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The Influence Of Communality And N On The Sampling Distributions Of Factor Loadings

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The Influence Of Communality And N On The Sampling Distributions Of Factor Loadings

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Content This dissertation has b een m icrofilm ed exactly as received 6 7 - 1 3 ,0 3 9 PENNELL, R oger John, 1941- THE INFLUENCE OF COMMUNALITY AND N ON THE SAMPLING DISTRIBUTIONS OF FACTOR LOADINGS. U n iversity o f Southern C alifornia, P h.D ., 1967 P sych ology, gen eral U niversity M icrofilms, Inc., A nn Arbor, M ichigan U N IV ERSITY O F S O U T H E R N C A L IF O R N IA TH E G R A D U A T E S C H O O L U N IV E R SIT Y PA R K L O S A N G E L E S , C A L IF O R N IA 9 0 0 0 7 This dissertation, written by Roger .John ..Pennell............. under the direction of h%3..~Dissertation C om 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 jgp a g f e a z e s f e s f e g ...... Dean Date. June 8 ? 1967 ACKNOWLEDGMENTS The author wishes to thank the members of his committee: Norman Cliff (Chairman), Robert Priest, Gerald Rigby. Special thanks are due Norman Cliff, principal investigator for NSF Grant GB 4230, in which the author was involved; the computer facilities of Western Data Processing Center and Health Sciences Com puting Facility, University of California, Los Angeles, sponsored by NIH Grant FR-3; and the Computing Sciences Laboratory at the University of Southern California. Finally, particular thanks are owed to my wife for her support and for her endurance. TABLE OF CONTENTS Page ACKNOWLEDGMENTS ...................................... ii LIST OF TABLES ........................................- v LIST OF FIGURES.................. vii Chapter I. INTRODUCTION .................................. 1 ( Previous Approaches to the Sampling-Error Problem in Factor Analysis II. POSSIBLE INDEPENDENT VARIABLES FOR THE STUDY OF SAMPLING ERRORS IN FACTOR ANALYSIS .......................... 5 III. THE EXPERIMENTAL HYPOTHESIS ............... 8 Untenability of Factor Strength and Loading Size Communality Factor Loading Distributions as a Function of Communality IV. LITERATURE REVIEW ........................... 15 The Communality Problem Rank and Communality Squared Multiple Correlations (SMC) Relations to the Present Research V. SAMPLING PROCEDURES ........................ 21 iii I VI. METHOD 2 5 The Population Factor Matrices Computer Programs Experimental Design and Analysis Test for Elongated Factor Loading Distributions Additional Observations Analysis of Variance for N and Communality Regression Analysis of Empirical Functions Results of the Additional Observations Analysis of Variance for Elongated Factor Loading Distributions Mean Sample Factor Loadings Generalization of the Experimental Results VII. RESULTS 3 2 VIII. DISCUSSION 5 3 IX. SUMMARY 6 2 REFERENCES 6 3 APPENDIX 68 iv LIST OF TABLES i Table Page 1. Factor Matrices Illustrating the Non invariance of Factor Patterns under Rota t io n..................................... 10 2. Factor Matrix Which Would Produce SMCs Which Were Both Good and Bad Estimates of Actual Communality ...................... 18 3. Replicated Observations with Factor Matrices of Different Dimensionality . . 31 4. Summarization of Factor Loading Frequencies.................................. 32 5.. Standard Deviations for the Experimental Tests (Three Replications) ............... 35 6. Analysis of Variance T a b l e ............ 36 7. Regression Coefficients for Predicting Standard Deviations from 1/nT N .......... 42 8. Regression Coefficients and Intercept Values for Predicting Standard Deviations from Communality ............... 45 9. Standard Deviations of Nonzero Loadings from Matrices with Different Dimensions.................................. 45 10. Analysis of Variance Table, Dependent Variable: Standard Deviation Ratios . . 50 11. Tabulation of Mean Factor Loadings Which Were Biased from Their Corresponding Population Value ("a”) . . 52 v 12. Size of Loading Necessary to be Significantly Different from Zero at 5% Level (Two Tailed)............ 58 13. Size of Loading Necessary for Difference to be Significant at the 5% Level (Two Tailed)................. 59 vi LIST OF FIGURES Figure . Page 1. Hypothetical Distributions of Test Points for Three Tests in Two Dimensional Space ........................... 12 2. Frequency Count of Randomly Generated Factor Loadings .............................. 33 3. Standard Deviations as a Function of 1/n/ N - Nonzero L o a d i n g s................. 37 4. Standard Deviations as a Function of 1/\J N - Zero Loadings................. 38 5. Standard Deviations as a Function of Communality - Nonzero Loadings .......... 39 6. Standard Deviations as a Function of Communality - Zero Loadings.......... 40 7. Regression Coefficients to Use in Pre-__ dieting Standard Deviations from 1 / \l N. . 43 8. Regression Coefficients and Intercept Values to Use in Predicting Standard > Deviation from Communality ............. 46 9. Standard Deviation Ratios: Zero Loading to Nonzero Loading.... ........... . 49 10. Distributions of Factor Loadings with N Increasing from Top to Bottom (right) and Communality Increasing from Top to Bottom (left)................. 55 vii / CHAPTER I INTRODUCTION This study will employ a Monte Carlo approach in attempting to demonstrate the functional dependence of factor-loading sampling error on the population communality of the test or variable on which a loading occurs. As such, factor matrices containing the experimental tests in question (i.e., tests with varying communalities) will be repeatedly sampled, factored, and rotated, and the results tallied over the loadings on the experimental tests. The variability of the loadings will constitute the dependent variable. The effect of an additional independent variable t will also be observed, N, the number of persons upon which each sampling is based. The necessity of a Monte Carlo approach to the sampling error problem in factor analysis results from the inadequacy of the model in dealing with questions of a statistical nature. This is illustrated by the inability of the model to provide methods of statistically differ entiating between two different loadings or between a given positive loading and a zero loading. For instance, 1 2 if an investigator chooses to replicate a factor solution on what is ostensibly the same population, he would have considerable difficulty in asserting whether or not the original study had in fact been replicated. Indices of congruence (e.g., Tucker, 1951) reflect the gross similar ity of factors; however, an approach so general obscures data of a potentially valuable nature, i.e., whether or not "significant” changes have occurred in the individual factor loadings. If the results of a factor analysis could be dealt with on a more quantitative basis, many hy potheses for subsequent experimentation would doubtless be generated. Previous Approaches to the Sampling-Error Problem in Factor Analysis Historically the sampling-error problem in factor analysis has received attention in two areas. Originally investigators seemed most interested in which methodologi cal procedures gave rise to results most in accord with what was thought to be a correct solution. In these cases a correct solution was usually taken to be data which were recognized to be stable and empirically verified. Exam ples of such data are Thurstone's PMA battery (1947), Holzinger and Swineford's "24 psychological variables" (1937), and Mullen's "eight physical variables" (1939) pre sented in Harmon (1960). An example of this class of study would be the rotation of one of the above factor so lutions by a number of analytic methods and then comparison of the results of each to the "correct" solution. Examples of studies of this kind are by Bechtoldt (1961) , Wrigley (1959) , and Tyler and Michaels (1958) . While an approach of this kind is admittedly crude, it has, for example, enabled investigators to eliminate orthogonal quartimax (Neuhaus and Wrigley, 1954) as a generally efficient method of factor rotation. As will be recognized, the purpose of the studies cited above is to discriminate between the effectiveness of various tools in arriving at a good factor solution. The effectiveness is evaluated using the resulting sampling errors as the criterion. The major drawback with this method is that one is forced to rely upon the adequacy with which the stable and empirically verified solution approximates the population. If the sample is inadequate in this sense it will not provide a valid criterion for the purpose of discriminating between methodological tools. To implement the second approach the investigator repeatedly simulates the actual gathering of data, applies a factor-analysis and rotation procedure to each simula tion, and then observes the actual sampling error in terms of the parameters input to the system. As previously out lined, this is the approach we shall use here, where com munality and N will be the parameters we are interested in. 4 The advantage this method has over the previous one is that it is forced to specify the population with which it is dealing, and it is therefore able to independently vary population values. Furthermore, since the population is known, we need not rely upon the adequacy of an empirically obtained representation of the population. Examples of this approach would be Joreskog (1963) , Hamburger (1965) , Pennell (1966), and Cliff and Pennell (1967). i CHAPTER II POSSIBLE INDEPENDENT VARIABLES FOR THE STUDY OF SAMPLING ERRORS IN FACTOR ANALYSIS There are four parameters which would be obvious first guesses as important determiners of factor loading stability. The first is N, second is communality, third is the size of the factor loading, and fourth is the size or strength of the factor. (Hamburger [1965] explored all these independent variables except factor strength plus including the variables of rotational procedure and popu lation factor pattern complexity.) The type of factor pattern seemed to produce a monotonic, asymptotic function of stability, with strong simple structure showing the most stability, and with stability decreasing as a function of complexity. Two sample sizes were employed, 100 and 400, and it was found that the larger sample size produced the smallest sampling errors. Although the effects of communality and loading size were not planned independent variables, the effects of these variables could be grossly assessed. The results, however, indicated no trend for I either communality or loading size. 6 Cliff and Pennell (1967) devised a population fac tor matrix such that when it was repeatedly sampled inde pendent evaluations of the effect of communality, loading size, and factor strength could be made within the single matrix. Their results very clearly indicated that larger communalities produced greater stability (smaller sampling errors) and that stronger factors produced a similar re sult. Concerning loading size it was found that larger loadings produced greater stability about as often as did smaller loadings, and therefore, in the context of their study, loading size was rejected as a potential independent variable. They utilized a multiple regression approach in determining that nearly 85% of the variance in their ob tained standard errors could be predicted using only two independent variables. These two predictors turned out to be variables related to factor strength and communality. They did not vary N. It is of course natural to assume that the size of the sample upon which obtained factor loadings are based would be an important independent variable, since N is such a salient parameter in statistical analysis. Lending support to these notions and to the previously enumerated results of Hamburger are the results of a series of unpub lished studies conducted by the author at the University of Southern California. Here it was found that when the same population factor matrix is repeatedly sampled under I 7 each of a number of levels of N, the resulting sampling errors are almost exactly a straight line function of 1/ nTnT That is to say that as the function l/*/~N~decreases the sampling errors associated with any given loading decreases toward zero. In summary, sample size has been found to be a consistent and strong influence whenever employed as an independent variable; the indications are similar for fac tor size, although its influences have been studied only once. The evidence is not entirely clear for communality since only one out of three studies have found a clear effect. On the other hand, no study which investigated loading size was able to demonstrate that it was related to obtained sampling errors, independently of communality. CHAPTER III THE EXPERIMENTAL HYPOTHESIS As previously outlined, the effects of the inde pendent variables of communality and N will provide the primary topic of investigation in this research. In this chapter the rationale for including the variable of commu nality shall be explored, as well as indicating why neither factor strength nor loading size was included. The previously outlined evidence concerning N should pro vide a foundation for its inclusion here as an independent variable. The previous evidence was of a "greater than," "less than" nature, and therefore a more quantitative assessment of its effect shall be attempted here. Let us now turn to loading size and factor strength. Untenability of Factor Strength and Loading Size Consider any factor matrix and its individual fac tor loadings. It is well known (e.g., Horst, 1966; Harmon, 1960) that a transformation such as A = FT, where F is factor matrix, T is a row-wise orthogonal transformation matrix, and A is the transformed factor matrix, does not change the theoretical properties of the factor model, whether conceptualized in terms of basic structure (Horst, 1966) or principal components (Hoteling, 1933). Since, in terms of basic structure R = PDP' where D is diagonal and contains the eigenvalues of R, and P is a column matrix of orthogonal eigenvectors. Then F = PD4 and A = PD*T AA» = PD4TT' D4P 1 = PUP' = R, since TT1 = I. This simply means that F is unique to within an orthogonal transformation. Since this is true, it means that we can produce a substantial change in the size of any given loading by a simple rotation. Further more, the size or strength of the factors can just about as readily be manipulated. As an example consider the simple 2 x 2 factor matrix in the upper portion of Table 1. The first factor accounts for the most variance, and the first loading on this factor is the largest in magnitude. Apply a 45 degree clockwise rotation and the factor matrix in the lower portion of Table 1 is obtained. Now the second factor accounts for the most variance and the designated loading from above is now smaller and equal in magnitude to the other loadings. Considerations such as these reveal 10 the questionable status of loading size and factor strength as predictive variates. Table 1 Factor Matrices Illustrating the Non-Invariance of Factor Patterns under Rotation Before 45 Degree Rotation I II .60 .00 .30 .30 After 45 Degree Rotation 1 H .42 .42 .00 .42 This conclusion is consonant with the experimental findings concerning loading size; however, it does not agree with the findings of Cliff and Pennell (1967) con cerning factor size. This conflict is readily obviated by first considering communality. Communality In Table 1 it can be seen that while such para meters as factor strength and loading size are not invari ant from rotation to rotation, the communality of each of 11 the variables remains the same in both cases. Why this is true is obvious when considering that any n x m factor matrix can be considered to be a representation of n vec tors in an m dimensional space where the factor loadings are merely coordinates on dimensions. Under a rotation the only change is in the coordinates (factor loadings) which describe the invariant property of the test--the length of the vector in space. It can easily be seen from this that what is traditionally called communality is mere ly the square of the euclidian distance from the origin to the termini, of the test vector in the m dimensional (com mon factor) space. Thus it is proposed that if any prop erty of the factor matrix can function as an independent variable it must be the variable of communality, since it is the only invariant parameter. Factor Loading Distributions as a Function of Communality Figure 1 will illustrate two points of interest concerning communality; one is in the realm of conjecture and the other is based on limited factual evidence. The circular figures labeled 1, 2, and 3 represent hypothetical distributions of factor loadings over samples from a two- dimensional factor domain. The spread of the marginal distributions from Tests 1 and 2 represent the result found by Cliff and Pennell (1967), i.e., the variability of sampled factor loadings decreases as a direct function of / 11 the variables remains the same in both cases. Why this is true is obvious when considering that any n x m factor matrix can be considered to be a representation of n vec tors in an m dimensional space where the factor loadings I are merely coordinates on dimensions. Under a rotation the only change is in the coordinates (factor loadings) which describe the invariant property of the test--the length of the vector in space. It can easily be seen from this that what is traditionally called communality is mere ly the square of the euclidian distance from the origin to the termini of the test vector in the m dimensional (com mon factor) space. Thus it is proposed that if any prop erty of the factor matrix can function as an independent variable it must be the variable of communality, since it is the only invariant parameter. Factor Loading Distributions as a Function of Communality Figure 1 will illustrate two points of interest concerning communality; one is in the realm of conjecture and the other is based on limited factual evidence. The circular figures labeled 1, 2, and 3 represent hypothetical distributions of factor loadings over samples from a two- dimensional factor domain. The spread of the marginal distributions from Tests 1 and 2 represent the result found by Cliff and Pennell (1967), i.e., the variability of sampled factor loadings decreases as a direct function of Fig. 1 Hypothetical Distributions of Test Points for Three Tests in Two Dimensional Space. t-* N> 13 communality. However, this result has not always been found (Hamburger, 1965; Pennell, 1966). The reason for this lack of unanimity might be due in part to how the relative shape of the distributions acts under increasing communality. The conjecture is that the distributions do not merely become more densely distributed around the m)-dimensional mean, but rather the points become more densely distributed in a direction colinear with the test vector. This notion is illustrated in Fig. 1 by the rela tive elongation apparent for Test 2. If the above conjecture about the relative shape of factor loading distributions is true, the consequences for computing standard deviations are immediate. To com pute the standard deviation from the marginal distribution of Test 3 on, say, the first factor would lead one to the conclusion that the loadings on Tests 2 and 3 had dif ferent standard deviations on the first factor, even though jthe two-dimensional distributions and communalities were identical. What Hamburger (1965) and Pennell (1966) both did was to average variance indices for rows across columns of the factor matrix. This procedure so inexorably con founds zero loadings and complex tests that it is no won der that there was a failure to uncover a communality effect. If the present notions are correct, what is re quired is to re-orient the reference axis so that each 14 distribution successively has an axis placed through it before variance measures are computed. In general it would be only under these circumstances that standard deviations of marginal distributions could be systematically related to the distance of the distribution from the origin. / It can now be seen that there is a high correlation between the amount of variance on a factor, i.e., factor size, and the extent to which the distributions of test termini are colinear with that factor. Therefore, if the above considerations were true, it would immediately follow that factor size and small variance of corresponding factor loadings would be associated. To recapitulate, the effects of communality are hypothesized to be twofold. First, the experimental hy- I pothesis asserts that standard deviations vary inversely as a function of communality. This is merely the notion that distributions of factor loadings become more dense as com munality increases. Second, it is asserted that not only do the distributions become more dense, but that they also become flattened in a direction colinear with the test vector. CHAPTER IV LITERATURE REVIEW Since the problem of communality has occupied such a central position in factor-analytic methodology, let us turn to the consideration of communality in a definitional and conceptual sense. The Communality Problem The factor model, as formalized by Thurstone (1935, 1947), hypothesized that any obtained correlation matrix of mental abilities could be more parsimoniously expressed in terms of a smaller number of less complex primary mental abilities or factors. The initial assump tion was that an empirical correlation matrix was non singular solely because the measures were not error free. The model hypothesized that were these measures free of error, the matrix would be singular or of rank of, say, p, where n is the order and p<n. This implies that the matrix would then be expressible in terms of p orthogonal vari ables. The model then assumes something analogous to the Spearman-Brown formulations and attempts to obtain the true variance of each test minus the error and specific 15 variance. In order to formalize these notions consider the matrix equation FF' = R - U2. Here F is a factor matrix of rank p, R is the correlation matrix, and U2 is the diagonal matrix of specific variance of each test. Since F is of rank p, the product of FF' is also of rank p. If the model holds, R - U is the matrix we wish to determine in order that it be expressible as the product of two matrices of rank p. This then is the communality problem, or the problem of finding . Rank and Communality It has been deduced first by Thurstone (1935), then by Ledermann (1937), that a correlation matrix of order n is reducible to rank m where m is given by (1) m<£(2n+l - n/ 8n+l) The general approach one takes in developing this equation is setting up linear equations which must be satisfied if R is to be reducible to rank by adjusting the diagonals. Guttman (1958) points out that none of the above authors intended the formula to take the form of a theorem, but that somehow or other it eventually gained that status, and that it was ultimately assumed that a correlation matrix could always be reduced to that m. That this is decidedly not the case is illustrated by Guttman with several I 17 convincing examples. Kaiser (1959) concludes that equa tion (1) holds with probability zero, since correlations are continuous random variables, which implies that empiri cal matrices do not have unique minimum rank. In addition, Guttman (1958) has presented examples of matrices which can be reduced to rank no less than n-1 regardless of what choices are made for the diagonals. Furthermore, arguments are presented, stemming from the concepts of image analysis (Guttman, 1953), that for as many matrices as can properly be reduced to small rank there must be an equal number which can be reduced to at most a relatively large rank. Apart from theoretical considerations, all investi gators will readily agree that it is impossible, in all but the case of the carefully contrived matrix, to find diago nal values which will preserve the gramian properties of the matrix and simultaneously reduce it to minimum rank. As a consequence, the notion of communality seems to be substantially more of a heuristic concept than an empiri cally workable one. This observation is verified by refer ring to Wrigley's (1959) Monte Carlo study of an order eleven correlation matrix obtained by Burt (1952). Among other results, Wrigley found when he iterated communalities for hypotheses of one, two, and so on through eleven common factors that there was little in the way of differ ences between the resulting sets of communalities that would enable one to choose among the more reasonable 18 hypothesis of m = 3 to 9. He concluded that the only chance for objectivity is in using squared multiple corre lations (SMC) as the communality estimates. Squared Multiple Correlations (SMC) I The reason that squared multiple correlations are not unanimously used for the diagonal elements in factoring is that such elements are in practice neither theoretically / nor conceptually the same thing as communalities. Consider the population factor matrix presented in Table 2. It can easily be shown that the SMC's for the computed correlation matrix are rather good estimates of the population commu nalities for variables one through four, but very poor for variable five. Also Guttman (1956) has shown that the population SMC is the lower bound for the population commu- nality, and that it becomes an increasingly better estimate only as n grows large. Table 2 Factor Matrix Which Would Produce SMC1s Which Were Both Good and Bad Estimates of Actual Communality I II 1 .4 1 .0 2 .0 . 