 3 3 1 -3 3 5 1 1 0 1 0 2 0 0 3 2 6 -3 3 0 2 1 2 3 2 1 0 3 3 2 1 -3 2 5 2 2 5 6 2 5 1 2 3 1 6 -3 2 0 2 2 4 5 2 5 2 3 3 1 1 -3 1 5 3 7 7 2 3 6 6 1 3 0 6 -3 1 0 4 2 5 7 4 2 2 3 3 0 1 -3 0 5 3 1 4 5 3 2 2 1 2 9 6 -3 0 0 9 13 6 8 7 13 4 1 2 9 1 -2 9 5 5 4 4 3 4 4 0 2 2 8 6 -2 9 0 6 3 7 3 5 4 4 0 2 8 1 -2 8 5 5 6 1 4 5 4 1 1 2 7 6 -2 8 0 2 3 0 4 1 4 0 1 2 7 1 -2 7 5 0 0 1 0 1 0 0 0 2 6 6 -2 7 0 2 1 2 1 1 1 1 1 2 6 1 -2 6 5 0 1 2 0 2 1 0 0 2 5 6 -2 6 0 0 1 0 0 0 1 0 0 2 5 1 -2 5 5 1 0 0 0 0 0 0 0 274 TABLE F FREQUENCY DISTRIBUTION FOR THE TOTAL SCORE ON THE SCAT AND TWO CRITERIA OF SUCCESS T score (Raw score) 24 unit criterion 6 0 unit criterion Less 2 than .0 2.0 or better Less 2 than .0 2.0 or better A B A B A B A B 101-105 1 0 0 0 0 0 0 0 96-100 1 0 2 3 0 1 1 2 91-95 1 0 4 0 2 0 2 0 8 6 -9 0 2 2 2 3 0 2 2 3 81-85 1 1 6 6 2 4 3 1 76-80 2 6 5 4 1 8 4 0 71-75 3 3 7 6 5 5 2 3 6 6 -7 0 3 4 3 8 2 4 3 4 6 1 -6 5 5 5 6 5 5 6 2 2 56-60 7 5 5 3 7 4 1 1 51-55 4 4 3 5 3' 5 0 0 46-50 7 8 5 3 6 8 4 0 41-45 7 3 2 4 4 3 1 2 36-40 2 4 0 2 2 3 0 1 31-35 1 0 2 0 3 0 0 0 26-30 0 2 0 0 0 1 0 0 21-25 1 1 0 0 0 1 0 0 275 TABLE G FREQUENCY DISTRIBUTION FOR THE SPEED OF COMPREHENSION SCORE ON THE READING TEST AND TWO CRITERIA OF SUCCESS S score (Scaled score) 24 unit criterion 60 unit criterion Less 2 than .0 2.0 or better Less 2 than .0 2.0 or better A B A B A B A B 97-98 0 0 0 1 0 0 0 1 83-84 0 0 0 1 0 0 0 1 81-82 0 0 0 0 0 0 0 0 79-80 1 0 0 0 0 0 0 0 77-78 0 0 0 1 0 1 0 0 75-76 0 0 1 1 1 0 0 1 73-74 0 0 3 0 0 0 3 0 71-72 0 0 1 1 1 1 0 0 69-70 0 0 1 1 0 0 0 0 6 7 -6 8 1 1 1 0 1 1 1 0 6 5 -6 6 2 0 3 1 2 1 1 0 63-64 5 2 2 3 2 4 1 0 6 1 -6 2 4 1 4 4 2 2 2 1 59-60 3 4 3 6 3 8 2 2 57-58 2 6 11 4 4 6 5 2 55-56 4 4 6 5 4 4 3 2 53-54 6 10 6 7 4 7 5 3 51-52 6 3 4 5 7 4 0 1 4 9 -5 0 6 4 6 9 5 6 2 4 47-48 2 5 0 0 1 5 0 0 45-46 5 4 0 1 5 3 0 0 43-44 0 1 0 1 0 0 0 1 41-42 1 2 0 0 0 1 0 0 39-40 0 0 0 0 0 0 0 0 37-38 0 1 0 0 0 1 0 0 276 TABLE H FREQUENCY DISTRIBUTION FOR TOTAL SCORE ON THE READING TEST AND TWO CRITERIA OF SUCCESS 24 unit criterion 60 unit criterion Total score Less than 2.0 or 2.0 better Less than 2.0 or 2.0 better (Scaled score) A B A B A B A B 89-90 0 0 0 1 0 0 0 1 75-76 1 0 0 1 0 0 0 1 73-74 2 0 0 2 1 2 0 0 71-72 0 0 3 1 1 0 1 1 69-70 1 2 2 2 0 3 2 0 67-68 2 0 3 3 1 1 2 0 65-66 5 2 3 4 4 4 1 0 63-64 0 1 5 4 0 4 4 1 61-62 2 1 7 1 4 1 2 0 59-60 4 5 10 3 5 5 6 2 57-58 3 8 5 5 1 8 3 4 55-56 5 4 2 10 5 4 0 5 53-54 9 6 7 6 8 7 2 0 51-52 5 7 ' 5 6 5 8 2 2 49-50 1 5 0 2 1 6 0 1 47-48 7 3 0 1 6 0 0 1 45-46 1 0 0 0 0 0 0 0 43-44 0 2 0 0 0 0 0 0 41-42 0 2 0 0 0 0 0 0

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Creator Derian, Albert Steven (author)

Core Title 'Cost' And 'Utility' In The Prediction Of The Successful Junior College Student

School School of Education

Degree Doctor of Education

Degree Program Education

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag education, educational psychology,OAI-PMH Harvest

Format dissertations (aat)

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Lefever, David Welty (committee chair), Calvert, Leonard (committee member), Carnes, Earl F (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-233431

Unique identifier UC11358110

Identifier 6106281.pdf (filename),usctheses-c18-233431 (legacy record id)

Legacy Identifier 6106281.pdf

Dmrecord 233431

Document Type Dissertation

Format dissertations (aat)

Rights Derian, Albert Steven

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA