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An Empirical Investigation Of F-Test Bias, Disproportionality, And Mode Of Analysis Of Variance

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An Empirical Investigation Of F-Test Bias, Disproportionality, And Mode Of Analysis Of Variance

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Content This dissertation has been — microfilmed exactly as received 67— 13,029 HOPKINS, Bobby Ray, 1932- AN EMPIRICAL INVESTIGATION OF F-TEST BIAS, DISPROPORTIONALITY, AND MODE OF ANALYSIS OF VARIANCE. University of Southern California, Ph.D., 1967 Statistics University Microfilms, Inc., Ann Arbor, M ichigan AN EMPIRICAL INVESTIGATION OP F-TEST BIAS, DISPROPORTIONALITY, AND MODE OF ANALYSIS OF VARIANCE toy Bobby Ray Hopkins A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Educational Psychology) June 1967 UNIVERSITY O F S O U T H E R N CALIFORNIA T H E G R A D U A T E SC H O O L U N IV E R SIT Y PA R K LO S ANGF:LES, C A L IF O R N IA 9 0 0 0 7 This dissertation, written by ....... BOBBY_RAY, HOPKINS........... under the direction of h^-.P—Dissertation Com mittee, and approved by all its members, has been presented to and accepted by the Graduate School, in partial fulfillment of requirements for the degree of D O C T O R O F P H I L O S O P H Y c . .... Dean Date June..l967............... DISSERTATION COMMITTEE TABLE OP CONTENTS LIST OF TABLES ..................................... Chapter I. INTRODUCTION .............................. The Problem Hypotheses Definition of Terms Delimitations Organization of the Remaining Chapters II. REVIEW OF THE LITERATURE ................ The Method of Fitting Constants The Weighted Squares of Means Method The Method of Unweighted Means The Method of Expected Numbers Tsao1s Method of Expected Equal Frequencies Kramer's Modified Weighted Squares of Means Method Summary III. PROCEDURES ................................ Sampling Procedures Samples Analyses of the Samples Statistical Treatment of the Sampling Distributions General Statistical Procedures Procedures Related to Hypotheses 1.1, 2.1, 3.1, 4.1, 5.1, and 6.1 Procedures Related to Hypotheses 1.2, 2.2, 3.2, 4.2, 5.2, and 6.2 Procedures Related to Hypotheses 1.3, 2.3, 3.3, 4.3, 5.3, and 6.3 IV. FINDINGS .................................. Findings Related to the Method of Fitting Constants Chapter Page v Hypothesis 1.1 Hypothe sis 1.2 Hypothesis 1.3 Findings Related to the Method of the Weighted Squares of Means Hypothesis 2.1 Hypothesis 2.2 Hypothesis 2.3 Findings Related to the Method of Unweighted Means Hypothesis 3.1 Hypothesis 3.2 Hypothesis 3.3 Findings Related to Snedecor's Method of Expected Numbers Hypothesis 4.1 Hypothe sis 4.2 Hypothe sis 4.3 Findings Related to Tsao1s Method of Expected Equal Frequencies Hypothe sis 5.1 Hypothesis 5.2 Hypothesis 5.3 Findings Related to Kramer's Modified Method of Weighted Squares of Means Hypothe sis 6.1 Hypothesis 6.2 - Hypothesis 6.3 V. SUMMARY, FINDINGS AND CONCLUSIONS, AND RECOMMENDATIONS .............................. 99 Summary Area of Investigation Procedures Findings and Conclusions iv Chapter Page Recommendations Related to the Statistical Analysis of Dispropor tionate Data Recommendations for Further Research APPENDIX......................................................110 BIBLIOGRAPHY................................................. 119 LIST OF TABLES Table Page 1. A Set of Subclass Numbers for Pattern 5 . . . . 24 2. Empirical Estimates of Level of Significance for Pattern 5 for Unweighted Means Method . . 25 3. Empirical Estimates of Level of Significance for Pattern 5 for Snedecor1s Method of Expected Numbers ............................. 29 4. Summary of F-Test Bias— Review of Literature.......................................33 5. Patterns of Subclass Frequencies Employed in this Study................. .................. 36 6. Probabilities Based on the Binomial Distribution............................. .. . 45 7. Probabilities Based on the Poisson Distribution .................................. 47 8. Confidence Intervals for Selected Proportions...................................54 9. Summary of F-Test Bias for the Method of Fitting Constants ............................. 56 10. Summary of F-Test Bias for the Method of Weighted Squares of Means...................62 11. Summary of F-Test Bias for the Method of Unweighted M e a n s ............................65 12. Proportion of F Values Greater Than F^gg for Unweighted Means Method................ 66 13. Proportion of F Values Greater than F^gg for Unweighted Means Method..... .............. 67 14. Proportion of F Values Greater than F for Unweighted Means Method .... *....... 68 15. Summary of F-Test Bias for Snedecor's Method of Expected Numbers .............. 72 vi Table Page 16. Proportion of F Values Greater than F _ for Expected Numbers Method . . . . '.......... 74 17. Proportion of F Values Greater than F^gcj for Expected Numbers Method . . . . ’.......... 75 18. Proportion of F Values Greater than Fgg for Expected Numbers Method ....* .......... 76 19. Scattergram of and%^/N Versus the Proportion Exceeding F^go for Snedecor1s Method ....................... 79 20. Descriptive Data on the Proportions of F Values Exceeding Six Checkpoints for Snedecor1s Method .............................. 80 21. Regression Equations to Estimate the Actual Alpha Level in Snedecor’s F-Tests for Main E f f e c t s .................. 82 22. Estimated Levels of Significance in Snedecor1s F-Tests for Main Effects ......... 83 23. Summary of F-Test Bias for Tsao's Method of Expected Equal Frequencies ................ 88 24. Proportion of F Values Greater than F.90 for Tsao ' s Method.............................. 89 25. Proportion of F Values Greater than F for Tsao 1 s Method...................*.......... 90 26. Proportion of F Values Greater than F for Tsao's Method...................*.......... 92 27. Summary of F-Test Bias for Kramer's Modified Weighted Squares of Means Method ............ 96 CHAPTER I INTRODUCTION The procedures of analysis of variance originated with R. A. Fisher, Chief Statistician, Rothamsted Experi mental Station, Harpenden, England (5, 31) . These procedures included the analyses of factorial designs employing equal numbers of observations within each of the subclasses. Since, for this condition, the two variances involved in the F statistic are independent and unbiased estimates of the population variance, his methods yield a valid and unbiased F-test (6, 18, 24, 25). This method of analysis found a ready application to early agricultural experimentation since it was feasible to assign an equal number of experimental units to each of the treatment combinations. Difficulties soon became apparent, however, when the analysis of variance procedures were incorporated into the statistical arsenal of the biometrician, psychologist, and educator. In 1933 Brandt stated: "Equal cell frequencies are frequently physically impossible to obtain, as for instance, in a study of the effects of sex and litter size on the birth weights of pigs." (5:265) Such unequal replication within the subclasses sometimes results in non-orthogonal data so that the Fisher formulas for the sums of squares for main effects and interactions do not add to the sum of squares representing variation among the cell means (18, 26) . In some early experimentation, the sum of squares due to interaction were computed to have negative values (5:167; 29:383). A negative sum of squares would yield an imaginary standard deviation, a value which is obviously not permissible as an estimate of population variability. For disproportionate subclass numbers, the sums of squares for the row and column effects are not independent (14). A variance computed among the weighted row means will not be free from the column effect. As a result, a modified method of analysis must be employed. In order to utilize the powerful and facile analysis of variance tool, several computing methods have been proposed modifying the formulas presented by Fisher (12). The comparable merits and demerits of these proposed methods are generally unknown as the degree of disproportionality and/or sample size increases. The Problem The experimenter in educational research is often faced with situations involving disproportionate subclass frequencies. Typically he is not at liberty to randomly assign the students to the various experimental conditions although this would be desirable from the researcher's standpoint. The disruptive effect on the normal ongoing educational program and the administrative detail demanded by such a procedure seem to preclude its recommendation by all concerned. Experimental educational programs and research projects normally extend over a period of several weeks or months. During this interval, the school population is continuously changing. In some of the lower income communities, where many of the programs of the Elementary and Secondary Education Act of 1965 are in operation, the attrition due to the transientness of the participating families is substantial. As a result some degree of disproportionality is introduced into an otherwise equal or proportional classification of the experimental units. Ex post facto research is also of marked interest and importance to educators. Kerlinger states: "If a tally of sound and important studies in psychology, sociology, and education were made, it is likely that ex post facto studies would outnumber and outrank experimental studies." (22:37 3) Many of the important educational research problems do not lend themselves to experimentation. Variables such as intelligence, aptitude, home background, parental attitudes, teacher personality, sex, and health cannot be manipulated and randomly assigned to the participating students. Even so, the interrelationship between these variables, and the correlations between such variables and educational variables are of interest to both the practicing educator and the theorist. The practicing educator may envision a program to modify certain variables such as parental attitudes and health conditions through long-range efforts. The theorist is highly interested in the development of intelligence, motivation, and personality. Therefore, the experimenter is sometimes concerned about possible interaction effects between correlated factors for which it may be not only extremely difficult but quite unrealistic to obtain equal cell sizes. For example, one possible area of investi gation is the exploration of the interaction of IQ and socioeconomic status with various methods of instruction. In this and similar investigations, a high degree of disproportionality of cell frequencies is likely to exist. In certain situations there are positive features of disproportionate cell frequencies. Cox (8:35) suggests that it is proper to devote more subjects to the control group than to the treatment groups when two or more experimental groups are to be compared with the control group. This procedure tends to yield a more stable and accurate estimate of the population parameters of the control group which enters into each of the comparisons. Then too, Horsnell (19:134) ascertained that in certain situations involving heterogeneous variances, unequal cell sizes may have a beneficial effect upon the power of the test and/or the probability of a Type I error. Box (3) and Scheffe (27:331-364) corroborate Horsnell's finding regarding the relationship between heterogeneity of variance, unequal cell sizes, and the probability of a Type I error. The frequency of the foregoing circumstances in current educational research seems to suggest the need for efficient and trustworthy methods of handling non-orthogonal data which preserve the advantages accrued through the application of factorial analysis of variance. Almost two decades ago, Johnson commented on the need as it existed at that time. "There is an urgent need, therefore, for a systematic formulation of methods of attacking problems when unequal replication in the subclasses occur." (20:261) Similarly, Tsao observed the situation at hand and came to much the same conclusion. The analysis of variance and covariance methods usually deal with data composed of equal or propor tionate numbers of observations in the subclasses. In fields connected with human beings, such as education and psychology, unequal representation in each cell of the multiple-classification of data is of common occurrence. Even in fields such as agriculture and biology it is sometimes unavoidable or desirable to have disproportionate numbers of observations in the subclasses. Therefore, the need is very urgent for a systematic formulation of general methods of attacking the problems under such conditions. (33:107) After two decades, Bradu's comment affirms a similar state of affairs. "No general practical method for analysis of a non-orthogonal p-factor layout seems to have been found so far.” (4:88) As one surveys the educational research scene today, the situation appears little different from that referenced by Johnson, Tsao, and Bradu. When Johnson made the above-mentioned comments, however, several modified methods of analysis had been proposed: (1) the method of fitting constants (36), (2) the method of weighted squares of means (36), (3) the method of unweighted means (36), (4) the method of expected subclass numbers (28), and (5) the method of expected equal frequencies (33). These same methods are extant today and are commonly preferred and recommended in prominent texts of statistical methodology (16, 27, 34, 35) . Another computing method— Kramer’s modified squares of means (23) — has been available since 1955 but apparently little notice has been given to it in the literature. Evidently the concern expressed by Johnson, Tsao, and Bradu was not due to the paucity of available methods, but to their appropriateness and interpretability when applied in certain circumstances. Snedecor (29:379, 385) indicated that the unweighted means analysis may yield biased results if the subclass numbers are more than slightly unequal. Gourlay (15) concluded that the use of Snedecor's method of expected numbers may result in some degree of positive bias. Tsao presented his approximate method in an attempt to eliminate the bias in Snedecor's approach but Gourlay (15) feels that Tsao's adjustment will, on the average, increase rather than decrease the resulting bias in the F statistic. The method of fitting constants gives unbiased tests of the main effects only when no inter action exists (1, 29). Yates (36:57) warned that under certain conditions the method of the weighted squares of means method will become increasingly inappropriate as departure from equal cell frequencies becomes great. A consideration of the above comments tends to give one the impression that there is a significant degree of uncertainty with respect to several of the proposed methods when substantial disproportionality of cell..sizes prevails. Obviously, a number of authorities suspect that the sampling distributions of the F statistic, for certain methods under certain conditions, do not closely approximate the theo retical central F-distributions. If this is true, the use of the standard F table is inappropriate and an unreliable guide to the interpretation of the obtained statistic. It seems probable that all of the above mentioned methods yield sampling distributions which cldsely conform to the theoretical central F-distributions under conditions of slight disproportionality. This assurance is vitiated, however, as the degree of disproportionality becomes great. It also seems likely that the magnitude of the discrepancies between the theoretically obtained F-distributions and the empirically obtained F-distributions is related to the sample size involved since larger sample sizes give more stable estimates of the population parameters. An investigation of the interrelationship among (1) the computing method, (2) the degree of disproportionality, (3) sample size, and (4) the magnitude and direction of any resulting bias, seems highly desirable to provide some indication of the comparative merits and demerits inherent in the various methods. Hypotheses Three basic hypotheses were examined for each method of analysis under consideration in this study. 1.0 Hypotheses Concerning the Method of Fitting Constants 1.1 For each combination of sample size and degree of disproportionality involved in this study, the empirically obtained sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F.901 F. 95 1 F.975' F.991 F.995' and F .999* 1.2 For each sample size, across all levels of disproportionality, the empirical sampling distributions will not differ significantly from the corresponding central F-distributions in the j proportion of F values which exceed the following; values: F>go. F#95. F>g75, F>ggi F>gg5, and F.999- 1.3 For each degree of disproportionality, across all levels of sample size, the empirical sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F#go> F_g5, F>g75. F_gg, F>gg5, and F.999• Hypotheses Concerning the Method of the Weighted Squares of Means 2.1 For each combination of sample size and degree of disproportionality involved in this study, the empirically obtained sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F>g0i F .95' F .975' F .991 F.995' and F.999* 2.2 For each sample size, across all levels of disproportionality, the empirical sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F 9qi F g^, F ^51 F.99' F. 9951 and F.999 * 2.3 For each degree of disproportionality, across all 10 levels of sample size, the empirical sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F.go1 F .95' F .975‘ F.99‘ F.995' F.999* 3.0 Hypotheses Concerning the Method of Unweighted Means 3.1 For each combination of sample size and degree of inequality involved in this study, the empirically obtained sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F gQi f .95‘ f .975' F.99‘ F.995‘ and F .999" 3.2 For each sample size, across all levels of inequality, the empirical sampling distributions will not differ significantly from the cor responding central F-distributions in the pro portion of F values which exceed the following values: F>90' F.95> F,975. F.99* F .995> and F .999 * 3.3 For each degree of inequality, across all levels of sample size, the empirical sampling distribu tions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following 11 values: F 9Q1 F^,., F ^ , F_99, F ^ , and F .999' 4.0 Hypotheses Concerning Snedecor1s Method of Expected Numbers 4.1 For each combination of sample size and degree of disproportionality involved in this study, the empirically obtained sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F ggi F.95' F. 9751 F.99' F.995' andF.99g- 4.2 For each sample size, across all levels of disproportionality, the empirical sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F.90' F.95' F.975‘ F.99' F.995' and p.999• 4.3 For each degree of disproportionality, across all levels of sample size, the empirical sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: Fgo' F>g5. F.975' F>99< F.995' and F.999• 12 5.0 Hypotheses Concerning Tsao1s Method of Expected Equal Frequencies 5.1 For each combination of sample size and degree of inequality involved in this study, the empirically obtained sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F gg, F.95' f.9751 F . 99 ' F.995' and F .999 * 5.2 For each sample size, across all levels of inequality, the empirical sampling distributions will not differ significantly from the corres ponding central F-distributions in the proportion of F values which exceed the following values: F . 901 f .95' F .975' F .99' F .995' an<^ - F„999* 5.3 For each degree of inequality, across all levels of sample size, the empirical sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F go 1 F 95' F 975' F 99' F .9951 and F .999‘ 6.0 Hypotheses Concerning Kramer's Method of Modified Weighted Squares of Means 6.1 For each combination of sample size and degree of disproportionality involved in this study, the 13 empirically obtained sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F .90 ' F.95' F.975‘ F .99' F.995' an<^ F .999* 6.2 For each sample size, across all levels of disproportionality, the empirical sampling distributions will not differ significantly from the corresponding central F-distributions in the proportion of F values which exceed the following values: F .90' F .95' F .975' F .99‘ F.995' and- F .999 * 6.3 For each degree of disproportionality, across all levels of sample size, the empirical sampling distributions will not differ signifi cantly from the corresponding central F- distributions in the proportion of F values which exceed the following values: F .90' F .95' P .9751 F .99' F.9951 andF.999* Definition of Terms For the purposes of this study, the following denotations have applied. Disproportionality. "When the numbers of observa tions in the subclasses of a multiple classification are 14 not proportional to the marginal totals, the subclass numbers and the data are called disproportionate." (13:115) Orthogonality. The condition under which the estimate of any one of the parameters of a statistical model is uncorrelated with that of any other parameter. As a consequence the total sum of squares for a multifactor layout may be partitioned into independent components such that the sums of squares attributable to the main effects added to the sums of squares attributable to the inter action effects equal the sum of squares between cells. Fixed-effects model. Hays defined this model as one that is "distinguished by the fact that inferences are to be made only about differences among the J treatments actually administered, and about no other treatments that might have been included." (18:357) Positive bias. This term describes the situation where the empirically obtained sampling distribution includes a greater proportion of large F values than does the theoretical F-distribution. Negative bias. The term which describes the situation where the empirically obtained sampling distribu tion includes a smaller proportion of large F values than does the theoretical F-distribution. 15 Monte Carlo method. The procedure which utilizes a computer to generate empirical sampling distributions for the desired statistic under specified conditions. Delimitations The scope of this study was limited to the 2 x 2 fixed-effects statistical model. This model was thought to be an appropriate selection due to its frequent application in educational research. Another relevant feature of this design is that a single degree of freedom is associated with each of the estimated effects and is thus the lower bound in factorial complexity. The following computing methods were examined in this study: (1) the fitting constants method, (2) the weighted squares of means method, (3) the method of unweighted means, (4) Snedecor's method of expected numbers, (5) Tsao's method of expected equal frequencies, and (6) Kramer's modified weighted squares of means method. Five different sample sizes were included ranging from a total of twelve to a total of 192 for the four cells. Sample sizes of twenty-four, forty-eight, and ninety-six were also investigated. The dimension of disproportionality was comprised of ten levels, and was measured by the ratio of chi square to sample size. The magnitude of the disproportionality ratios included 0.00 (both equal and proportional cell 16 frequencies), 0.0625, 0.0988, 0.1111, 0.1250, 0.1600, 0.2570, 0.3086, and 0.4444. The specific configurations of the cell frequencies are described in Chapter III. An investigation of the 288 combinations of method employed, sample size involved, and degree of dispropor tionality present was made to ascertain the interrelation ship among these factors. Organization of the Remaining Chapters Chapter II presents a review of the literature pertaining to F-test bias as it relates to the computing methods which are under consideration in this study. Chapter III describes the procedures employed in conducting this study, including the use of the computer, considerations relative to sample size and disproportion ality, and the statistical treatment of the data. All of . the information needed to replicate the study is contained in this chapter. Chapter IV discusses the findings of the present investigation with an evaluation in terms of the research hypotheses. Chapter V presents a summary of the study and relates general conclusions and recommendations which can be drawn from the findings. CHAPTER II REVIEW OF THE LITERATURE It was the purpose of this chapter to present a review of the literature pertaining to F-test bias as it relates to the following analysis of variance computing methods: (1) the method of fitting constants, (2) the weighted squares of means method, (3) the method of unweighted means, (4) the method of expected numbers, (5) Tsao's method of expected equal frequencies, and (6) Kramer's modified weighted squares of means method. The Method of Fitting Constants Yates (36) presented simultaneously three methods of dealing with problems of disproportionate cell frequencies. One of these was his method of fitting constants. This method is based on the statistical model, Xij-^ = M + a^ + bj + where a and b correspond to the two main effects and ej_jk is an error term. The computational formulas are reported in the Appendix. Kempthorne (21), Scheffe (27), and Graybill (16), among others, have presented extensive mathematical treatments of this method. 17 18 Concerning the rationale underlying this method, Yates wrote: The general rule applicable to any groups of fitted constants is to find the part of the sum of squares accounted for by fitting all the constants and deduct from it the part of the sum of squares accounted for by fitting all the constants except those to be tested. (36:63) The least squares criterion is used to determine the set of best fitting constants which minimize the residual sum of squares. The increment added to the residual sum of squares when all of the constants are fitted except the constant for factor A, is the sum of squares attributable to this factor. Of particular interest is the observation that there is no term in the statistical model which represents the interaction effect. However, Steel and Torrie (32:258), Snedecor (29:388), and Winer (35:226) have indicated that an efficient and unbiased estimate of the interaction sum of squares can be obtained with little additional effort. Thus an efficient and unbiased test of interaction effects is available through the use of the fitting constants method. Steel and Torrie (32) and Snedecor (29) report that if the condition of no interaction prevails in the popula tion of interest, the method of fitting constants also yields unbiased estimates and tests for the main effects. The situation with respect to a population where interaction 19 exists presents a more formidable barrier. Goulden (14: 313) states: "Basically the fitting of constants is a procedure enabling us to set up a set of adjusted values, Y, corresponding to the original values such that in the adjusted values the interaction is zero." Concerning this situation and in reference to the method of fitting constants based upon the statistical model as given above, Steel and Torrie state: "If interaction is present but Eq. (13.3) has been the description, then any main effect sum of squares is only an approximation, increasingly poor as the disproportion increases." (32:258) Snedecor (29:390) suggests that when interaction is shown to be present in the population, no further steps in the analysis should be taken. Winer (35:226) mentions that should interaction be significant, the primary interest is focused upon the simple main effects rather than the overall main effects. It appears possible that if the assumption of no interaction seems questionable, a more appropriate model and method should be sought. Steel and Torrie (32:258) and Snedecor (29:390) recommend a preliminary test of interaction before applying this method of analysis. In reference to this preliminary test of interaction effects, Goulden seems to suggest considerable flexibility. Possibly we should not take the evidence of the significance of the interactions too literally. Even if the interaction is significant, if we have good 20 reason to believe that the interaction in the population is negligible, the constants are still efficient estimates of the main effects. (14:335) These recommendations seem somewhat contradictory and insufficient. No discussion of the Type I or Type XI errors relative to these conditions is given. It is well known that with small samples (and little power) the degree of interaction must be substantial in order to be ascer tained by the F-test even when a liberal alpha level of significance is selected (9:94-100). The effect of a slight degree of interaction upon the estimates and tests for main effects seems to be a moot point. Tsao questions the applicability of Yates' method of fitting constants from another approach. The restrictions which were used appear to assume that each subclass of a classification contributes the same amount of variation whatever the number of observations in the subclasses. We may call these restrictions 'the unweighted means.1 Under such conditions the frequencies of all subclasses in the population are assumed to be equal. But this assump tion may not be fulfilled in all cases. (33:108) The Weighted Squares of Means Method The second method that Yates (36) proposed for dealing with non-orthogonal data is known as the method of weighted squares of means. This method is a development from the procedures for the analysis of variance of a single classification with unequal numbers in the various classes as described by Fisher (12). Concerning the 21 rationale of this method (where Q represents the sum of squares between classes for the one-way layout) Yates stated: We are thus led to the more general case where an estimate of a variance is required from differences of a set of numbers whose variances are known fractions of that variance. From the above expression for Q this is seen to be the weighted sum of the squares of the deviations from the weighted mean of the numbers, divided by p-1, the weights being equal to the - reciprocals of the known fractions. (36:56) The computing formulas for this method are presented in the Appendix. According to Federer (10:126) and Anderson and Bancroft (1:278) the method of the weighted squares of means is especially appropriate when interaction is present. Under such conditions this procedure yields estimates of the parameters which are both unbiased and efficient. There seems to be some question about the indiscrim inant application of the method of weighted squares of means. Steel and Torrie (32:265) state that unless interaction does exist in the population, the estimates and tests of the main effects are inefficient. With reference to this situation Yates wrote: "The method of weighted squares of means can be regarded as an approximation to the rigorous method of least squares. The approximation will become less close as the class numbers become more unequal." (36:57) Snedecor (28:389) commends the method as being easy to compute but feels that it is somewhat 22 limited in the conditions under which it can be used. The Method of Unweighted Means The third method that Yates (36) proposed for analyzing disproportionate data is the method of unweighted means. This is an approximate method based on the assumption that the variances of all the subclass means are equal. Winer (35:222) notes that in essence the method of unweighted means considers each cell in the experiment as if it contained the same number of observations. Relative to the underlying rationale, Yates wrote: If the differences between the various class numbers are ignored, all the sub-class means being assumed to have a variance equal to the mean of all the true variances, the methods appropriate to the case of equal class numbers may be employed. (36:65) Snedecor (29:385) and Winer (35:224) state that under conditions of near-equal cell sizes, when it can be assumed that the population size for the cells are approximately equal, the method of unweighted means is appropriate. The applicability of this method when these conditions do not obtain is seriously questioned. Snedecor (28:389) relates that if the main class numbers differ decidedly, much information is lost by the reduction to a single item in each subclass. Concerning this limitation Yates (36:65) cautioned: "This approximation is only useful when the class numbers do not vary greatly." No practical criterion for the degree of inequality permissible was 23 given. The computational formulas are presented in the Appendix. Gosslee and Lucas (13) investigated the disturbance to the level of significance of additive sums of squares methods of analyzing disproportionate data when interaction was present in the population sampled. The model considered was a 3 x 3 layout with two patterns of subclass frequen cies. Empirical sampling studies were conducted through the use of a computer program designed to generate 400 F values using the unweighted means formulas. The disturbance to the ten, five, and one per cent levels of significance was judged to be moderate. However, sampling errors for empirically derived sampling distributions consisting of 400 F values is pertinent in this context. It seems likely that the number of F values which exceed the critical value will vary considerably from sample to sample even though the same population is involved. Gosslee and Lucas (13) also generated a sampling distribution of 6000 F ratios for one 3 x 3 pattern of frequencies. The population sampled was one in which interaction was present. Table I presents the information on sample size and cell configuration. Table II summarizes the results of this investigation. 24 TABLE 1 A SET OF SUBCLASS NUMBERS FOR PATTERN 5 Column 1 Column 2 Column 3 Row 1 1 1 1 Row 2 3 6 10 Row 3 2 20 20 SOURCE: David G. Gosslee and Henry L. Lucas, "Analysis of Variance of Disproportionate Data When Interaction is Present," Biometrics, March, 1965, pp. 115- 33. 25 TABLE 2 EMPIRICAL ESTIMATES OF LEVEL OF SIGNIFICANCE FOR PATTERN 5 FOR UNWEIGHTED MEANS METHOD Nominal Level of Effect Significance Unweighted Means Method 5% 6.62 Rows 1% 1.80 5% 4.77 Columns 1% 1.20 5% 5.83 Interaction 1% 1.72 SOURCE: David G. Gosslee and Henry L. Lucas, "Analysis of Variance of Disproportionate Data When Interaction is Present," Biometrics, March, 1965, pp. 115- 33. 26 The Method of Expected Numbers Shortly after the development of the three preceding methods by Yates, Snedecor (28) proposed another method to handle disproportionate data known as the method of expected numbers. He relates that this method is essen tially an extension of Yates 1 method of unweighted subclass means to include the situation where the population subclass numbers are assumed to be proportional to the marginal totals. The computational formulas are presented in the Appendix. Concerning the rationale underlying this method Snedecor wrote: The next point to be noted in connection with theory is that neither the subclass means nor the class numbers are disturbed during adjustment to expected values. The subclass means are known to be efficient estimates of the variable under investigation. The change from actual to expected numbers is merely a matter of weighting, usually of minor consequence unless carried to extremes. (28:391) In essence, the weighting procedures transform the obtained cell frequencies into proportional frequencies which yield orthogonal data and obviate the difficulties inherent in disproportionate subclass numbers. Commenting on the weighting and adjustment procedures Tsao wrote: In retaining the original variance within subclasses, the variances for any other components are worked out with the proportionate expected subclass numbers and expected sums or means in the usual manner. As we know, the original variance for "within" comes from the 27 original data of unequal or disproportionate frequen cies. The writer questions the validity of retaining the "within" variance derived from the original data, while the other variances are derived from the adjusted data. (33:108) Snedecor (28:389-392) indicated that this method of analysis should have a wide applicability to experimental situations since the population from which a sample is drawn is often assumed to have proportional subclass numbers. The obtained disproportionality in these cases is attributable to sampling variation. As a guide to when the method could be applied, Snedecor suggested the use of the chi square criterion to indicate significant departure from proportionality. He felt that unless the value of chi square was great, the method of expected numbers could be relied upon to give satisfactory results. Apparently many educational researchers did concur since in 1955, according to Gourlay (15:242) this was by far the most popular method of analyzing investigations in educational research, Comstock (7:335) investigated the mean square for interaction as computed by the method of expected numbers in a 2 x p layout and concluded that some bias did exist. The Snedecor formulas tended to give a slight overestimate of the mean square for interaction but he felt that the overestimate was neither serious nor frequent. Gourlay (15:227-248) investigated F-test bias for several experimental designs and methods of analysis. He 28 suggests that the use of Snedecor1s method will normally produce a positively biased test of interaction, and that bias, if present, will usually be positive for the main effects. Gosslee and Lucas (13:115-133) made a study of the analysis of variance of disproportionate data when inter action is present in the population being sampled. They indicated that the expected value for the interaction mean square may be too large or too small depending on the pattern of the subclass numbers. They also conclude that the bias in the method of expected numbers causes the expected mean square for main effects to be too large. These conclusions were supported by their Monte Carlo study of the sampling distribution of the F statistic yielded by the method of expected numbers. The sampling distribution generated consisted of 6,000 F ratios sampled from a population where interaction existed. The pattern of subclass frequencies employed was identical to that indicated in Table 1. As shown in Table 3, Snedecor1s method yielded moderately biased results. Tsao1s Method of Expected Equal Frequencies Tsao (33) proposed a procedure to be applicable to situations in which the assumption of equal subclass frequencies seems justifiable even though the obtained cell frequencies may vary to some extent. These procedures are 29 TABLE 3 EMPIRICAL ESTIMATES OF LEVEL OF SIGNIFICANCE FOR PATTERN 5 FOR SNEDECOR’S METHOD OF EXPECTED NUMBERS Effect Nominal Level of Significance Snedecor1s Method of Expected Numbers 5% 7.70 Rows 1% 2.00 5% 9.03 Columns 1% 2.50 5% 4.45 Interaction 1% 1.08 SOURCE: David G. Gosslee and Henry L. Lucas, "Analysis of Variance of Disproportionate Data When Interaction is Present," Biometrics, March, 1965, pp. 115- 33. 30 known as Tsao's method of expected equal frequencies, and involve making an adjustment in the cell sums, cell sums of squares, and the sum of squares within cells. The computational formulas are presented in the Appendix. The rationale underlying this technique is that the obtained cell sum and sum of squares per experimental unit is an adequate approximation to the cell sum and sum of squares per experimental unit that would have been obtained had the cell frequency been of mean value (33:320). Thus subclass sums and sums of squares for cells containing fewer than the mean number of experimental units are weighted to yield adjusted sums and sums of squares which are proportionally greater than the obtained values. Likewise, a decremental weighting is employed for the more populous subclasses. Concerning the applicability of this method, Ferguson wrote: "The method of expected equal frequencies is simple and may be usefully applied where the numbers of observations in the cells do not differ very much." (11:320) He suggested the use of the chi square criterion at the .01 level as a guide to whether the cell frequencies depart significantly from equality. Tsao's method has received little investigation as judged by the literature to date. The important considera tions of bias, power, and applicability are of undetermined 31 disposition. Neither Tsao nor Ferguson deal with the question of possible bias inherent in this method of analysis for non-orthogonal data. Presumably they consider that the use of these procedures will yield insignificantly biased F-tests when the subclass frequencies do not significantly differ from equality. Gourlay investigated the possibility of bias in Tsao's procedures in relation to the bias in Snedecor's method and concluded: "For Eisenhart's Model I, bias, if present, will normally be positive. When Tsao's modifica tion of Snedecor1s method is applied, it would appear that the bias will on the average be increased." (15:247) Kramer's Modified Weighted Squares of Means Method Kramer (23) proposed a procedure for analyzing disproportionate data which is known as the modified weighted squares of means method. One motivation behind the development of this method was to circumvent the very tedious and laborious computational demands of the method of fitting constants when the classifications involved are large, and simultaneously to provide a technique of greater efficiency than Yates' weighted squares of means. As its name implies, the rationale underlying Kramer's procedures closely parallels that of Yates' weighted squares of means method. Kramer's modification 32 pertains to the weighting of cell means. Whereas Yates' method employs the unweighted cell means for the estimates of the class means, Kramer's procedures give differential weighting to the subclass means more in proportion to the numbers on which they are based. The computational formulas are presented in the Appendix. In reference to the applicability of this method Kramer wrote: "This new method is proposed for cases in which p and q are large and when interaction can be assumed absent or negligible." (23:445) The advantages of employing this method rather than that of fitting constants are the practical considerations of ease and simplicity in computations. However, it does not provide a test for interaction effects. As with the fitting constants method, it seems likely that these procedures would become increasingly inappropriate as the degree of interaction increased. Actually very little is known as to how closely the results from these two methods agree since Kramer1s method has received little mention in the literature to date. Summary A model is presented in Table 4 which gives a summary of the findings of the relevant published literature with respect to F-test bias in each of the methods under consideration. TABLE 4 SUMMARY OF F-TEST BIAS— REVIEW OF LITERATURE Method of Analysis Main Effects When Inter action Present Main Effects When No Inter action Present Interaction Effect Yates 1 Fitting Constants Possible Bias Unbiased Unbiased Yates' Weighted Squares of Means Unbiased Possible Bias Does Not Apply Yates' Unweighted Means Possible Bias Possible Bias Possible Bias Snedecor's Expected Numbers Possible Bias Possible Bias Possible Bias Tsao's Expected Equal Frequencies Possible Bias Possible Bias Possible Bias Kramer's Modified Weighted Squares of Means Possible Bias Unbiased Does Not Apply u> u> CHAPTER III PROCEDURES The purpose of this chapter was to describe the following: (1) the sampling procedures employed, (2) the samples selected for study, (3) the analyses of the samples, and (4) the statistical treatment of the resulting sampling distributions. Sampling Procedures A Monte Carlo study was performed utilizing the IBM 7094 computer through the courtesy of Western Data Processing Company. A program was written to generate random numbers from a normal distribution. These numbers were then assigned to the desired subclasses of a 2 x 2 layout in a variety of patterns and frequencies. The specific configurations of the samples included in this study are discussed in the following section. Samples A variety of sample sizes and configurations of subclass frequencies were included in this study. The dimension of sample size consisted of five levels: 12, 24, 34 48, 96, and 192. In every case each cell contained at least one number and in some cases as many as ninety-six numbers were assigned to one cell. In no case, however, did the total for the four cell frequencies exceed 192. The patterns of the subclass frequencies were analyzed in terms of the degree of disproportionality involved. In order to assess the degree of disproportionality, the chi square value denoting the departure from proportionality was computed. The degree of disproportionality was indicated by the ratio of this chi square value to the sample size. The degree of inequality of cell frequencies was also assessed and in a similar fashion— the ratio of the chi square value from equality to the sample size. The dimension of disproportionality consisted of ten levels: 0.0 (for both equal and proportional cell frequencies), 0.0625, 0.0988, 0.1111, 0,1250, 0.1600, 0.2571, 0.3086, and 0.4444. These figures represent a wide range of disproportionality. Table 5 presents the patterns of subclass frequen cies which were included in this study as well as the degree of disproportionality and inequality corresponding to each pattern. TABLE 5 PATTERNS OF SUBCLASS FREQUENCIES EMPLOYED IN THIS STUDY Pattern C Row 1 Column 1 e l l L o Row 1 Column 2 cation Row 2 Column 1 Row 2 Column 2 Degree of Dispropor tionality3 Degree of Inequality*5 A 3k 3k 3k 3k 0.0000 0.0000 B lk 2k 3k 6k 0.0000 0.3889 C 2k 2k 2k 6k 0.0625 0.3333 D 2k 3k 5k 2k 0.0988 0.1667 E 4k 2k 2k 4k 0.1111 0.1111 F lk 3k 5k 3k 0.1250 0.2222 G lk lk lk 9k 0.1600 1.3333 H 2k 5k 4k lk 0.2571 0.2778 I lk 2k 8k lk 0.3086 0.9444 J lk 5k 5k lk 0.4444 0.4444 aThe ratio of the chi square value from proportionality to sample size. ■^The ratio of the chi square value from equality to sample size. OJ ov 37 Analyses of the Samples After a set of random numbers had been generated and distributed according to the desired configuration of cell frequencies, the data were submitted to analysis by each of the six methods herein under consideration. The appendix supplies the formulas required by the various methods of analysis. The F values for the main effects and interaction were computed from a random sample for the fixed-effects model and were stored in the computer memory. This process was continued until 1,000 sets of random numbers had been analyzed for that particular pattern of frequen cies. The F values for each combination of method and effect were then classified by the computer into the following categories: (1) greater than F gg of the central F-distribution, (2) greater than F 95 of the central F-distribution, (3) greater than F 975 of the central F-distribution, (4) greater than F gg of the central F-distribution, (5) greater than F.ggs of the central F-distribution, (6) greater than F ggg of the central F-distribution. The above procedure was performed for each of the forty- eight combinations of sample size and degree of dispropor tionality. In order to enhance the precision of the empirical 38 sampling distributions, the set of procedures described above were replicated. Thus a total of 1,536,000 F values were computed to form 768 sampling generations each involving 2,000 F values. Statistical Treatment of the Sampling Distributions A total of 768 sampling distributions, each consisting of 2,000 F values, were generated by the Monte Carlo methods described in the foregoing sections. The 2,000 F values comprising a sampling distribution were categorized according to whether they exceeded the value corresponding to the following points of the central F-distributions: F.gO' F .95' F.975' F .99' F.995' and F 999. The purpose of this section was to describe the statistical treatment of these data to determine if the obtained proportions falling into the various categories exceeded the expected proportions for the corresponding central F-distributions. These procedures are described according to the following organization: (1) general statistical procedures, (2) procedures related to hypo theses 1.1, 2.1, 3.1, 4.1, 5.1, and 6.1, (3) procedures related to hypotheses 1.2, 2.2, 3.2, 4.2, 5.2, and 6.2, and (4) procedures related to hypotheses 1.3, 2.3, 3.3, 4.3, 5.3, and 6.3. General Statistical Procedures Since all of the random numbers were drawn from the same population, the expected values for the means of the cells, rows, and columns are identical for all samples. Any differences that may occur between the cells, rows, or columns are totally attributable to sampling variability or to bias inherent in the particular method of analysis involved. the proportion of P values expected to exceed certain values when orthogonal data are subjected to the standard analysis of variance techniques. It is to be expected that empirically determined sample values of these proportions (p) will randomly deviate from the population parameters (P). Statistical procedures are necessary to determine if the observed deviation exceeds that expected by chance. The degree of variation or deviation to be expected is indicated by the magnitude of the standard error of a proportion. The formula for the standard error of a proportion is The theoretical central F-distribution describes where P represents the population proportion, Q = (1-P)/ and n represents the sample size (17:175). If an infinite number of samples were drawn from the above distribution, 40 95 per cent of the resulting values for p (sample proportion) would fall in the interval with limits of P i 1.96SEp. If one sample is selected there is a 95 per cent probability that the obtained proportion would lie in this interval. Assuming a valid null hypothesis (the empirical F-distributions do not differ from the corresponding F-distributions at the specified points) a 95 per cent confidence interval was determined for each of the following population proportions: P = .10, P = .05, P = .025, P = .01, P =.005, and P = .001. In a particular instance, if the value of p is computed to be a number falling within the corresponding confidence interval, the null hypothesis will be retained as plausible. Conversely, an obtained value falling outside the confidence interval will result in a rejection of the null hypothesis as an adequate description of the relationship existing between the two distributions in question. It appears that the above procedure would be sufficient if a single value of p were involved since the probability of making a Type I error would be identical to the adopted .05 level of significance. However, when several tests are to be made the probability of rejecting a true null hypothesis increases rapidly since chance is operative in each test. This increase is evident when considered in the context of the binomial theorem and the 41 expansion of the binomial (.05 + . 95)-^, where k indicates the number of tests performed. The nth term in the expansion of the binomial indicates the probability of making or committing (k - n + 1) Type I errors. The probability of at least one alpha error is given by the formula A' = 1 - (1 - A)k where A is the level of significance adopted for each individual test, A 1 is the probability of making at least one Type I error, and k is the number of tests to be performed (35:69). Thus with a large number of tests to be made, the probability approaches certainty that the study will yield some statistically significant results. For a .05 test-wise level of significance if k = 100, the experiment-wise level of significance soars to approximately .994 and the number of decisions that can potentially be incorrect due to Type I error can be relatively large. As indicated in the preceding subsections, forty- eight combinations of sample size and degree of dispro portionality were investigated in this study. Each combination was analyzed by six methods, and each method yielded three F-distributions: F-test for row effect, F-test for column effect, and F-test for the interaction between rows and columns. (No interaction data were obtained for the weighted squares of means method or for Kramer's modified weighted squares of means method.) 42 Finally, each F-distribution was dichotomized in six different ways. Thus 4,608 sample proportion values were tested for a significant deviation from the expected proportion based on the central F-distribution. A substantial number, 230, would be expected to reach statistical significance at the .05 level through the operation of chance alone. Obviously then, a few scattered statistically significant deviations may mean little more than that randomness is operative in the processes of gathering data. Isolated instances of a significant difference cannot be relied upon to denote an actual difference in the underlying distributions. In order to be conclusive, a consistency or a pattern of statistically significant differences must be evident. An effort was made to statistically ascertain the patterns of significant results and thus to add a measure of interpretability to these data. These procedures are described in the following subsections. Procedures Related to Hypotheses 1.1, 2.1, 3.1, 4.1, 5.1, and 6.1 The 768 distributions were partitioned into sixteen sets of forty-eight corresponding to mode of analysis and effect under consideration. One partition consists of the forty-eight distributions generated by the method of fitting constants for the row effect. Another is made up of the set of forty-eight distributions generated by the 43 method of fitting constants for the row effect. Another is made up of the set of forty-eight distributions generated by the fitting constants method for the column effect. A third partition consists of the forty-eight distributions generated by the method of fitting constants for the interaction effect. Three other partitions derive from each of the methods of unweighted means, Snedecor's method of expected numbers, and Tsao's method of expected equal frequencies. Yates' weighted squares of means method and Kramer's modified weighted squares of means method each provide only two sets of forty-eight distributions since no interaction data were collected for these methods. Hypothesis 1.1 relates the three partitions involving the method of fitting constants; hypothesis 2.1 relates to the two partitions involving the weighted squares of means method, hypothesis 3.1 relates to the three partitions involving the unweighted means method, hypothesis 4.1 relates to the three partitions involving Snedecor's method of expected numbers, hypothesis 5.1 relates to the two partitions involving Kramer's modified weighted squares of means method, and hypothesis 6.1 relates to the three partitions involving Tsao's method of expected cell frequencies. For each distribution within the partition, six tests were conducted to determine if the sample proportion of large F values deviated significantly from the parameter value. Although an isolated instance of a significant difference may not be taken too seriously, a series of deviations may be quite conclusive. It seems likely that if the distributions do differ, they will differ in more than one of the checkpoints. Under the null hypothesis, for a given distribution specified in advance, it is unlikely that all six of the tests will be statistically significant. Employing the binomial theorem, such a "coincidence" would occur by chance approximately once in sixty million trials. The probability that at least five of the six tests would be significant by chance is approximately .000002. Table 6 gives the probabilities associated with the occurrence of zero, one, two, three, and four significant results from a group of six tests. For the purposes of this study the following decision rules were followed: (1) whenever at least four of the six tests proved to be statistically significant, it was concluded that the upper regions of the two distributions are not coincident and that the proportion of F values that exceed the points in question do differ, (2) whenever two or three of the six tests are significant, the null hypothesis was tentatively rejected but further corrobora tion of the discrepancies was required from the pattern of deviant distributions before the decision was finalized. It is important to be aware of the actual partition- wise level of significance regarding the decisions to be 45 TABLE 6 PROBABILITIES BASED ON THE BINOMIAL DISTRIBUTION Total Tests Number of Performed Number of Signi ficant Results Probability of Occur rence if ol = .05 0 .7738 1 . 2036 5 2 .0214 3 .0011 4 .0000 0 .7351 1 .2321 2 .0305 6 3 .0021 4 .0001 5 .0000 0 .5987 1 .3151 2 .0746 10 3 .0105 4 .0010 5 .0001 6 .0000 SOURCE: William H. Beyer (ed.), Handbook of Tables for Probability and Statistics (Cleveland, Ohio: The Chemical Rubber Co., 1966), pp. 41-43. / 46 made. For decision rule (1) the probability of obtaining by chance at least four significant tests out of six trials was .0001. Since forty-eight distributions are involved in each partition, the partition-wise alpha level resulting from the use of decision rule (1) is given by A' = 1 - (.9999)48 or A' = .0046. This probability of a Type I error is easily tolerated and decision rule (1) can be applied with confidence. However, decision rule (2) results in a partition-wise alpha level of A' = .78. Thus the proba bility of making a Type I error when decision rule (2) is applied necessitates further corroboration before one can confidently reject the null hypothesis. Further confirmation for differing distributions was given by an observation that an unusual number of significant differences within a particular level of disproportionality occurred. Each level of dispropor- tionality consists of five sample sizes and consequently thirty checkpoints for proportions. It seems reasonable to assume that for a given level of disproportionality whenever bias exists for one sample size, it would tend to exist for the other sample sizes. Table 7 indicates that for a group of thirty tests with the alpha level set at .05, six or more significant results will occur by chance less than one time out of a hundred. Decision rule (3) was that whenever six or more significant deviations occurred 47 TABLE 7 PROBABILITIES BASED ON THE POISSON DISTRIBUTION Total Number of Number of Signi- Probability of Occur- 'Tests Performed ficant Results rence if 0L= .05 0 .2231 1 .3347 2 .2510 3 .1255 30 4 .0471 5 .0141 6 .0035 7 .0008 8 .0001 9 .0000 3 .0002 4 .0006 5 .0019 6 .0048 7 .0104 8 .0194 9 .0324 10 .0486 11 .0663 12 .0829 13 .1956 14 .1024 300 15 .1024 16 .0960 17 .0847 18 .0706 19 .0557 20 .0418 21 .0299 22 .0204 23 .0133 24 .0083 25 .0050 26 .0029 27 .0016 28 .0009 29 - .0004 30 .0002 SOURCE: William H. Beyer (ed.), Handbook of Tables for Probability and Statistics (Cleveland, Ohio: The Chemical Rubber Co., 1966), pp. 41-43. 48 within one level of disproportionality, the null hypothesis that the proportions do not differ was rejected. The partition-wise alpha level resulting from the use of decision rule (3) is approximately .05. Also, by decision rule (2), each of the five distributions within a particular level of disproportionality had a probability of rejection of .03. In a group of five distributions the probability that the two will deviate significantly by chance is approximately .01. Decision rule (4) was that whenever at least two of the five distributions within a particular level of disproportionality deviated from the theoretical distributions, the null hypothesis that the proportions do not differ was rejected. The partition-wise alpha level resulting from the use of decision rule (4) is approximately .10. Finally, the partition as a whole was examined for a greater-than-chance number of significant proportions. Each partition consisted of 288 tests for proportions so that the expected number of proportions significant at the .05 level was fifteen. As indicated in Table 7, a Poisson distribution with N = 300 and P = .05 yields a standard deviation of approximately 3.9. Therefore, the 95 per cent confidence interval for the number of significant proportions ranges from seven through twenty-three. If a particular partition yielded more than twenty-three significant proportions, it was concluded that something 49 other than chance was operative and that a further examina tion of the data was warranted. Procedures Related to Hypotheses 1.2, 2.2, 3.2, 4.2, 5.2, and 6.2 It seems possible that the empirical distributions may differ from the theoretical ones in the proportion of large F values above some of the six checkpoints but not others. For example, it may be that the proportion of F values which exceed F is comparable for the two distributions but that the proportions above F differ grossly. The hypotheses of this subsection relate to the examination of these possibilities. Each sample size consists of ten levels of dispro portionality. Obviously, there are ten tests concerning the proportion exceeding each of the six specified values on the central F-distribution. Under the null hypothesis, three of the ten proportions above a specific point will differ from the corresponding central F values approxi mately four times in 1000 trials. Decision rule (5) was that whenever three of the ten proportions exceeded a particular distribution point, the null hypothesis was rejected. The partition-wise alpha level resulting from the use of decision rule (5) was computed to be approxi mately .056. The consistency of the decision to reject or to retain the null hypothesis was explored both within the sample size for adjacent checkpoints and between adjacent 50 sample sizes within levels of disproportionality. Procedures Related to Hypotheses 1.3, 2.3, 3.3, 4.3, 5.3, and 6.3 The comments which were made in the preceding subsection regarding the possibility that a pair of distributions may differ at some checkpoints but not others, are applicable here. The objective of the hypotheses of this subsection relate to the comparability of the distributions within a particular level of dispropor tionality. It seems plausible that for a given level of disproportionality, sampling distributions which vary only in sample size would tend to show some similarity in the pattern of deviations from the theoretical distributions. Thus if there is a bias at the .05 level for sample size twenty-four, a similar condition may exist for sample size forty-eight. Since each level of disproportionality consists of five sample sizes, there are five sample proportions corresponding to each checkpoint. Under the null hypo thesis, three of the five tests would reach significance by chance approximately once in a thousand cases. Therefore, decision rule (6) was that whenever three of the five tests reached statistical significance, the null hypothesis of coincident distributions was rejected. There was a total of fifty decisions to be made according to the above rule so that the partition-wise alpha level is approximately .05. As with the other conclusions, supporting relationships derived from adjacent sample sizes and degrees of disproportionality were explored. CHAPTER IV FINDINGS The purpose of this chapter was to present the findings of the study and to make an assessment of the hypotheses relative to the obtained data. In presenting these results, the following organization has been employed: 1. Findings related to the method of fitting constants. 2. Findings related to the method of the weighted squares of means. 3. Findings related to the method of unweighted means. 4. Findings related to Snedecor1s method of expected numbers. 5. Findings related to Tsao's method of expected equal frequencies. 6. Findings related to the method of Kramer's modified weighted squares of means. Findings Related to the Method of Fitting Constants Hypothesis 1.1 The underlying objective of this hypothesis was to determine whether the obtained sample proportions of F 52 53 values exceeding certain critical points were significantly different from the expected proportions based on the null hypothesis that the corresponding distributions were coincident. In adherence to the procedures which were outlined in Chapter II, 95 per cent confidence intervals were determined for the samples whose proportion parameters are .10, .05, .025, .01, .005, and .001. For comparative purposes, the 99 per cent confidence intervals were also computed. Table 8 specifies the limits of these sets of confidence intervals. To accomplish the stated purpose, each sample proportion was observed in relation to the corresponding confidence interval. If the span of the interval encompassed the obtained values, the null hypothesis was retained as a plausible description of the distributions in question. If the value of p (sample proportion) deviated from P (population parameter) the null hypothesis was tentatively rejected as inadequate. The results of the 144 x 6 or 864 statistical tests for the method of fitting constants are portrayed in Table 9. In order to facilitate the interpretation of Table 9 and the succeeding tables, the following features are emphasized: (1) the dimension of sample size is ordered vertically, (2) subclass patterns are ordered from left to right according to the degree of dispropor tionality present, and (3) the six spaces within each combination of sample size and subclass pattern are TABLE 8 CONFIDENCE INTERVALS FOR SELECTED PROPORTIONS Parameter Value 95 Per Cent Confidence Interval 99 Per Cent Confidence Interval P = o 1 — i « .0869 4 -4 P = .1131 .0827 < p = .1173 P = .05 .0404 4 P = .0596 .0374 p = .0626 P = .025 .0181 < P = .0319 .0160 < p = .0340 P = .01 .0057 « £ . 