5 3 . 5 .0 4 . 0 .4 5 . 0 . 8 19 It is considerations similar to those presented which have led many to question the fundamental concepts of small rank and common factor analysis. The concern with communalities and primary mental factors has obscured the fact that what we are dealing with is a-mathematical tool which will answer the questions we ask of it. Therefore, since SMC's are the theoretically best diagonals, they should be used when no prior information about the vari ables exists or when the maximum likelihood methods of Lawley (1940) or Rao (1955) are not being used. The ques tion of the number of factors, on the other hand, is nothing more than a formulation of objectives which should be readily answerable in terms of utility and parsimony. Relations to the Present Research What implications do the above conclusions have in terms of factor loading stability? First, if one is at tempting to assess the effects of communality on the samp ling error of factor loadings, one needs to indicate which of the numerous possible meanings for communality he shall employ. As shall be outlined later, the procedure we shall employ here will be to estimate communalities with the SMC, even though the population communality is known (i.e., the sum of squared loadings for a test in the population factor matrix). In this case we will have three reasonable "communalities" to choose from--the population communality, 20 the communality estimate, and the final computed communal ity (i.e., the sum of squared loadings for a test in the sample factor matrix). In a sense we shall operationalize ourselves out of this perplexity by meaning the population communality when we use an unmodified "communality.” The above definition of communality tacitly avoids considerations such as the following. The author has found in unpublished research that it is possible to devise cor relation matrices so that for some given variable the SMC's are the same across matrices even though the population loadings are of a different size. The method of doing this involves juggling the relative sizes of the correlations. Now if two of these correlation matrices are repeatedly sampled and factored, and if variance measures are computed for this variable for each of the two sets of samples, there is evidence that, even though the mean of the sample factor loadings for that variable are the same for each of the sets, and therefore the mean sample communalities are the same, the variances are smaller for those variables with the larger population loadings. This would result in tests of equal communality having different sampling characteristics. These are perhaps but a few of the "worries" one might express about the formulation as it stands. I CHAPTER V SAMPLING PROCEDURES In order to obtain measures of factor loading vari ability, many samplings from a hypothesized population factor domain must be made. Not only must the procedure adopted be efficient in terms of computing time, but it needs to be theoretically sound and to conform to the clas sical notions of sampling in terms of error variance. There have been a number of methods proposed to accomplish this sampling task; one, suggested by Kaiser and Dickman (1962), is based on the notion that the standard score / matrix, Z, can be considered to be the product of two other matrices, viz.: Z = FX, where X is an m x N factor score matrix and F is n x m. Here n stands for the number of variables and m for the number of factors. In order to sample F one only needs to generate random-normal-deviates (RND) to insert as values in X and then ZZ' = FXX’F'= R where R is the sampled correlation matrix with communalities 21 22 in the principal diagonals. This is easily seen from the fact that £XX' = I. This method has the disadvantage that for large N and four or five factors one needs to generate a tremendous number of RND’s. Joreskog (1963) used a vari ant of this approach in testing the adequacy of his factor analysis procedure under assumptions of different popula tion factor loadings and different distributions of factor scores. Hamburger (1965) used a procedure based on the no tion that an error perturbed correlation matrix could be generated by hypothesizing random departures in the popula tion factors from the assumptions of variance equal to unity and covariance equal to zero. Specifically, the pro cedure involves generating a symmetric matrix, A, of order m + n of RND' s , each weighted by 1 /' s / N-1. This matrix is the matrix of random factor correlations (both common and specific) weighted by the standard error of a correlation. This matrix is then pre- and post-multiplied by a diagonal matrix, S, where each element Sjj is 1 + RND/n/ 2N, where 1/ n/ 2N is taken to be the standard error of a factor stan dard deviation. The operations' thus far produce C = SAS' where C is the factor variance-covariancq matrix. Then R = DFCF’D where F is now n x m+n, and where D is diagonal. While this method appears to work in practice it is not known 23 whether the generated correlation matrices conform to the expected distributions. Browne (1965) has derived a procedure for sampling which is efficient and has been shown to be theoretically sound. The procedure has also been described by Cliff and Pennell (1967) and will be briefly recounted here. If H is a random Wishart matrix (i.e., the scatter matrix associ ated with N observations from an n dimensional, normally distributed random variable, it is known that it can be expressed in the form of H = TT1 where T is lower triangular and composed of elements t^j = RND, for i<j; t^^ a random chi variate with d.f. = N-i and t.jj = 0 for i>j. The population correlation matrix can be factored by the square root method (cf, Dwyer, 1945) to produce r = nn*. Finally, let (2) R = DftHft’D where D is diagonal, and R is the sample correlation matrix. In particular, Browne (1965) has shown that the Wishart property assures that R is the maximum likelihood estimate of R. If (2) were changed to R* = 1, N then R* would be the maximum likelihood estimate of the population variance-covariance matrix, in this case the correlation matrix. It is the maximum likelihood property that makes the Browne (1965) procedure preferable to Hamburger's (1965), and the relative ease with which each sampling can be accomplished makes it preferable to the Kaiser and Dickman (1962) procedure. CHAPTER VI METHOD The Population Factor Matrices In order to make a controlled observation of the effects of communality on factor loading variability, a univocal test having the proper squared loading was se lected and input to a program which randomly selected a test number, i.e., a position in a factor matrix, inserted the experimental test in the selected position, and then randomly generated the factor loadings for the other vari ables. This was then the population factor matrix employed for generating the sample correlation matrices, where we are interested in the sampling characteristics of only one variable. One observation for each communality - N combina tion permits their separation by a complete crossing of the design. Moreover, this method of constructing population factor matrices has two other advantages. First, the ex perimenter was relieved of trying to construct appropriate population factor matrices, with the resulting bias inher ent in such construction. Second, since the resulting 25 matrices, except for the experimental variable, at random took the form of simple structure, complex structure, and intermediate structures, the type of matrix, in terms of analysis of variance, was a random effect and therefore generalizable. It should be noted that, strictly speaking, the results of the study will be generalizable only to samples from populations having properties similar to the one contrived for use here. Computer Programs A program to accomplish this task was written in such a way that the generated factor matrix could be output on tape to be immediately read by a following program. The program had as input one univocal variable with the proper communality. The loadings needed to fill out the remaining variables were generated from a uniform distribution on [-.2, 1.0]. For two or more factors, this allows a commu nality larger than 1.0. In order to prevent this, another random number was generated from [.2, 1.0] to determine the communality of each random variable. If aj represents a generated loading on a particular variable, then a*j = V X i/ S a2. (a..) (j = 1, . . . , m ; i = 1, . . . , n) where aj is the adjusted loading and was the randomly generated communality. Then 27 It should be noted that the expectation of aj depends on the number of factors. — A program specially written to generate correlation matrices according to the method of Browne (1965) was pro grammed to run on the IBM 7090-7094. It was written in such a way that the population factor matrix generated by the previous program was read from a scratch tape, post multiplied by its transpose, had unities inserted in the principal diagonals, and then served as the population correlation matrix employed to generate the sample corre lation matrices. To implement Browne's procedure a MAP subroutine (Kronmal, 1964) was employed to generate RND's. This method of RND generation had previously been found to be highly efficient as well as unbiased (Cliff and Pennell, 1966). The generated sample correlation matrices from the above program were also output onto scratch tape to be im mediately read by a factor analysis program designed to operate sequentially on correlation matrices. The program first computed SMC's to serve as communality estimates and then factored each matrix using an extremely fast, non iterative roots and vectors subroutine ("JACVAT," 1966). The resulting principal factor matrices were similarly out put on tape to be read by the following program. The next step was a program designed to perform a least-squares fit of a factor matrix to a fixed target 28 (Cliff, 1966). This program read in the previously ob tained principal axis solutions and fit them to the gener ated, population factor matrix which had been saved on tape. This program also operated sequentially on any num ber of matrices, and similarly output the resulting factor matrices on tape. At this point summary statistics were computed for factor loadings over blocks of samples. For instance, for the experimental test standard deviations, root-mean- squares, ranges, and other summary statistics were computed for each loading over a block of samples. For the purpose of investigating standard errors, the standard deviation was the statistic used. The standard deviation of the sample loadings for the experimental test will therefore constitute the dependent variable. Experimental Design and Analysis , The results of the last program were analyzed in the following manner. For each level of the independent variable of communality a univocal variable with the proper sum of squares was input to the population factor matrix generator, 100 samples based upon one of the levels / of the other independent variable of N was generated, fac tored, matched, and then summarized, thus providing infor mation concerning one communality - N combination. For each such combination three observations were made 29 utilizing for each a different population factor matrix. Thus there were 5x5x3 individual population factor ma trices generated, for each of which 100 samples were analyzed. It should be pointed out that the target used for the rotations included the univocal representations of the experimental variables. Thus, when standard deviations were computed for the experimental loadings the respective distributions were in a position analogous to variables one or two in Fig. 1. The levels of N were chosen at 100, 150, 300, 600, and 2500. These particular levels were chosen because they are approximately linear in the function 1/n N7 Levels of communality were chosen to span the appropriate domain,- and hence were .1, .3, .5, .7, and .9. The size of the population factor matrices was chosen to be n = 12 and m = 2. The number of factors was chosen to be two mostly for convenience, while n was chosen to be as large as was thought feasible in terms of practical computing time, since computation time of the eigenroots and eigenvec tors routine goes up as something like the cube of the size of the matrix. On the other hand, a large number of vari ables is desirable in terms of getting good communality estimates from SMC's. The results were analyzed in a two-way analysis of variance where the dependent variable was first the 30 standard deviation associated with the non-zero loading on each experimental variable, and second,the standard devia tion associated with the zero loading on each experimental variable. Test for Elongated Factor Loading Distributions It will be recalled that one of the main points of contention of the present study was that communality effects should be analyzed only from univocally represented variables, since it was posited that distributions of fac tor loadings become increasingly elliptical as a function i of increasing communality. This hypothesis will be tested in the following manner. For each experimental variable i the ratio of the standard deviation corresponding to the zero loading to the standard deviation of the positive loading was formed. These ratios for each replication and in each cell then constituted the dependent variable for an additional analysis of variance. These ratios will reflect the trend of the density of points within the dis tributions. If the distributions are circular the cor responding ratio will be close to 1.0; if they become more elliptical, the ratio will increase. Additional Observations Since one might be wary of generalizing the results of the study to factor matrices of any other than 12 x 2, it was decided to make additional observations on factor 31 matrices of different sizes. Table 3 lists the additional observations that were made. Since it is thought that the number of tests and the number of factors do not contribute to sampling variance of tests, the dependent variables (standard deviations) for these additional observations should merely replicate the observations made with the standard size factor matrix. 1 Table 3 Replicated Observations with Factor Matrices of Different Dimensionality N Communality New Dimensions of Factor Matrix 1 100 . 1 12 x 5 2 150 .3 14 x 3 3 300 . 5 16 x 4 4 600 . 7 18 x 5 5 2500 .9 20 x 6 I CHAPTER VII RESULTS The 75 generated factor matrices containing the randomly placed experimental variables are listed in the Appendix. The standard deviations as well as information concerning the mean sampled loading is also given for each of the loadings. In Fig. 2 the distribution of randomly generated loadings is graphed. The mean of the distribu tion is .39, where this value was computed utilizing the mean of the respective interval times the frequency. The frequency of loadings can be summarized as in Table 4. Table 4 Summarization of Factor Loading Frequencies Interval Per-Cent Occurrence .Of|a 1<.3 36 .35|a|<.6 30 .65|a|f1.0 34 32 Frequency Fig. 2 Frequency Count of Randomly Generated Factor Loadings 180- 160- 140- 120- 100- 80. 60- -.6 -.2 *0 Factor Loading 34 Analysis of Variance for N and Communality Since the population factor matrices used were 2 X 12, and since all experimental variables were univo- cally represented, there was one zero loading and one non zero loading. The standard deviations from the three replications of each N - communality observation can there fore be analyzed separately for the zero and nonzero load ings. The raw data upon which the following analyses of variance were based can be found in Table 5. The results of the two analyses of variance are presented in Table 6. Both the main effects and the interaction are highly sig nificant in both the analyses. It is interesting that while the Fs for N remain approximately the same for the zero and nonzero loadings, the influence of communality is strongest for the nonzero loadings. More of this will be said later. Figures 3 through 6 will permit examination of the mean values from the analysis of variance. In Fig. 3 the mean standard deviations, i.e., the mean over the three replications for any given cell, is graphed as a function of 1/^1 N. Except for one cell the consistency of the cell means for each of the communalities is striking. The larger the sample size the smaller the standard deviations of the resulting factor loadings, irrespective of the com munality. Figure 3 also indicates the reason for the significant interaction effect. For small N's the standard Table 5 Standard Deviations for the Experimental Tests (Three Replications) Zero Loadings N . 