1 1 ^ .0143 .0043 < p = .1057 P = .005 .0019 4 < p = .0081 .0009 4 p = .0091 P = .001 .0000 c p = .0024 .0000 4 p = .0028 i on 55 ordered according to the checkpoints Fggi ^ 95> ^ 975' F gg1 ^.9951 and F ggg- (That is, the first space corresponds to the test at F^gg1 and the sixth space corresponds to the test for F ggg.) Row effects. As indicated in Table 9, of the 288 proportion tests for row effects, seven showed statistical significance at the .05 level. This number is less than that expected by chance but within the interval of high likelihood. Only one of the forty-eight distributions differed at more than one point and none differed in more than three of the six checkpoints. Pattern F, with a sample size of ninety-six (PpSgg), differed at three points: F .975‘ F gg . and F_gg5. As indicated by decision rule (2), such an occurrence is not conclusive due to the large number of distributions under consideration. Sup portive evidence must be sought from independent distribu tions which are comparable as to sample size and degree of disproportionality. Table 9 reveals that within pattern F for other sample sizes, there are no additional instances of significance. Further, it shows that within sample size ninety-six for neighboring levels of disproportionality, there are no significant deviations. A consideration of the above conditions led to the retention of the null hypothesis 1.1 as it relates to the distributions of F values for row effects. TABLE 9 SUMMARY OF F-TEST BIAS FOR THE METHOD OF FITTING CONSTANTS Pattern of Subclass Frequencies Sample F-Test Size A B C D E F G H I J 12 :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: 24 :::::: :::::P :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: Row 48 :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: 96 :::::: :::::: :N:::: :P:::: :::::: ::PPP: :::::: :::::: :::::: :::::: "1 Q O • « • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • _ _ _ _ _ _ • • • • • « _ _ _ _ _ _ • • • • • • j. y m • • • • • • • • • • • * • • • • • • • • • • • * • • • • • • • • • • » • • • • • • • • • • • • • 12 :::::: :::P:: :::::: ::N::: :::::: :::::: :::::: :::::: :::::: :::::P 24 :::::: :::::: :::P:: :::::: :::::: :::::: :::::: :::::: :::::: :::::: Column 48 :::::: :::::: :::P:: :::::: :::::: :::::: :::::: :::::: :::::: :::::: 96 :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: ::::P: :::::: 192 :::PP: :::::: :::::: :::::: :::::: ------ ::N::: ------ :::::: 12 :::::: :::::: :::::: :::::: :::::: :::::: :::::: N::::: :::::: :::::: 24 :::::: :::::: :::::: NN:::: :::::: :P:::: :::N:: :::::: :::::: P:P::: Inter- 48 :::::: :::::: :::::: ::P::: :::::: :::::: :::::: :::::: :::::P P::::: action 96 :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: P::PP: 192 :::::: :::::: :::N:: :::::: :::::P P:::::-------::::::-------PP: ::: KEY: P represents a positive bias significant at the .05 level. N represents a negative bias significant at the .05 level. : represents no significant difference. indicates no data obtained. NOTE: The six spaces in each combination of pattern and size represent the significance tests at the points: F.go' ^ 95' ^.975' F^gg< ^,995' E\ggg. ( J 1 ov 57 Column effects. Table 9 reveals that of the 288 proportion tests performed under column effects, ten reached statistical significance. This number is well within the interval of high likelihood with a valid null hypothesis. Only one of the distributions showed more than one deviation in the six tested. (pattern A for sample size 192) deviated in two points, but this is certainly a result of sampling variation since pattern A is made up of equal cell frequencies whose sampling distribu tions of the F statistic are known to be the central F distributions. The null hypothesis 1.1, as it relates to the column effect, was acceptable as a result of these findings. Interaction effects. As indicated in Table 9, nineteen of the 288 tests for proportions under interaction effects reached statistical significance. This number exceeds the expected number of 14.4 by a small amount but it is well within the region of high likelihood. The scattering of significant differences in subclass patterns A through I seems highly consistent with expectation as a result of chance under a true null hypothesis. However, within pattern J, the aggregate of significant deviations are not easily attributable to chance. With an alpha level of .05, eight significant differences from a group of thirty is an atypical occurrence. Based on the Poisson 58 distribution as an approximation of the binomial theorem (30), such an event has a probability of approximately .0002. The null hypothesis 1.1 was rejected for the following tests: PJS24 for F 99' anc^ F 975' PJ^48 ^or F .901 PJS96 for f.90' f.991 and F>995. and PjS192 for F>90 and ^,95- The null hypothesis was accepted as it relates to the other distributions of F values for interaction ef fects . Hypothesis 1.2 Row effects. The rows corresponding to sample sizes 12, 24, 48, and 192 contained at most one instance of a significant deviation out of the sixty proportion tests conducted at each level of sample size for row effects. Obviously, no pattern of significant results appeared within these rows. The row corresponding to sample size ninety-six contained five significant differences. Two of the significant deviations occurred at F 9^, and one each at F 975# F 99, and F 995. Since random effects offers a plausible explanation for such an aggregate of differences, the null hypothesis 1.2 was retained as it relates to the distributions of F values for row effects. Column effects. Table 9 indicates that one of the sixty proportion tests performed in each row corresponding to sample sizes of twenty-four, forty-eight, and ninety-six reached statistical significance. The row corresponding to 59 a sample size of twelve produced three significant deviations but no more than one for any of the six points under consideration. The row corresponding to a sample size of 192 yielded four differences with only two of these, PB and PH, deviating at any one checkpoint. (Also, pattern B consists of proportional frequencies whose F statistics are known to be distributed as the central F-distributions.) Consequently, the null hypothesis 1.2 was accepted as it relates to the distribution of F values for column effects. Interaction effects. Table 9 portrays the 19 obtained significantly different proportions which were obtained from the F-distributions for interaction effects. As many as six deviations occurred within one sample size but in no case did more than two differences occur corresponding to the same checkpoint. Consequently, the null hypothesis 1.2 was accepted as it relates to the distribution of F values for interaction effects. Hypothe sis 1.3 Row effects. Table 9 displays the fact that the seven obtained significantly different proportions are rather widely scattered over the various levels of disproportionality and checkpoints under consideration. The null hypothesis 1.3 was accepted as it relates to the distribution of F values for row effects. 60 Column effects. The ten significantly different proportions were again widely scattered over the levels of disproportionality and checkpoints of interest. In only three cases did as many as two instances of a significant difference occur within the same subclass pattern, and only once did the two deviations coincide as to the checkpoint involved. The null hypothesis 1.3 was accepted as it relates to the distributions of F values for column effects. Interaction effects. None of the subclass patterns yielded either a large number of differences or a consistent pattern in the occurrence of the differences except pattern J. Table 9 portrays the fact that there were eight deviations from the central F-distribution within pattern J, four of which were concentrated at the checkpoint F. 90' The low probability of such an event resulted in the rejection of the null hypothesis 1.3 for pattern J at the point F go- The four other deviations within this pattern give added assurance that the empirical F-distributions reveal an actual difference in the underlying distributions rather than a chance event. The null hypothesis 1.3 for the other levels of disproportionality and distribution points was accepted as it relates to the distributions of F values for interaction effects. 61 Findings Related to the Method of the Weighted Squares of Means Hypothe sis 2.1 Row effects. As indicated in Table 10, eleven of the 288 proportion tests for row effects revealed a statistical significance at the .05 level. This is less than the number to be expected by chance under the null hypothesis but well within the interval of high likelihood. Only one of the forty-eight distributions, PpSgg1 showed a discrepancy at more than one of the six points tested, and it differed for two of the points— F.975» anc^ F .99* Supportive evidence was lacking, however, when other distributions were observed within pattern F or within sample sizes of ninety-six in adjacent subclass patterns. Therefore, the eleven observed instances of statistical significance are deemed as attributable to random effects. The null hypothesis 2.1 was accepted as it relates to the distributions of F values for row effects. Column effects. Table 10 discloses that fifteen of the 288 proportion tests under column effects reached statistical significance. Again this number approximates the result which would be expected due to chance vari ability in the selection of the sets of random numbers. No more than two points of any distribution reached statis tical significance at the .05 level, and in no case was TABLE 10 SUMMARY OF F-TEST BIAS FOR THE METHOD OF WEIGHTED SQUARES OF MEANS Pattern of Subclass Frequencies F-test Sample Size A B C D E F G H I J 12 24 48 96 Row 192 12 24 Column 48 96 192 :::PP: :::::: :::::: :::::: :::::: :::::: ------ ::N::: ------ KEY: P represents a positive bias significant at the .05 level. N represents a negative bias significant at the .05 level. : represents no significant difference. indicates no data obtained. NOTE: The six spaces in each combination of pattern and size represent the significance tests at the points: F 901 F 951 F 9751 F 991 F 995' F 999* t o 63 corroborative data found in the adjacent distributions. The null hypothesis 2.1 was accepted as it relates to the distributions of F values for column effects. Hypothesis 2.2 Row effects. Of the eleven statistical differences for row effects, six were scattered widely over four of the five rows representing sample size. Although five dif ferences appeared in the row corresponding to sample size ninety-six, no more than two involved the same point of the F-distribution. The null hypothesis 2.2 was accepted as it relates to the distributions of F values for row effects. Column effects. Table 10 displays the fact that the fifteen statistical differences were dispersed throughout five sample sizes and six distribution points. No more than two differences occur within the same row and involving the same distribution point. Consequently, the null hypothesis 2.2 was accepted as it relates to the distributions of F values for column effects. Hypothesis 2.3 Row effects. The eleven statistical differences are located so that no more than two differences exist within the same row and involving the same distribution point. The null hypothesis 2.3 was accepted as it relates to the distributions of F values for row effects. 64 Column effects. The fifteen statistical differences extend over eight of the ten subclass and five of the distribution points. No more than two of the differences appeared in the same sample size involving the same distribution point. The null hypothesis 2.3 was accepted as it relates to the distributions of F values for column effects. Findings Related to the Method of Unweighted Means ' Hypothesis 3.1 Row effects. A comparison of Table 10 with Table 11 discloses that the identical pattern of significant proportions was produced by the weighted squares of means and the unweighted means for row and column effects. Consequently, the decisions that were made for the two methods are identical. The null hypothesis 3.1 was accepted as it relates to the distributions of F values for row effects. Further data concerning the output from these two methods is given in Tables 12, 13, and 14. Table 12 presents the numerical values of the sample proportions exceeding F^gg f°r t*1® row, column, and interaction effects. For the row and column effects, the obtained values seem to indicate a random variation about the parameter value of .10. Only one of the forty-eight values TABLE 11 SUMMARY OF F-TEST BIAS FOR THE METHOD OF UNWEIGHTED MEANS Pattern of Subclass Frequencies F-Test Sample I Size A B C D E F G H I J 12 24 48 96 192 T O .....P * • • • Kf...... Row 24 48 96 192 12 24 48 96 192 Column . . .pp. •. KTKT..... Inter action KEY: P N represents represents a positive bias a negative bias significant at significant at the .05 level, the .05 level. : represents no significant difference. indicates no data obtained. NOTE: The six spaces in each combination of pattern and size represent the significance tests at the points: F^gg1 F 951 F 9751 F.99‘ F 9951 F 999 cn TABLE 12 PROPORTION OF F VALUES GREATER THAN F gQ FOR UNWEIGHTED MEANS METHOD Sample Size Pattern of Subclass Frequencies F-Test A B C D E F G H I J 12 .091 .104 .094 .094 .096 .094 .108 .096 .103 .106 24 .107 .090 .100 .091 .098 .092 .110 .096 .094 .090 Row 48 .099 .098 .088 .105 .108 .100 .110 .097 .109 .102 96 .095 .095 .084* .110 .104 .104 .098 .090 .089 .096 192 .091 .094 .090 .091 .105 .110 ---- .090 ---- .098 12 .094 .106 .101 .096 .101 .104 .104 .094 .094 .106 24 .104 .104 .104 .090 .097 .098 .108 .106 .114* .092 Column 48 .089 .098 .102 .097 .104 .105 .104 .108 .100 .098 96 .088 .109 .102 .099 .095 .100 .100 .104 .099 .113 192 .102 .102 .103 .098 .094 .090 ---- .088 ---- .096 12 .097 .111 .094 .098 .088 .108 .110 .086* .104 .108 24 .100 .094 .096 .085* .102 .112 .098 .100 .106 .114* Inter 48 .102 .093 .096 .108 .104 .098 .104 .099 .104 .117* action 96 .092 .098 .100 .099 .102 .105 .099 .100 .100 .122* 192 .094 . 100 .092 .102 .106 .114* ---- .101 ---- .120* KEY: * represents a significant difference at the .05 level. indicates no data obtained. NOTE: The 95% confidence interval for p ranges from .087 to .113 inclusively. as as TABLE 13 PROPORTION OF F VALUES GREATER THAN F g5 FOR UNWEIGHTED MEANS METHOD Sample Size Pattern of Subclass Frequencies F-Test A B C D E F G H I J 12 .048 .050 .046 .045 .050 .048 .060 .046 .052 .043 24 .050 .042 .050 .044 .044 .050 .052 .050 .048 .046 Row 48 .050 .050 .044 .050 .052 .049 .057 .052 .050 .052 96 .050 .048 .040 .061* .050 .058 .047 .048 .046 .054 192 .045 .048 .044 .046 .051 .050 ---- .047 ---- .050 12 .051 .055 .044 .040 .050 .060 .058 .048 .054 .051 24 .044 .058 .057 .044 .048 .054 .046 .058 .058 .050 Column 48 .041 .044 .056 .056 .054 .045 .054 .054 .054 .052 96 .045 .058 .053 .052 .048 .051 .048 .056 .054 .054 192 .059 .058 .052 .048 .045 .044 ---- .040 ---- .047 12 .045 .053 .046 .051 .044 .055 .056 .052 .052 .052 24 .049 .051 .049 .038* .054 .064* .050 .046 .053 .060 Inter 48 .050 .042 .046 .057 .056 .052 .050 .052 .053 .058 action 96 .044 .047 .054 .045 .055 .051 .050 .050 .050 .056 192 .048 .050 .042 .049 .055 .056 ---- .052 ---- .068* KEY: * represents a significant difference at the .05 level. indicates no data obtained. NOTE: The 95% confidence interval for p ranges from .0404 to .0596 inclusively. < T \ TABLE 14 PROPORTION OF F VALUES GREATER THAN F gg FOR UNWEIGHTED MEANS METHOD F-test Sample Size Pattern of Subclass Frequencies A B C D E F G H I J 12 .012 .008 .010 .008 .008 .010 .010 .010 .009 .010 24 .010 .006 .007 .008 .010 .008 .009 .014 .011 .012 Row 48 .009 .010 .008 .008 .010 .010 .013 .016* .010 .010 96 .012 .015* .008 .012 .009 .016* .008 .012 .010 .010 192 .008 .012 .008 .006 .007 .010 ---- .008 ---- .009 [ 12 .014 .012 .010 .008 .012 .016* .012 .010 .009 .012 24 .010 .014 .010 .008 .009 .011 .009 .012 .015* .009 Column 48 .007 .007 .016* .011 .008 .010 .010 .010 .011 .008 96 .008 .016* .008 .011 .008 .008 .008 .012 .013 .010 192 .016* .012 .008 .008 .007 .009 ---- .008 ---- .010 12 .010 .010 .008 .008 .007 .011 .012 .008 .012 .014 24 .014 .012 .009 .010 .012 .014 .005* .008 .012 .014 Inter 48 .008 .008 .006 .013 .014 .011 .012 .014 .012 .010 action 96 .010 .010 .012 .008 .012 .008 .008 .012 .012 .015* 192 .010 .010 .004* .012 .013 .010 ---- .012 ---- .012 KEY: * represents a significant difference at the .05 level. indicates no data obtained. NOTE: The 95% confidence interval for p ranges from .0057 to .0143 inclusively. C F i oo 69 for the row effects fell outside the 95 per cent confidence interval of .087 to .113 although several sample propor tions approached significance. The obtained proportions under column effects also exhibited a single instance of significance. Table 13 gives data of a similar nature concerning the proportions exceeding ^,95 for the method of unweighted means (and consequently for the weighted squares of means). Again, random variability under a valid null hypothesis offers a plausible explanation of the results. Table 14 displays the numerical values of the sample proportions exceeding F 99 for the unweighted means analysis. No trend was discernible relating the conditions of sample size or disproportionality and the magnitude of the sample proportions for row or column effects. Column effects. The null hypothesis 3.1 was accepted as it relates to the distributions of F values for column effects. Interaction effects. A comparison of Table 9 with Table 11 reveals that the identical pattern of significant proportions was produced by the method of fitting constants and the method of unweighted means for interaction effects. Consequently, the decisions that were made for the fitting constants method are repeated here. The null hypothesis 70 3.1 was rejected for the following tests: pj^24 f°r F.90 and F_975/ PjS48 for F>90, pjS96 for F.90' F.99' and F.995» and PJS192 for and F^gg. The null hypothesis was accepted as it relates to the other distributions of F values for interaction effects. A consideration of the magnitude of the sample proportions within pattern J for interaction in Table 12 seems to confirm the rejection of the null hypothesis for the indicated conditions. Of the five sample proportions involved, two fall outside the 99 per cent confidence interval and two fall outside the 95 per cent confidence interval. Hypothe sis 3.2 Row effects. The null hypothesis 3.2 was accepted as it relates to the distributions of F values for row effects. Column effects. The null hypothesis 3.2 was accepted as it relates to the distributions of F values for column effects. Interaction effects. The null hypothesis 3.2 was accepted as it relates to the distributions of F values for interaction effects. Hypothesis 3.3 Row effects. The null hypothesis 3.3 was accepted as it relates to the distributions of F values for row effects. Column effects. The null hypothesis 3.3 was accepted as it relates to the distributions of F values for column effects. Interaction effects. The null hypothesis 3.3 was rejected for pattern J at the point was accepted for all other levels of disproportionality and distribution points. A slight bias seems possible within pattern J at F gg. Table 12 reveals that of the five sample proportions obtained, one falls outside the 95 per cent confidence interval; two are borderline cases. Findings Related to Snedecor1s Method of Expected Numbers Hypothesis 4.1 Row effects. As indicated in Table 15, 153 of the 288 proportion tests for row effects showed statistical significance at the .05 level. Since the number to be expected under a true null hypothesis is 14.4 with a standard error of 3.9, the obtained results indicate that something other than chance was operating. Of the forty- eight distributions generated, eighteen differed in all six TABLE 15 SUMMARY OF F-TEST BIAS FOR SNEDECOR'S METHOD OF EXPECTED NUMBERS F-Test Sample Size Pattern of Subclass Frequencies A B C D E F G H I J 12 PPPPPP PPPPPP PPPPPP PPPPPP 24 P: : PPPP:: PPPPP: PPPPPP PPPPPP PPPPPP Row 48 PP::: PPP PP :PP::: PPPPPP PPPPPP PPPPPP PPPPPP 96 PPPPP PP: :P PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP 192 P • • • • PPP a a a a PPP::: -------- PPPPPP PPPPPP 12 PPPPP: PPPPP: PPPPP: PPPPPP PPPPPP PPPPPP 24 • • :::: PPPPP: » a • • • a • a a a P: : PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP Column 48 N :::: PPPPPP PPPPP PPP PPPPP: PPPPPP PPPPPP PPPPPP PPPPPP 96 • a :::: :PPP:P PPPPP : PP PPPPP: PPPPPP PPPPPP PPPPPP PPPPPP 192 : :PP: ■ P::: PPPPP: a P a a a • J - a a a : P: PPPPPP PPPPPP PPPPPP 12 PPPPP: NNNN:: PPPPPP PP:P :: PPPPPP 24 PPPPPP NNNNN: PPPPPP PPPPP: PPPPPP Inter 48 PPPPP: PPPPPP PPPPPP NNNNN: PPPPPP PPPPPP PPPPPP action 96 PPPP:P PPPPP: NNNN:: PPPPPP PPPP:: PPPPPP 192 ID a a a a a 'DTD'D’ D'DTD PPPPPP PPPPPP PPPPPP JrJrirlrJrJr KEY: P represents a positive bias significant at the .05 level. N represents a negative bias significant at the .05 level. : represents no significant difference. indicates no data obtained. NOTE: The six spaces in each combination of pattern and size represent the significance tests at the points: F g g 1 ^95 ■ F^gygi F^ggi ^9951 f 999. 