1 . 3 C ommtina lity - 5 . 7 . 9 1 0 0 - O 9 S . HO . o 8 a .078 . 048 - 1 1 2 .097 .070 .0 64 .055 .10 3 .081 .075 .069 .057 ISO . O 9 7 .071 .066 .0 59 .055 .097 .077 .069 .054 .049 .082 .079 .0 66 .067 .068 3 0 0 .064 .0 50 .027 .0 34 .035 .057 .055 .046 . 04 2 .033 .063 .057 .048 .04 6 .027 6 0 0 .044 .031 .0 30 .027 .024 .04 0 .039 .028 .028 .021 .039 .032 .034 .025 .017 2 S O O .019 .020 .017 .014 .012 .024 .018 .016 .012 . O 1 O .025 .017 .013 .013 .014 Nonzero Loadings 1 O O - lO 1 .082 .071 .04 7 .0 14 .093 .077 .056 .0 29 .on .10 7 .076 .054 .0 39 .019 15 0 .076 . 066 .042 .027 .028 .080 .0 59 .053 .02 8 .015 . 096 . 060 .0 50 .027 .031 3 0 0 .059 .043 .027 .0 16 .007 .065 .043 .0 34 .020 .016 .066 .039 .039 .022 .008- 6 0 0 .043 .030 .026 .015 .004 .039 .031 .022 .012 .006 .046 .029 .026 .016 .007 2 5 0 0 .019 .015 .013 .008 .003 .024 -013 .012 .007 .00 3 .025 .017 . O 1 O .008 .004 tn 36 Table 6 Analysis of Variance Table Source Sum of Squares d.f. Mean Square F Nonzero Loadings N .021 , 4 .0053 265.** Communality .024 4 .006 300.** / Interaction .005 16 .0003 15.** Error .001 50 .00002 Total .051 74 Zero Loadings N . 043 4 .011 267 . 5** Communality .0079 4 . 002 50.0** Interaction . 0021 16 . 00012 3.0* Error .0019 50 .00004 Total .0539 74 *p<.01 **p<.001 Standard Deviation .104 .09 .08 .07 .064 .03 .02 .01 .10 Pig. 3 Standard Deviations as a Function of l/iN - Nonzero Loadings Standard Deviation 09. .08. .07- • 06. Ob- .03. 02- . 01. .10 Fig. if Standard Deviations as a Function of l/\lN - Zero Loadings 10 4 09. 08. 02- 01- 5 7 9 1 .3 Communality Fig. £ Standard Deviations as a Function of Communality - Nonzero Loadings Standard Deviation .10 .09 . 08. .07- •06 .05- • 01*.. .03- .02 .01 100 300 600 2500 .1 .3 .5 .7 .9 1.0 Communality Fig* 6 Standard Deviations as a Function of Communality - Zero Loadings - p . o 41 deviations are heavily conditional upon which communality level one is working with, but for, say, an infinitely large N, it appears as if the standard deviations would be zero independently of what the communality might be. In other words the functions are not parallel over the various levels of communality. The mean standard deviations for each level of N as a function of communality is graphed in Figs. 5 and 6. The standard deviations for the nonzero loadings are in Fig. 5 and for the zero loadings in Fig. 6. Comparing these two figures one can see why the mean square for com munality was so much smaller for the zero loadings. As the sample size increases the level of communality becomes increasingly less important in predicting standard devia tions; for the largest N the standard deviations for com munalities of .1 and .9 differ by only .011, whereas for nonzero loadings the difference is .02. It should be remembered that when speaking of zero loadings with certain communalities, we are referring to zero loadings occurring on variables with that communality. Regression Analysis of Empirical Functions I For each of the empirical functions in Fig. 3 a linear regression line was computed. The regression coef ficients and the intercept values for each level of com munality can be found in Table 7. Partial support for the 42 previous observations (that for infinitely large N the standard deviations would be zero) comes from inspecting the intercept values for each level of communality. Here it can be seen that as 1/n W approaches zero, i.e., N ap proaches infinity, each of the functions takes on a value very close to zero. Table 7 Regression Coefficients for Predicting Standard Deviations from l/'vTN, for Each Level of Communality Communality Level Regress ion Coefficient for Slope Intercept . 1 .995 .003 . 3 . 809 - .006 . 5 . 559 .003 .7 .373 - .001 .9 . 221 - .001 Since the intercept value for each of the functions is close to zero, it may be ignored for purposes of pre diction with no appreciable increase in error. If this is done the function in Fig. 7, which is the function produced by plotting regression coefficients against communality, can be used in determining the proper regression •H O 1.0 Communality Fig» 7 Regression Coefficients to Use in Predicting Standard Deviation from l/JIT 44 coefficient for a certain communality level. If we fit a regression line to this function, the result is (3) b = -.992C + 1.087 where b is the regression coefficient and C stands for com munality. In this way equation (3) can be used to obtain the regression coefficient for utilizing 1/nTN as the in dependent variable in predicting standard deviations. A straight line function was fit to each of the curves in Fig. 5 in the same manner as for Fig. 3. The re gression coefficients and intercept values for each level of N are presented in Table 8. Both sets of coefficients are graphed as a function of l/'V-N in Fig. 8. The extreme regularity of the data is evident from this figure. If we fit straight lines to each of the curves the result is (4) b = -.98/N - .005 (5) I = 1.13/N. Equation (4) can be used to determine the regression coef ficients (b) and equation (5) to determine the intercept (I) values to use in predicting standard deviations of non zero loadings from communality. Results of the Additional Observations The results from the additional observations enu merated in Table 3 are presented in Table 9. Also pre sented are the corresponding cell means from the original data. Here it is seen that the standard deviations from 45 Table 8 Regression Coefficients and Intercept Values for Predicting Standard Deviations from Communality for Each Level of N N Regress ion Coefficient Intercept 100 - .108 .113 150 - .077 .088 300 - .065 .066 600 - . 045 .047 2500 - .024 .024 Table 9 Standard Deviations of Nonzero Loadings from Matrices with Different Dimensions Dimensions n m Communality N Standard Deviation Cell Mean from n=12, m=2, Data with Same N and Communality 12 5 .1 100 . 115 . 100 14 3 . 3 150 .071 .062 16 4 .5 300 .040 .033 18 5 . 7 600 .015 .014 20 6 .9 2500 .009 .003 Intercept 10J 08 J 06J Ol|J 02 J 08 06 02 10 00 -.02 J co -.06 - I 10 A Fig.. 8 Regression Coefficients and Intercept Values to Use in Predicting Standard Deviation from Communality 47 the differently dimensioned matrices are consistently larger than for the 12 x 2 matrices. In terms of the vari ability obtained within the cells of the original data, all the newly obtained standard deviations except two, 16 x 4 and 18 x 5, are too large to have been a random sample from a population whose only parameters were N and commu nality. These results would seem to indicate that there is an additional source of variance other than communality and N which plays a part in determining how large standard deviations will be. It is encouraging, however, that with the wide variety of factor matrices employed, all of the resulting standard deviations differ little in actual mag nitude from the results of the original data. Analysis of Variance for Elongated Factor Loading Distributions Turning to the hypothesis that the distributions of factor loadings become more elongated as communality in creases, the ratio previously discussed, i.e., the ratio of the standard deviation associated with the zero loading to the standard deviation associated with the nonzero load ing, was formed for each observation. An additional anal ysis of variance was performed with the dependent variable taken to be this ratio. The analysis of variance based upon this rather unique dependent variable is justifiable on two counts. First, the analysis is intended primarily as a vehicle for summarizing the index (the ratios) of 48 elongation. Second, it is known that the analysis of vari ance is relatively insensitive to rather severe departures from equality of variance and normality (cf. Box, 1954). At the very least the claim of independence can be sub stantiated. Thus we again have a two-way, fixed-effects analysis of variance where the ratios of the three observa tions constitute the dependent variable. The results are presented in Table 10. The ratios for N were not signifi cant and indicate that for each communality level the respective ratios as a function of N are not significantly different. The F for communality is highly significant and indicates that the mean ratios for different communality levels are significantly different. In order to illustrate these effects one replication from selected cells was chosen in advance to have all rotated factor matrices printed. In this way the hundred observations in two- i dimensional space could be plotted. The selected cells were those with a communality of .5 and those with an N of 300. The resulting plots can be found in Fig. 9 where the first column represents the same communality of .5 with N increasing, and the second column represents the same N of 300 with increasing communality. From this figure it can be seen that as N increases (first column) the distri butions of points becomes more dense without changing the general circularity of the overall distribution. On the other hand, as communality increases (second column) the Ratio 49 600 100 2^00 300 3.0 150 1.0- Fig 9 Standard Deviation Ratios: Zero Loading to Nonzero Loading. 50 distributions become elongated; this is especially evident in the last two plots (communalities corresponding to .7 and .9, respectively). Table 10 Analysis of Variance Table, Dependent Variable: Standard Deviation Ratios Source Sum of Squares Mean Square F N . 27 .068 .75 Communality 60. 25 15.060 165.49* Interaction 15.73 .970 10.66* Error 4. 54 .091 Total 80 . 78 *p<.01 Mean Sample Factor Loadings A question of some interest is the extent to which the mean factor loadings over the hundred samples for each experimental loading approximated the actual population values. Table 11 summarizes this information by loading interval and by N, and indicates that in general the sample means were unbiased estimators of the population values. It can also be seen that there is a tendency for the lar ger loadings to have the greatest incidence of bias. - This is a rather common finding (e.g., Cliff and Pennell, 1966) and has been found to be related to the use of SMCs 51 for estimating communalities. As can be seen from Fig. 2, the density of loadings greater than .7 is relatively small, therefore when a relatively large loading does occur the likelihood of there being other variables which also have large loadings is small. In this event the SMC will tend to be an estimate of communality which is biased downward, thus causing a downward biasing of the resulting sampled means. From Table 11 there also appears to be a tendency for larger Ns to have a smaller incidence of bias. It should be noted that it was considered inappropriate to test either of these tendencies statistically, since in general a downward bias of a large loading was accompanied by an upward bias in the companion loading on that test. Table 11 Tabulation of Mean Factor Loadings Which WTere Biased From Their Corresponding Population Value (x "a") Population Factor Loading Interval Per Cent of Means with Bias Greater Than or Equal to .03 Percent of Means with Bias Greater Than or Equal to .05 .00< a <.10 4 2 .iof a <.20 4 1 .205 a <.30 3 1 .305 a <.40 3 0 .405 a <.50 2 1 .505 a <.60 1 1 .605 a <.70 7 3 .705 a <.80 8 2 .805 a <.90 15 6 .905 a 5100 44 19 Sample Size from Which Factor Load ings Were Obtained Percent of Means With Bias Greater Than or Equal to .03 100 29 150 26 300 15 600 13 2500 17 CHAPTER VIII DISCUSSION The results of the study, especially the consis tency of the plotted regression coefficients in Figs. 7 and 8, are highly encouraging in terms of potentially objecti fying results from factor studies. In only one of the 25 cells of the two-way design did the observations deviate from the trend of the other cells. This particular cell was the one represented by a communality of .9 and an N of 150. The divergent results from this cell are most clearly apparent in Figs. 4 and 5. Several hypotheses were generated in attempting to explain this result; how ever, none seemed to isolate influences which were present for this on? cell and absent for the others. In general, however, the notion that sampling errors of rotated factor loadings are predictable from certain parameters of the system seems supported. The results from the study demonstrate another advantage of developing factor-pure measures of psycholog ical traits. As we have seen, the analysis of variance based on ratios of standard deviations was highly 53 54 significant for communality. This means, as was posited before, that the constraint exerted on distributions of factor loadings as a function of communality is more severe on the positive axis than on the zero axis. This means that for a given communality a univocal representation will produce the smallest amount of sampling error for the nonzero loading--the loading we are usually most interested in. In other words, the confidence bounds around a non zero loading on a univocal variable are smaller than for a variable with the same communality but with a complex repre sentation. Fig. 10 represents these ratios as a function of communality. Here we can see that the effect we have been speaking of is not too strong until communality in creases beyond .5, and that for small communalities, say around .1, the distributions are approximately circular and hence whether the variables were univocal or complex would make little difference in terms of sampling errors. Generalization of the Experimental Results The author sees the results of this study as sup plying a unidirectional inferential system. This will be explained in the following manner. It will be recalled that in rotating each of the sample factor matrices, the actual population matrices were used as the target. This obviously results in the factor loading distributions being optimal in the sense of being as tight3v constrained as Fig. JO Distributions of Factor Loadings with N Increasing from Top to Bottom (right) and Communality Increasing from Top to Bottom (left). in c - n 56 I possible. For this reason an investigator cannot neces sarily conclude from the findings of this study that a loading he obtains empirically is large enough to be con sidered significantly different from zero, since our re- ' suits are optimum in the sense just discussed. In a like manner he would not be able to conclude statistically that two loadings were sampled from different populations. In other words, in a sense other than the statistical one the null hypothesis cannot be rejected. It is in not rejecting the null hypothesis that our results have the most value, i.e., a loading within the confidence interval of a zero loading in terms of our results would be in the confidence interval in terms of any results less optimum than ours. A similar conclusion would hold for two loadings thought statistically to arise from the same population. When one makes a decision to reject the null hypothesis one has generated an idea for more definitive experimentation, as opposed to reaching a sound statistical decision. The standard errors found to be associated with certain factor loadings in the current research should lend considerably more flexibility to factor-analytic studies. First, investigators will be better able to deal with zero loadings. By assuming that the obtained standard deviations represent standard errors of sampling distribu tions of factor loadings one can compute confidence bounds around zero loadings for various communalities. The null 57 hypothesis is that a given loading on a test is zero and that the remaining loadings define the communality. The entries in Table 12 indicate the absolute bound around a zero loading as a function of N and communality for 5% levels of significance. From this table it can be seen that the rule of thumb which only takes loadings over .30 as significant is far too restrictive for all cases other than for small N and communality. It should be remembered that these are optimum values, and that loadings less than the values reported should most assuredly not be used for inter preting factors, but rather considered hyperplane loadings. From Table 12 we see that quite small loadings can be sig nificantly different from zero. The variance accounted for by such small loadings is, however, quite small and one might wish to ignore them even though the population load ing was not zero. On the other hand, the capacity to recognize much smaller loadings as significant provides a larger number of hypotheses for further exploration. For example, small loadings which were significant might pro vide the impetus to design more valid measures of the traits indicated by the significant loadings, whereas pre viously these loadings would most likely have been ignored. Another advantage comes from our ability to evalu ate replications of factor studies. Table 13 presents some data on this topic. This table indicates that for an N of, say, 100, a replicated