'j t o 73 checkpoints and twenty-seven differed in at least three points. A trend relating the number of statistical differ ences and the degree of disproportionality is evident. As the departure from proportionality increases, the number of differences increases. Employing decision rule (3), the deviations in patterns, D, E, F, G, H, I, and J are deemed to reflect actual differences in the underlying distributions. Applying decision rule (2), the distribu tion Pcs24 judged to represent a true discrepancy between the empirical and the theoretical F-distributions. The null hypothesis 4.1 was rejected for the conditions indicated by the symbol P in subclass patterns C, D, E, F, G, H, I, and J under row effects in Table 15. Tables 15, 17, and 18 present the numerical values for the obtained proportions exceeding F 9q, F 95, and F^gg respectively. The magnitude of the discrepancy increases as the cell frequencies depart from propor tionality. In Table 15, for row effects, subclass patterns G, H, I, and J result in a positive bias as great as .12 or 120 per cent above the expected proportion of .10. Similarly, in Table 17, the proportion of F values exceeding F>95 ranges from a modest .04-.08 for patterns D, E, and F, to .09-.14 for patterns G, H, I, and J. The same tendency is observed in Table 18, where the proportion of F values exceeding P.99 maY as large as .057. TABLE 16 PROPORTION OF F VALUES GREATER THAN F>90 FOR EXPECTED NUMBERS METHOD F-test Sample Size Pattern of Subclass Frequencies A B C D E F G H I J 12 .090 .109 .108 .110 .110 .118* .156* .145* .184* .210* 24 .107 .097 .119* .112 .120* .116* .164* .154* .184* .217* Row 48 .099 .095 .110 .118* .124* .113 .170* .148* .202* .220* 96 .095 .093 .098 .128* .128* .128* .162* .146* .180* .216* 192 .091 .092 .106 .114* .122* .129* ---- .149* ---- .212* 12 .094 .106 .115* .110 .114* .138* .152* .156* .172* .212* 24 .104 .102 .114* .110 .116* .132* .152* .172* .202* .220* Column 48 .089 .086* .114* .120* .120* .145* .156* .182* .197* .220* 96 .088 .100 .112 .126* .108 .142* .142* .166* .197* .242* 192 .102 .102 .118* .112 .108 .129* ---- .156* ---- .226* 12 .097 .111 .093 .108 .100 .140* .065* .144* .121* .214* 24 .100 .094 .094 .098 .126* .149* .056* .160* .121* .222* Inter 48 .101 .093 .096 .120* .125* .136* .054* .174* .128* .230* action 96 .092 .098 .098 .114* .124* .138* .053* .170* .121* .226* 192 .094 .100 .090 .116* .132* .154* ---- .172* ---- .139* KEY: * represents a significant difference at the .05 level. indicates no data obtained. NOTE: The 95% confidence interval for p ranges from .087 to .113 inclusively. TABLE 17 PROPORTION OF F VALUES GREATER THAN F 95 FOR EXPECTED NUMBERS METHOD F-test Sample Size Pattern of Subclass Frequencies A B C D E F G H I J 12 .048 .054 .054 .055 .061* .060 .092* .080* .114* .130* 24 .050 .044 .063* .054 .058 .066* .098* .087* .115* .128* Row 48 .050 .044 .051 .064* .065* .063* .092* .087* .128* .142* 96 .050 .048 .048 .075* .065* .077* .090* .082* .108* .136* 192 .045 .054 .056 .060 .070* .066* ---- .090* ---- .138* 12 .051 .054 .053 .050 .063* .081* .091* .086* .102* .130* 24 .044 .054 .062* .056* .060 .076* .082* .100* .127* .138* Column 48 .041 .047 .062* .073* .068* .076* .092* .106* .132* .137* 96 .045 .053 .061* .066* .062* .077* .089* .104* .127* .164* 192 .059 .056 .068* .067* .064* .068* ---- .089* ---- .142* 12 .045 .053 .046 .057 .052 .076* .034* .076* .061* .134* 24 .049 .051 .046 .046 .066* .085* .020* .098* .064* .146* Inter 48 .050 .042 .046 .066* .066* .078* .023* .096* .065* .162* action 96 .044 .047 .052 .055 .066* .076* .019* .100* .063* .158* 192 .048 .050 .042 .058 .072* .092* ---- .101 ---- .168* KEY: * represents a significant difference at the .05 level. indicates no data obtained. NOTE: The 95% confidence interval for p ranges from .0404 to .0596 inclusively. TABLE 18 PROPORTION OF F VALUES GREATER THAN F^99 FOR EXPECTED NUMBERS METHOD F-test Sample Size Pattern of Subclass Frequencies A B C D E F G H I J 12 .012 .009 .012 .012 .012 .015* .023* .022* .028* .032* 24 .010 .010 .012 .012 .014 .017* .026* .030* .042* .044* Row 48 .009 .010 .008 .012 .014 .014 .026* .028* .043* .053* 96 .012 .015* .012 .016* .014 .024* .022* .029* .036* .057* 192 .008 .008 .010 .012 .014 .014 ---- .024* ---- .054* 12 .014 .014 .011 .008 .015* .022* .020* .022* .028* .038* 24 .010 .011 .015* .012 .014 .020* .022* .030* .042* .047* Column 48 .007 .007 .017* .016* .012 .018* .025* .033* .046* .054* 96 .008 .014 .015* .017* .014 .022* .026* .031* .046* .059* 192 .016* .010 .018* .010 .010 .018* ---- .023* ---- .052* 12 .010 .010 .008 .011 .010 .021* .005* .018* .015* .035* 24 .014 .012 .008 .013 .018* .023* .002* .026* .016* .056* Inter 48 .008 .008 .006 .016* .018* .022* .004* .032* .018* .059* action 96 .010 .010 .012 .010 .018* .018* .002* .034* .016* .059* 192 .010 .010 .004* .014 .020* .026* ---- .032* ---- .071* KEY: * represents a significant difference at the .05 level. indicates no data obtained. NOTE: The 95% confidence interval for p ranges from .0057 to .0143 inclusively. c r » 77 As stated in Chapter II, Snedecor's method of expected numbers has been recommended for situations in which the subclass frequencies do not significantly depart from proportionality as indicated by the chi square criterion. This situation exists for approximately 50 per cent of the conditions under consideration in this study when the .05 level of significance is selected for the chi square criterion. In Tables 15-18, these conditions are approximately tabled above the minor diagonal of the 5 x 10 matrix for each partition. The specific combinations of subclass pattern and sample size which satisfy the above requirement are as follows: subclass pattern A, all sample sizes; subclass pattern B, all sample sizes; P^S]^, PCS24» PCS48' pDS24» pDs12> peS12' pES24» pFs12/ PpS24' pGs12' PGS24' PHS12' and pls12* Tables 16-18 show that some of the above conditions result in sampling distribu tions which deviate markedly from the central F-distribu- tions. The proportion of F values greater than F go may be as high as .184, the proportion exceeding F g^ ranges from .041 to .114, and the proportion of F values above F gg may reach .028 and simultaneously satisfy the chi square criterion. Obviously, the utilization of the chi square criterion as a screening measure for the appli cability of Snedecor's method of expected numbers cannot be wholly relied upon in all cases. A cursory examination of Tables 16-18 seems to 78 indicate that the degree of bias that may result from the application of Snedecor1s method is not related to the sample size involved. Within each of the subclass patterns, the obtained proportions are strikingly homogeneous. Since the chi square criterion is directly related to the sample size, it is apparently inefficient as a guide for the applicability of Snedecor1s method of expected numbers. The measure of disproportionality employed in this study was the chi square value divided by the sample size. It appears that this measure of disproportionality could be more appropriately employed as a screening device than the recommended chi square test. Table 19 is a scattergram relating the two methods of indicating the degree of disproportionality to the proportion of F values exceeding F gQ. A perusal of this table makes it evident that the values bear a stronger relationship to the propor tions than do the chi square values. Table 20 presents the correlations relating the values of chi square and chi square divided by N to the proportion of F values exceeding the points F gg, F.95» f.97 5' f .99' f.995' and F.999- A comparison of the correlations involving these proportions clearly indicates that a stronger relationship exists for the y $ / N and proportion values than for the chi square and proportion values. This is especially true with reference to the main effects, where the median value for the correlation TABLE 19 SCATTERGRAM OF % 2 AND % 2/fi VERSUS THE PROPORTION EXCEEDING F 9Q FOR SNEDECOR'S METHOD x .22 .20 .18 o oo Ipgo o xxxo xoo o o °Px o F x X o °£x X X oxxxx X X “ “ E3fac X X X X X o X X X X °38£ X X X X £x X xx5 X X — O X X X _ _ .18— .16“ .iH . 12 .iqZ _ s 06 ".04 X 1 5 10 1 15 20 1 25 30 1 35 40 45 l 50 1 55 60 ! 65 70 75 80 85 I 1 .05 .10 .15 .20 .25 .30 .35 .40 .45 KEY: x = (a,b) where "a" is a value for %/fi and "b" is the proportion exceeding F.go- 0 = (a,b) where "a" is a value and "b" is the proportion exceeding F 9q. TABLE 20 DESCRIPTIVE DATA ON THE PROPORTIONS OF F VALUES EXCEEDING SIX CHECKPOINTS FOR SNEDECOR'S METHOD Checkpoint Means Standard Deviations Correlations with y}" Correlations with yJ"/H f.90 .1355 .0390 .54 .95 f.95 .0759 .0290 .57 .95 Row F.975 Effect -p .99 .0429 .0204 .0205 .0132 .60 .62 .93 .93 F. 995 .0122 .0090 .63 .90 p. 999 .0035 .0040 .60 .87 F. 90 .1395 .0415 .58 .97 F. 95 .0803 .0307 .59 .96 Column F.975 Effect p F. 99 .0461 .0221 .0212 .0134 .59 .60 .94 .92 F. 995 .0128 .0087 .59 .91 F. 999 .0034 .0036 . 66 .83 Chi Square 10.2757 15.5759 1.00 .60 Chi Square/N .1535 .1367 .60 1.00 oo o 81 between X2/N and proportion values is .93 while the median correlation between the chi square and proportion values is .60. Regression equations to estimate the actual level of significance for Snedecor1s F-tests for main effects were derived from the data on Table 20. No equations are given to estimate the actual alpha level for interaction effects because of the moderate correlation coefficients and the nature of the bias to fluctuate between positive and negative values. Table 21 presents the regression equations to estimate the proportion of F values exceeding the six distribution points under consideration. The dispropor tionality ratio of X3 /n functions as the independent variable. The regression equations to estimate the bias are readily obtained from the regression equation to estimate the actual alpha level by subtracting the nominal alpha level from the right member of each equation. The regression equations of Table 21 were employed to predict the actual level of significance for the disproportionality ratios which were used in this study. These estimates are given in Table 22 under the headings of patterns A through J. A comparison of the estimated values of Table 22 with the empirically obtained values of Tables 16-18 reveals that the uncertainty regarding the actual level of significance is severely attenuated. 82 TABLE 21 REGRESSION EQUATIONS TO ESTIMATE THE ACTUAL ALPHA LEVEL IN SNEDECOR'S P-TESTS FOR MAIN EFFECTS Nominal Alpha Level Regression Equation A = .10 > I I • to 00 (XVw - .15) + .1375 A = .05 "A" = .21 (y}/N - .15) + .0781 A = .025 "A" = .14 (X 2/f> - -15) + .0445 A = .01 "A" = .09 ( X V n - .15) + .0212 A = .005 "A" = .06 (X2 /W - .15) + .0125 A = .001 "A" = .03 ( y ? M “ .15) + .0034 KEY: "A" is the estimated level of significance %2 /N is a measure of the departure from proportional subclass frequencies TABLE 22 ESTIMATED LEVELS OF SIGNIFICANCE IN SNEDECOR'S F-TESTS FOR MAIN EFFECTS Nominal Alpha Level Subclass Patterns A B C D E F G H I J o 1 —1 • .094 .094 .110 .121 .124 .128 .137 .164 .178 .214 .05 .047 .047 .059 .068 .070 .072 .080 .099 .112 .139 .025 .024 .024 .032 .038 .039 .040 .046 .059 .067 .085 .01 .008 .008 .013 .017 .0018 .019 .022 .030 .036 .047 .005 .0035 .0035 .0071 .0095 .0101 .0107 .0131 .0185 .0221 .0299 .001 .0000 .0000 .0007 .0019 .0022 .0025 .0037 .0064 .0082 .0121 NOTE: These estimates were derived by employing the regression equations presented in Table 21. co U1 84 Column effects. Table 15 indicates that 183 of 288 proportion tests for column effects showed statistical significance at the .05 level. The pattern of significant differences generally corresponds with the pattern manifest for the row effects. Of the 48 empirical distributions, 20 deviated in all 6 of the checkpoints while 32 differed in at least 3 points. As in the data for row effects, a definite trend was evident. As the degree of dispropor tionality increases, the number of obtained deviations increases. A notable exception to this trend was presented by subclass pattern C which yielded a greater number of differences than did either pattern D or pattern E. Employing decision rule (3), the null hypothesis .4.1 was rejected for the conditions indicated by the symbol P in subclass patterns C, D, E, F, G, H, I, and J for column effects. Interaction effects. Table 15 indicates that 157 of the 288 proportion tests for interaction effects showed statistical significance at the .05 level. Approximately twenty tests revealed a negative bias and eighteen of these were located within subclass pattern G. Employing decision rule (3), the null hypothesis 4.1 was rejected for the conditions indicated by the symbols P and N in subclass patterns D, E, F, G, H, I, and J for interaction effects. 85 A perusal of Tables 16-18 reveals that the trend toward greater bias with greater disproportionality is not as evident in the interaction effects as it is for the row and column effects. Subclass pattern G produced a consistent yield of negative bias whereas the adjacent pattern resulted in substantial positive bias. The data from this study confirm the statement given by Gosslee and Lucas (13) that the bias for interaction effects is dependent upon the particular pattern of frequencies employed in the design. Hypothesis 4.2 Row effects. The null hypothesis 4.2 was rejected for all combinations of sample size and distribution point as it relates to the distributions of F values for row effects. Column effects. The null hypothesis 4.2 was rejected for all combinations of sample size and distribu tion point as it relates to the distributions of F values for column effects. Interaction effects. The null hypothesis 4.2 was rejected for all combinations of sample size and distribu tion point as it relates to the distributions of F values for interaction effects. 86 Hypothesis 4.3 Row effects. The null hypothesis 4.3 was rejected for the following conditions: subclass pattern D at F gg? pattern E at F go and F 95; pattern F at F q q, F 95/ F 975' and F gg; patterns G, H, I, and J for all six checkpoints. The null hypothesis was accepted for all other combinations of subclass pattern and distribution point for row effects. Column effects. The null hypothesis 4.3 was rejected for the following conditions: subclass pattern C at F^gg, F.95, F>975, F gg, and F 9957 pattern D at F>95; pattern E at F.go/ F>g5, and F>g75? pattern F at F>90, F>95, F>975, F 99, and Fggg; patterns G, H, I, and J for all six checkpoints. The null hypothesis was accepted for all other combinations of subclass pattern and distribution point for column effects. Interaction effects. The null hypothesis 4.3 was rejected for the following conditions: subclass pattern D at F.90; pattern E at F>90, F>95, F>975, F , and F^gg5? pattern F at all six checkpoints? pattern I at F gg, F 95, F 9-75, and F gg; pattern J at all six checkpoints. The null hypothesis was retained for all other combinations of subclass pattern and distribution point for interaction effects. 87 Findings Related to Tsao1s Method of Expected Egual Frequencies Hypothesis 5.1 Row effects. As indicated in Table 23, 236 of the 288 proportion tests for row effects showed statistical significance at the .05 level. All subclass patterns, with the exception of pattern A which consisted of equal cell sizes, show a greater number of differences than can be plausibly explained under the null hypothesis. Of the forty-eight empirical F-distributions, forty-two deviate in at least two points and all have confirmation from adjacent distributions that the discrepancies noted reflect a true difference in the underlying distributions. The null hypothesis 5.1 was rejected for the conditions indicated by the symbol P in subclass patterns B, C, D, E, F, G, H, I, and J for row effects. Table 24 gives the numerical values of the propor tions exceeding the F ^ q point of the F-distribution. There is a trend toward a greater bias in the F statistic as the disparity between cell frequencies increases. Also, a negative relationship seems to exist between sample size and degree of bias. The smaller samples result in a greater degree of bias. Table 25 displays the proportions of F values exceeding the F gg checkpoint. Again two trends were discernible: a positive relationship between magnitude of TABLE 23 SUMMARY OP F-TEST BIAS FOR TSAO'S METHOD OF EXPECTED EQUAL FREQUENCIES Pattern of Subclass Frequencies Sample --------------------------------------------------------------------------- F-test Size__^ __________£______£______5______ -______ ?______ ' L______ E______G 12 :::::P PPPP:: PPPP:: PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP 24 :::::: PPPPP: PP:::: PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP Row 48 :::::: PPPPPP PP:::: PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP 96 :::::: PP:::P PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP 192 :::::: PPPP:: PP:::: PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP--------------- 12 :::::: PPPPPP PPP::: PPPPPP- PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP 24 :::::: PP:P:: PPP:P: PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP Column 48 :::::: PPP::: PP:PP: PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP 96 :::::: :P:P:: PPPPP: PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP 192 :::PP: :P:::: PPPP:P PPPPPP PPPPPP PPPPP: PPPPPP PPPPPP--------------- 12 :::::: :PPP:: PPPP:: PPPPPP PPPPPP PPPPP: PPPPPP PPPPPP PPPPPP PPPPPP 24 :::::: PPPPP: P::::P PPPPPP PPPPPP PPPPP: PPPPPP PPPPPP PPPPPP PPPPPP Inter- 48 :::::: PPPPPP PPPPPP PPPPPP PPPPPP PPPP:: PPPPPP PPPPPP PPPPPP PPPPPP action 96 :::::: PPPPP: PP:::: PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP PPPPPP 192 :::::: PPPPPP PPPPPP PPPPPP PPPPPP PPPP:: PPPPPP PPPPPP------- ------ KEY: P represents a positive bias significant at the .05 level. : represents no significant difference. indicates no data obtained. NOTE: The six spaces in each combination of pattern and size represent the significance tests at the points: F.go1 F.95' F 975' ^.99' ^ 995' 999* 03 03 TABLE 24 PROPORTION OP F VALUES GREATER THAN F 9Q FOR TSAO'S METHOD Sample Size Pattern of Subclass Frequencies F-test A E D F H C B J I G 12 .090 .122* .134* .202* .205* .164* .236* .332* .357* .516* 24 .107 .122* .122* .160* .181* .154* .194* .253* .292* .346* Row 48 .099 .128* .124* .152* .178* .134* .192* .238* .265* .322* 96 .095 .129* .133* .171* .166* .118* .168* .221* .244* .286* 192 .091 .122* .117* .174* .165* .136* .184* .221* ---- ---- 12 .094 .128* .130* .209* .206* .167* .243* .321* .367* .493* 24 .104 .120* .116* .175* .208* .158* .198* .262* .314* .372* Column 48 .089 .120* .126* .182* .196* .142* .191* .238* .259* .312* 96 .088 .111 .124* .169* .175* .143* .184* .251* .266* .294* 192 .102 .111 .120* .160* .163* .147* .196* .232* ---- ---- 12 .097 .112 .131* .204* .202* .172* .240* .320* .376* .502* 24 .100 .130* .116* .191* .200* .152* .198* .277* .294* .370* Inter 48 .102 .126* .133* .174* .190* .144* .186* .246* .266* .325* action 96 .092 .127* .125* .170* .180* .140* .186* .240* .242* .290* 192 .094 .130* .130* .182* .174* .138* .192* .243* ---- ---- KEY: * represents a significant difference at the .05 level. indicates no data obtained. NOTE: The 95% confidence interval for p ranges from .087 to .113 inclusively. oo TABLE 25 PROPORTION OF F VALUES GREATER THAN F nc FOR TSAO1S METHOD . yo F-test Sample Size Pattern of Subclass Frequencies A E D F H C B J I G 12 .048 .068* .072* .121* .130* .090* .158* .224* .272* .419* 24 .050 .067* .065* .101* .120* .088* .119* .178* .210* .272* Row 48 .050 .072* .070* .104* .108* .071* .112* .160* .191* .240* 96 .050 .068* .080* .108* .098* .068* .110* .148* .166* .211* 192 .045 .070* .062* .106* .098* .074* .100* .146* ---- ---- - 12 .051 .075* .072* .126* .132* .096* .151* .230* .265* .394* 24 .044 . 066* .069* .112* .130* .088* .132* .182* .226* .285* Column 48 .041 .071* .068* .112* .128* .086* .117* .160* .184* .234* 96 .045 .062* .074* .110* .110* .082* .119* .160* .184* .208* 192 .059 .064* .066* .091* .098* .086* .118* .146* ---- ---- 12 .045 .063* .071* .128* .122* .096* .154* .229* .287* .418* 24 .049 .069* .056 .119* .122* .086* .122* .186* .220* .272* Inter 48 .050 .073* .080* .103* .118* .080* .112* .182* .176* .234* action 96 .044 .066* .068* .096* .110* .082* .116* .266* .164* .220* 192 .048 .072* .070* .115* .108* .074* .114* .172* ---- ---- KEY: * represents a significant difference at the .05 level. indicates no data obtained. NOTE: The 95% confidence interval for p ranges from .0404 to .0596 inclusively. o 91 bias and the degree of disproportionality in cell frequen cies, and a negative relationship between sample size and magnitude of bias. The numerical values of the obtained proportions exceeding P 95 for row effects ranged from .045 for P^192 to -419 for Pq^12' Again, Table 26 reveals a similar finding in the proportions of F values which exceed the F.gg distribution point. The obtained propor tions ranged from a minimum of .008 for P^Syg2 to a maximum of .245 for Pq^12’ T^e same trends observable from the previous tables are also present in Table 26. As stated in Chapter II, Tsao and Ferguson suggest that the method of expected equal frequencies could use fully be applied to situations which do not significantly depart from equal cell sizes as measured by the chi square criterion at the .01 level. Approximately 33 per cent of the conditions investigated in this study fulfill this requirement. The specific combinations of subclass pattern and sample size which do not depart significantly from equal cell sizes are as follows: subclass pattern A, all sample sizes; PES12, PES24, PES48, PDS12, PDS24, PpS12, PpS24, ^h^12' ^0*^12• 12• ^j^12* Tables 24—26 reveal that some of the above conditions have sampling distribu tions which deviate markedly from the central F-distribu- tion. The proportion of values greater than F ranged from .09 to .332; the proportion of values greater than F_g5 was as high as .23; and the proportion of values TABLE 26 PROPORTION OF F VALUES GREATER THAN F gg FOR TSAO’S METHOD F-test Sample Size Pattern of Subclass Frequencies A E D F H C B J I G 12 .012 .016* .016* .033* .042* .024* .048* .091* .142* .245* 24 .010 .016* .014 .037* .044* .023* .034* .068* .093* .152* Row 48 .009 .018* .014 .032* .038* .019* .038* .062* .084* .117* 96 .012 .013 .019* .041* .034* .019* .039* .064* .068* .106* 192 .008 .015* .014 .024* .032* .018* .037* .058* ---- ---- 12 .014 .017* .014 .044* .044* .022* .056* .095* .131* .232* 24 .010 .015* .014 .039* .046* .030* .049* .074* .105* .163* Column 48 .007 .011 .020* .030* .039* .025* .036* .063* .084* .116* 96 .008 .015* .017* .037* .039* .024* .045* .062* .074* .099* 192 .016* .010 .016* .027* .028* .022* .042* .053* ---- ---- 12 .010 .016* .020* .038* .032* .020* .051* .096* .140* .252* 24 .014 .022* .018* .044* .038* .025* .046* .086* .100* .151* Inter 48 .008 .020* .021* .035* .038* .020* .036* .079* .082* .116* action 96 .010 .016* .012 .027* .038* .028* .039* .068* .076* .108* 192 .010 .020* .020* .037* .036* .015* .040* .076* ---- ---- KEY: * represents a significant difference at the .05 level. indicates no data obtained. NOTE: The 95% confidence interval for p ranges from .0057 to .0143 inclusively. < x > t o 93 greater than F_gg reached .096 while simultaneously satisfying the recommended criterion. Obviously, this criterion for the applicability of Tsao's method of expected equal frequencies cannot be relied upon in all cases. Tables 24-26 seem to indicate that whenever the pattern of subclass frequencies depart from equality, Tsao's method of analysis will likely incur a measure of positive bias. When there is a substantial departure from equal cell sizes, the magnitude of the positive bias for larger sample sizes is less than for smaller sample sizes. Column effects. The configuration of statistical differences for column effects closely approximate that for row effects. Table 23 presents the information that thirty-three of the forty-eight generated distributions differed in all six checkpoints and forty-three differed in at least two points. The null hypothesis 5.1 was rejected for the conditions indicated by the symbol P in subclass patterns B, C, D, E, F, G, H, I, and J for column effects. Interaction effects. As indicated in Table 23, 237 of the 288 proportion tests for interaction effects reached statistical significance. Of the forty-eight empirical distributions, thirty-three differed in all six 94 checkpoints and forty-three differed in at least two points. The null hypothesis 5.1 was rejected for the conditions indicated by the sumbol P for all subclass patterns. Hypothe sis 5.2 Row effects. The null hypothesis 5.2 was rejected for all combinations of sample size and distribution point as it relates to the distributions of F values for row effects. Column effects. The null hypothesis 5.2 was rejected for all combinations of sample size and distribu tion point as it relates to the distribution of F values for column effects. Interaction effects. The null hypothesis 5.2 was rejected for all combinations of sample size and distribu tion point as it relates to the distribution of F values for interaction effects. Hypothesis 5.3 Row effects. The null hypothesis 5.3 was rejected for the following conditions: pattern E at F F 95' F.975, and F #gg; pattern D at F and F patterns F, H, C, B, J, I, and G for all six checkpoints. The null hypothesis was accepted for all other combinations of subclass pattern and distribution point for row effects. 95 Column effects. The null hypothesis 5.3 was rejected for all following conditions: pattern E at F^g, F.95» and F 99; pattern D at F^gQ, F>g5, F,975/ F>gg, and F>ggs; patterns F, H, C, B, J, I, and G for all six distribution points. The null hypothesis was accepted for all other combinations of subclass pattern and distribution point for column effects. Interaction effects. The null hypothesis 5.3 was rejected for the following conditions: pattern E for F 9q, F.95' F.975' f.99• and F .9957 pattern D for F_go, F.95* F.975' f.99' and F .999 * pattern C at F>g0, F>g5, F g75, F gg, and F 995; patterns F, H, B, J, I, and G for all six checkpoints. The null hypothesis was retained for all other combinations of subclass pattern and distribution point for interaction effects. Findings Related to Kramer's Modified Method of Weighted Squares of Means Hypothesis 6.1 Row effects. Table 27 shows that ten of the 288 proportion tests for row effects reached statistical significance. No more than one discrepancy was found in any distribution except PpSgg which produced three deviations. However, corroborative evidence from within pattern F or from the adjacent patterns was lacking to disallow the null hypothesis. The null hypothesis 6.1 was TABLE 27 SUMMARY OF F-TEST BIAS FOR KRAMER'S MODIFIED WEIGHTED SQUARES OF MEANS METHOD Sample Size Pattern of Subclass Frequencies F-test A B C D E F G H I J 12 24 48 96 192 1 O Row -PPP........ .....- ............... J-6 24 48 96 192 Column M KEY: P represents a positive bias significant at .05 level. N represents a negative bias significant at .05 level. — -- indicates no data obtained. : represents no significant bias. NOTE: The six spaces in each combination of pattern and size represent the significance tests at the points: F^go'^F i F 5751 ^ 99 1 F 9951 ^.999* VO 97 accepted as it relates to the distributions of F values for row effects. Column effects. The thirteen statistical differ ences for column effects are scattered generally throughout the various subclass patterns, sample sizes, and distribu tion points. No more than two discrepancies were obtained for any one distribution. There is no indication to suggest that anything other than chance effects are responsible for these data. The null hypothesis 6.1 was accepted as it relates to the distributions of F values for column effects. Hypothe sis 6.2 Row effects. The ten statistical differences are widely spaced throughout the sample sizes and distribution points. No more than two differences exist within any sample size for a particular checkpoint. Consequently, the null hypothesis 6.2 was accepted as it relates to the distributions of F values for row effects. Column effects. The thirteen statistical differ ences exhibit no definite concentration related to sample size. The null hypothesis 6.2 was accepted as it concerned the distributions of F values for column effects. Hypothe sis 6.3 Row effects. No more than two differences corres pond with respect to both subclass pattern and distribution point. Consequently, the null hypothesis 6.3 was accepted as it relates to the distributions of P values for row effects. Column effects. Since eight of the ten subclass patterns and five of the six checkpoints are represented by the collection of statistical differences, no consis tency or configuration in the results occurred. The null hypothesis 6.3 was accepted as it relates to the distribu tions of F values for column effects. CHAPTER V SUMMARY, FINDINGS AND CONCLUSIONS, AND RECOMMENDATIONS Summary Area of Investigation The major purpose of this study was to assess the F-test bias, under the null hypothesis, of six commonly used analysis of variance methods under varying degrees of disproportionality, and differing sample sizes. A 2 x 2 fixed-effects factorial model was employed to supply F- tests for row, column, and interaction effects. The dimension of sample size consisted of five levels: 12, 24, 48, 96, and 192. In every case each cell contained at least one number and in some cases as many as ninety-six numbers were assigned to one cell. The factor of disproportionality comprised ten levels ranging from equal cell sizes to extreme disproportionality. Procedures A computer program was written to generate random numbers from a normal distribution, to assign the numbers to the subclasses according to the desired pattern, and to 99 100 process these data by the following six analysis of variance methods: (1) fitting constants, (2) weighted squares of means, (3) unweighted means, (4) Snedecor's expected numbers (5) Tsao's expected equal frequencies, and (6) Kramer's modified weighted squares of means. These procedures were repeated 2,000 times for each of the forty-eight combina tions of sample size and degree of disproportionality, thus yielding forty-eight empirical sampling distributions of the F statistic for row, column, and interaction effects for each of the methods under consideration. The total number of F values involved was 1,536.000. The upper region of each generated sampling distribu tion and the upper region of the corresponding central F- distribution were compared to assess their conformity. The proportions of F values exceeding the points F.90‘ F.95‘ F.975' F 99' F 995’ an(^ F.999 t*ie central F-distribution were computed and observed in relation to the 95 per cent confidence intervals for P = .10, .05, .025, .01, .005, and .001. A sample proportion falling outside the confidence interval resulted in a tentative rejection of the null hypothesis of coincident distributions. Due to the large number of tests that were made, a discernible pattern or consistency in the statistically significant proportions was required for a permanent rejection of the null hypothesis. 101 Findings and Conclusions Fitting constants method. With one exception, the empirical sampling distributions which were generated for the row, column, and interaction effects did not signifi cantly differ from the corresponding central F-distribu- tions in the proportion of F values exceeding the six distribution points under consideration. The sampling distributions for the interaction F-test values tended to be positively biased for conditions of extreme dispropor tionality . For all practical purposes, the method of fitting constants is deemed to be without serious bias. There is neither an important nor a substantial relationship among sample size, degree of disproportionality, and direction or magnitude of bias. The standard F table may safely be referenced to assist in the interpretation of an obtained F statistic. Method of weighted squares of means. For each sample size and degree of disproportionality involved in this study, the empirically obtained sampling distributions did not differ significantly from the corresponding central F-distributions in the proportion of F values which exceeded the six distribution points in question. The method of the weighted squares of means is judged to be an adequate mode of analysis for main effects. 102 The standard F table may safely be employed to assist in the interpretation of an obtained F statistic. There is no important interrelationship among the dimensions of sample size, disproportionality, and F-test bias for main effects in a 2 x 2 factorial layout. Unweighted means method. It was found that the F values for row and column effects as computed by the unweighted means and the weighted squares of means methods were identical. Moreover, the F values for interaction effects as computed by the unweighted means and the fitting constants methods were identical. The findings concerning the various sampling distributions were given in the preceding sections. For varying degrees of disproportionality and differing sample sizes in a 2 x 2 factorial layout it is concluded that there is no significant disturbance to the alpha level of the F statistic for main effects as computed by the unweighted means method of analysis. Moreover, no important interrelationship among the dimensions of sample size, disproportionality, and bias was discernible. The unweighted means method was observed to possess the following desirable features: 1. The computational demands are slight. 2. It furnishes an unbiased F-test for both main effects and interaction effects so that the 103 standard P table may appropriately be referenced to assist in the interpretation of an obtained F statistic. 3. Since the fitting constants method is reported to provide an efficient F-test for interaction effects, and since the methods of fitting constants and unweighted means give identical results for inter action effects, the unweighted means method also provides an efficient F-test for interaction effects. 4. Since the weighted squares of means is reported to provide an efficient test for the main effects whenever interaction is present in the population, and since the methods of unweighted means and the weighted squares of means give identical results for main effects, the unweighted means also provides an efficient F-test for main effects when interaction is present in the population. Snedecor1s method of expected numbers. There were 114 sampling distributions involving disproportionate data, ninety of which differed significantly from the central F-distributions in at least two points. Moderate to severe bias results from the application of Snedecor1s method of expected numbers to situations in which the cell frequencies depart from proportionality. 104 For main effects, the bias is always positive and the actual level of significance may substantially exceed the nominal level. For interaction effects the bias may be positive or negative. A strong positive relationship exists between degree of disproportionality and magnitude of bias but there is, apparently, little or no relation ship between sample size and magnitude of bias. Regression equations were developed to give an estimate of the actual working level of significance. The utilization of these equations severely attenuates the uncertainty with regard to the risk of Type I errors. Snedecor1s recommended use of the chi square criterion as a guide to the applicability of the method of expected numbers is not an efficient or good one for small or moderate sample sizes. A more suitable criterion and one which provides a more accurate estimate of the inherent bias is the ratio of chi square to sample size. Tsao's method of expected equal frequencies. Of the 129 sampling distributions involving unequal cell frequen cies, 100 differed statistically from the corresponding central F-distributions in the proportion of F values exceeding all six of the distribution points tested. In addition, all distributions differed in at least one test. Based on these findings, Tsao's method of expected equal frequencies is judged to be inferior. It is 105 extremely sensitive to a departure from equal cell sizes. The bias is positive for all effects and is positively related to the degree of inequality of cell sizes, and negatively related to sample size. Subclass patterns employing cell sizes which differ markedly result in extreme bias. Ferguson's suggestion that Tsao's method could be usefully employed when the subclass frequencies do not depart significantly from equality, as measured by the chi square criterion, is clearly indefensible and insufficient for small sample sizes. Kramer's modified weighted squares of means method. For each sample size and degree of disproportionality involved in this study, the empirically obtained sampling distributions did not differ significantly from the corresponding central F-distributions in the proportion of F values which exceeded the six distribution points in question. It is concluded that this method is adequate with regard to unbiased F-tests for main effects. No significant interrelationship was discerned among the variables of sample size, disproportionality, and degree of bias. Recommendations Related to the Statistical Analysis of Disproportionate Data The following recommendations are made with respect to the analysis of variance for a 2 x 2 layout involving disproportionate subclass frequencies. 1. With regard to unbiased main effects, the following methods are recommended: (a) the unweighted means, (b) fitting constants, (c) the weighted squares of means, (d) Kramer's modified weighted squares of means, and (e) Snedecor1s expected numbers in conjunction with the regression equations which were developed. 2. With regard to unbiased interaction effects, the following methods are recommended: (a) the unweighted means, (b) fitting constants, and (c) Snedecor1s method of expected numbers when the ratio of chi square to sample size in less than one-tenth. 3. With regard to practicability and versatility, the method of unweighted means is recommended. The computational demands are minimal and no assumption regarding the presence or absence of interaction is required. Recommendations for Further Research There appears to he a need for information on the comparability of the various methods of analysis with respect to the power involved. Which methods are most efficient in ascertaining a difference? Does a relationship exist between the power of the test and the degree of disproportionality present? Is there an interaction of sample size with mode of analysis in the efficiency of the F-tests? Robustness studies for the analysis of variance of disproportionate data seem desirable. What is the extent of the bias in the F-tests when the population sampled is characterized by non normality? What influence does hetereogeneity of variance have upon the F-test? Do the various methods of analysis differ with respect to their applicability when the underlying assumptions of normality and homogeneity of variance are not satisfied? How do the variables of sample size and degree of disproportionality relate to the robustness of the F statistics? Further research is needed to provide information on the F-test bias for the various methods of analysis for more complex multifactor and multi level designs. Does the method of unweighted means retain the property of unbiasedness when the factors consist of several levels? For example, does a 3 x 3 x 4 design result in F-test bias when subjected to the unweighted means analysis? Do the various methods of analysis involve F-test bias for main effects when interaction is present in the population? Answers to questions concerning the independence of effects seem desirable. Is the derived F- statistic for an effect dependent upon the presence or absence of one or more other effects? APPENDIX COMPUTATIONAL FORMULAS Symbolism "j"— represents the column number in the layout. "k"— represents the replicate number in the cell. "r"— represents the number of rows in the layout. "c"— represents the number of columns in the layout. "X"— represents a raw score. "X^j"— represents a cell mean. "X^."— represents a row mean. "X.j"— represents a column mean. "A"— represents the main effect for rows. "B"— represents the main effect for columns. "n-•"— represents the cell frequency for the cell in The remaining symbols which require a definition are defined in the context of the method in which they are employed. row i and column j. Fitting Constants Differences in cell means: X X 2j 110 Weighting factors nljn2j w . = D n, . + n„ . ID 23 Adjusted sum of squares due to interaction: ffw.d.)2 SS1 = 5 w-d? - --------- ab D D sw. Adjusted sum of squares due to factor A: SSi = SScells ' SSAb - SSb Adjusted sum of squares due to factor B: SSb = SScells " SSab - SSa The subsequent computations follow the general analysis of variance formulas which are presented in a later subsection. Weighted Squares of Means Within row cell differences: d. = X._ - x._ i ll i2 Unweighted row and column means: Row and column weighting factors: wi Vj 2(JL_) 3E ( —^—) j nij 1 ij Sum of squares due to factor A: c2 ( J w ^ . ) 2 SS= = c23Ew . X? . - i--------- a . i x ^ l 2 w i 1 Sum of squares due to factor B: r2 (*v.X. . )2 o —o i 3 3 SS, = r 2v'X. . - J---------- b a 3 3 2v. j j j The subsequent computations follow the general analysis of variance formulas presented later in this section. Unweighted Means Harmonic mean: n. = ___£C_ h z2<j=-> ij ij Unweighted row and column means: 113 Grand mean: 2X. . X.. = Sums of squares due to factor A and factor B: SSa = nhc2(X±. - X..)2 SSb = nhrI(X.j - X. .) 2 Sum of squares due to interaction: SSab = - V - *-j + ^->2 The subsequent computations follow the general analysis of variance formulas. Expected Numbers Adjusted cell frequencies: nij(adj) ni *n* - j n. . Adjusted cell sums: ^ Xijk(adj) ~ nij(adj)Xij After the adjusted cell numbers and cell sums have been found, the subsequent computations are performed as usual, employing these adjusted values in the general analysis of variance formulas. Tsao's Expected Equal Frequencies Mean cell size: 2Zn. . . . i] n = U 1 rc Adjusted sura of squares attributable to the cells SS! . = n 2(X. - X. .) 2 i: n. . !Dk ^-D ID Row and column means: 2X. . 2x. . X.. - L l l i c Dr Grand mean: _ X.. = ±2__1 rc Adjusted sum of squares within cells: SS' = «SS' within ij ij The subsequent computations follow the general analysis of variance formulas presented in the following subsection. Kramer's Modified Weighted Squares of Means Row and column means: 115 Row and column weights: 2 W. . = n * Sum of squares attributable to factor A: SS = 2rw. .xj? a .11 aw. .x!.)2 A 1 1 i 2W. i Sum of squares attributable to factor B: ssw = : aw. x. .)2 T j - * j The subsequent computations follow the general analysis of variance formulas. General Analysis of Variance Formulas.— All of the foregoing methods of analysis employ many of the general formulas for the analysis after the adjustments or modifi cations unique to each method have been made. The general formulas which are necessary to complete these analyses are given below. Not all of the formulas listed find applica tion in every method of analysis. Sum of squares between cells: SS ij ij Sum of squares attributable to factor A s.?xijk)2 <«IX )2 SSa = 2 3k_______ - ijk 13k i n-i 2Zn. . - * - • • • in ID J Sum of squares attributable to factor B esx..,)2 (ssix.. >2 SS = I ik J-3k _ jjk ^ b j nM » nij ij Total sum of squares: (Z££X.. )2 qq = JJS — ijk________ total. ijk ijk ££ni. Sum of squares within cells: SSwithin “ SStotal SScells Mean square for factors A and B: SS qq MS = MS - _ a r-1 k c-1 Mean square for interaction: SS , ab MS . = _________ ab (r-1) (c-1) Mean square for within: SS within MS . , . =---------- within £2n^.-rc ij 117 F statistic for factors A and B: MSa MSb F* MSwithin b MSwithin F statistic for interaction: MSab F , =---------- ab MS • within BIBLIOGRAPHY BIBLIOGRAPHY 1. 2. 3. 4. 5. 6. 7. 8 . 9. 10. 11. Anderson, R. L. and T. A. Bancroft. Statistical Theory in Research. New York: McGraw-HillBook Company, Incorporated, 1952. Beyer, William H. (editor). Handbook of Tables for Probability and Statistics. Cleveland, Ohio: The Chemical Rubber Company, 1966. Box, G. E. P. "Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems," Annals of Mathematical Statistics, XXV (1954) , 290-302. Bradu, Don. "Main-effect Analysis of the General Non- orthogonal Layout with any Number of Factors," Annals of Mathematical Statistics, XXXVI (Febru- ary, 1965) , 88-97. Brandt, A. E. "The Analysis of Variance in a 2 x s Table with Disproportionate Frequencies," Journal of the American Statistical Association, XXVIII (1933), 164-73. Brownlee, K. A. Statistical Theory and Methodology in Science and Engineering. 2nd edition. New York: John Wiley and Sons, Incorporated, 1965. Comstock, R. E. "Overestimation of Mean Squares by the Method of Expected Numbers," Journal of the American Statistical Association, XXXVIII (1943), 335-40. Cox, D. R. Planning of Experiments. New York: John Wiley and Sons, Incorporated, 1958. Edwards, Allen L. Experimental Design in Psychologi cal Research. Revised edition. New York: Holt, Rinehart and Winston, 1960. Federer, Walter T. Experimental Design. New York: The MacMillan Company, 1955. Ferguson, George A. Statistical Analysis in Psycho logy and Education. 2nd edition. “ New York: McGraw-Hill Book Company, Incorporated, 1966. 119 120 12. 13. 14. 15. 16. 17. 18. 19. 20. 21 . 22. 23. 24. Fisher, R. A. The Design of Experiments. Edinburgh: Oliver and Boyd, 1935. Gosslee, D. G., and H. L. Lucas. "Analysis of Variance of Disproportionate Data when Interaction is Present," Biometrics, XXI (March, 1965), 115- 133. Goulden, Cyril H. Methods of Statistical Analysis. New York: John Wiley and Sons, Incorporated, 1952. Gourlay, Neil. "F-Test Bias for Experimental Designs in Educational Research," Psychometrika, XX (1955), 227-248. Graybill, Franklin A. An Introduction to Linear Statistical Models. New York: McGraw-Hill Book Company, Incorporated, 1961. Guilford, J. P. Fundamental Statistics in Psychology and Education. 4th edition. New York: McGraw- Hill Book Company, Incorporated, 1965. Hays, William L. Statistics for Psychologists. New York: Holt, Rinehart and Winston, 1963. Horsnell, G. "The Effect of Unequal Group Variances on the F-test for Homogeneity of Group Means," Biometrika, XL (1953), 128-136. Johnson, P. 0. Statistical Methods in Research. New York: Prentice-Hall, 1949. Kempthorne, Oscar. The Design and Analysis of Experiments. New York: John Wiley and Sons, Incorporated, 1952. Kerlinger, Fred N. Foundations of Behavioral Research. New York: Holt, Rinehart and Winston, Incorporated, 1964. Kramer, C. Y. "On the Analysis of Variance of a Two-way Classification with Unequal Subclass Numbers," Biometrics, XI (1955), 441-452. Li, C. C. Introduction to Experimental Statistics. New York: McGraw-Hill Book Company, Incorporated, 1964. 121 25. 26. 27. 28. 29. 30. 31. 32 . 33 . 34. 35. 36. McNemar, Quinn. Psychological Statistics. 3rd edition. New York: John Wiley and Sons, Incorporated, 1962. Ostle, Bernard. Statistics in Research. Ames, Iowa: The Iowa State College Press, 1954. Scheffe, Henry. The Analysis of Variance. New York: John Wiley and Sons, Incorporated, 1959. Snedecor, G. W. "The Method of Expected Numbers for Tables of Multiple Classification with Dispro portionate Subclass Numbers," Journal of the American Statistical Association, XXIX (1934), 389-393. ________. Statistical Methods. Ames, Iowa: Iowa State University Press, 1956. Spiegel, Murray R. Statistics. New York: Schaum Publishing Company, 1961. Stanley, Julian C. "The Influence of Fisher's 'The Design of Experiments' on Educational Research Thirty Years Later," American Educational Research Journal, III (May, 1966), 223-230. Steel, Robert G. D., and James H. Torrie. Principles and Procedures of Statistics. New York: McGraw- Hill Book Company, Incorporated, 1960. Tsao, Fei. "General Solution of the Analysis of Variance and Covariance in the Case of Unequal or Disproportionate Numbers of Observations in the Subclasses," Psychometrika, XI (June, 1946), 107- 128. Walker, Helen M., and Joseph Lev. Statistical Inference. New York: Holt, Rinehart and Winston, Incorporated, 1953. Winer, B. J. Statistical Principles in Experimental Design. New York: McGraw-Hill Book Company, Incorporated, 1962. Yates, F. "The Analysis of Multiple Classification with Unequal Numbers in the Different Classes," Journal of the American Statistical Association, XXIX (1934), 51-66.

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Creator Hopkins, Bobby Ray (author)

Core Title An Empirical Investigation Of F-Test Bias, Disproportionality, And Mode Of Analysis Of Variance

Degree Doctor of Philosophy

Degree Program Educational Psychology

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

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Advisor Metfessel, Newton S. (committee chair), Magary, James F. (committee member), McIntrye, Robert B. (committee member), Wolf, Richard M. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-140465

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