loading would have to be 4 - J •H i — I r t C 3 £ £ o CJ Table 12 Size of Loading Necessary to be Significantly Different from Zero at 5% Level (Two-Tailed) N 50 100 150 200 250 300 400 600 800 1000 .05 .299 .211 .172 .149 .134 .122 .106 .086 .075 .067 .10 .291 .206 .168 .146 .130 .119 .103 .084 .073 .065 .15 .283 .200 .164 .142 .127 .116 .100 .082 .071 .063 .20 .276 .195 .159 .138 .123 .113 .098 .080 .069 .062 .25 .268 .190 .155 .134 .120 .110 .095 .078 .067 .060 .30 .261 .184 .151 .130 .117 .107 .092 .075 .065 .058 .35 .253 .179 .146 .127 .113 .103 .090 .073 .063 .057 .40 .246 .174 .142 .123 .110 .100 .087 .071 .061 .055 .45 .238 .168 .138 .119 .107 .097 .084 .069 .060 .053 .50 .231 .163 .133 .115 .103 .094 .082 .067 .058 .052 .55 .223 .158 .129 .112 .100 .091 .079 .064 .056 .050 .60 .216 .152 .125 .108 .096 .088 .076 .062 .054 .048 .65 .208 .147 .120 .104 .093 .085 .074 .060 .052 .047 .70 .201 .142 .116 .100 .090 .082 .071 .058 .050 .045 .75 .193 .137 .111 .097 .086 .079 .068 .056 .048 .043 .80 .185 .131 .107 .093 .083 .076 .066 .054 .046 .042 .85 .178 1 26 .103 .089 .080 .073 .063 .051 .045 .040 U1 00 59 smaller than .23 to be significantly different from .5. A difference larger than this would provide an excellent hy pothesis as to the difference between the two replicated populations. On the other hand, a loading of .24 or .25 would provide definite support for an affirmative replica tion. For larger Ns quite small differences become signif icant. For instance, for an N of 600 two loadings of .74 and .80 would provide evidence for rejecting the null hypothesis. Table 13 Size of Loading Necessary for Difference to be Significant at the 5% Level (Two-Tailed) Comparison Loading .90 .80 .70 .60 .50 'lOO . 79 .63 .49 .36 .23 150 • 0 0 .67 . 54 .41 . 29 N 300 CO • .71 . 59 .47 .35 600 . 86 . 74 . 62 .51 .39 2500 • 0 0 CO .77 .66 . 55 .45 As an example, an investigator might administer a number of paper and pencil tests of personality traits to a sample of persons from a typical college sophomore 60 population. After factoring the matrix of intercorrela tions among these measures and performing a suitable rota tion, the investigator would have an objective means of deciding which of the n loadings on each factor to use for purposes of interpretation (i.e., factor interpretation means placing a verbal label on the factor which appears descriptive of the measures which exhibit a significant loading on that factor). By entering Table 12 with the number of subjects in his sample he can determine the mini mum size for any given loading so that it can be assumed significantly different from zero, and therefore appropri ate to utilize for the purpose of interpretation. Further more, suppose the investigator administered the same battery of tests to subjects sampled from a population of homosexuals. After obtaining the rotated factor matrix from this sample, the investigator could determine those characteristics which normals and homosexuals have in com mon (obviously lkmited by the scope of the original mea sures) by referring to Table 13. Furthermore, hypotheses could be generated as to those characteristics which dif ferentiate normals and homosexuals. These hypotheses would result from those loadings between the two studies which were found to be significantly different, and all n x m characteristics could be evaluated as to whether or not they were shared by normals and homosexuals. It should be pointed out that the hypothesis we are 61 testing in factor analysis is a rather complex one that involves the entire factor matrix. Testing hypotheses about single loadings presupposes that we have focused our attention on the complex hypothesis as opposed to a poten tial hypothesis about an individual loading. In this case we have permitted the constraint of the entire hypothesis to act upon each loading indiscriminately. Only in this case is it at all reasonable to hypothesis test on indi vidual loadings. These illustrations indicate that the results of the current study provide a basis for relatively efficient, multivariate hypothesis testing. Factor-analyzed measures from two experimental groups can easily be evaluated to obtain good hypotheses for potentially valuable experi mentation, and unlikely hypotheses left until all other avenues have been explored. CHAPTER IX SUMMARY One hundred sample correlation matrices from each of three replications from the 25 observations of five levels each of communality and N were obtained. The popu lation correlation matrices were constructed by a random procedure and included the appropriate experimental tests representing the appropriate communality - N combination. The main results and conclusions were 1) larger communalities produce smaller variation in sampling distributions of factor loadings, 2) larger N's produce smaller variation in samp ling distributions of factor loadings, 3) the effect of communality was found to be greater on nonzero loadings than on zero loadings, whereas 4) the influence of N produced equal constraint for both zero and nonzero loadings, 5) standard deviations from factor matrices of different orders were found to be slightly larger than those from which the above results were obtained, 6) implications for psychological research were discussed. REFERENCES REFERENCES Bechtoldt, H. P. An empirical study of the factor analysis stability hypothesis. Psychometrika, 1961, 26, 405-432 . Box, G.E.P. Some theorems on quadratic forms applied in the study of analysis of variance problems: II. Effect of inequality of variance and of correlation of errors in the two-way classification. 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The effect upon the communalities of changing the estimate of the number of factors. British Journal of Statistical Psychology, 1959, 12, 35-54. APPENDIX Generated Factor Matrices Corresponding to the Three Replications of each N - Communality Combination, Presented with Standard Deviations (SD) Corresponding to Each Generated Loading (LD) (All decimal points omitted. An asterisk indicates that the loading and the sampled mean differed by at least .05). 68 Communality = .1 N = 100 Replication 1 Replication 2 Replication 3 I II I II I II LD SD LD SD LD SD LD SD LD SD LD SD 1 616 043 658 041 245 093 607 069 -300 082 746 044 2 641 065 -028 087 379 072 606 068 271 106 145 100 3 601 033 752 030 703 053 -107 083 490 052 680 042 4 089 092 618 064 407 094 381 096 316 107 000 103 5 -120 086 700 058 Oil 072 809 048 796 032 429 050 6 825 030 303 055 316 093 000 112 922 020 228 056 7 827 038 -139 073 421 063 703 046 902 019 381 041 8 854 027 263 052 850 026 444 043 -090 089 509 083 9 316 101 000 098 769 044 108 075 143 097 250 091 10 739 042 489 048 419 066 468 074 619 041 622 041 11 055 122 028 121 359 088 156 100 442 042 807 032 12 113 065 863 029 781 044 131 072 109 062 879 028 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .1 N = 150 Replication 1 I LD SD ■245 384 701 826 502 577 316 069 054 034 030 062 040 076 606 804 465 994 697 059 024 063 012 044 II LD SD -635 645 468 250 390 661 000 051 047 045 051 073 037 097 206 450 248 •001 ■023 077 038 089 044 068 Replication 2 II LD SD LD SD Replication 3 LD SD 222 004 792 ■562 901 403 316 099 083 031 066 022 046 080 349 478 715 225 160 086 073 032 083 097 145 100 486* 070 087 063 -086 084 322 035 912* 031 000 097 138 093 278 089 531 043 189 110 111 100 229 099 470 051 006 080 261 105 •060 097 220 102 316 096 163 059 622* 067 621* 048 178 103 160 113 II LD SD 120 101 602 053 651 047 166 086 447 071 341 084 000 082 818 039 •006 091 566 048 366 084 243 104 o 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .1 N = 300 Replication 1 Replication 2 Replication 3 I II II I II LD SD LD SD LD SD LD SD LD SD LD SD 156 062 516 031 704 027 316 065 113 064 296 056 679 026 305 050 940* 018 354 034 758 013 456 052 300 056 671 026 Oil 045 000 057 292 055 464 040 554 029 150 066 ■124* 036 804 022 632 016 087 053 658 035 702 016 560 031 291 055 256 067 530* 049 316 066 509 054 059 075 039 062 ■019 080 650 039 529 039 685 018 671 034 502 054 336 073 190* 056 000 063 322 059 069 079 436 052 141 084 161 051 ■620 •007 089 694 767 ■041 064 317 315 316 041 063 055 026 026 043 050 029 057 059 270 551 058 041 465 330 487 672 302 786 637 901 124 000 047 056 049 025 039 026 035 014 062 064 249 075 052 050 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .1 N = 600 Replication 1 I LD SD 204 030 552* 027 527 030 476 028 316 043 008 031 612 025 362 036 616 028 716 022 476 032 702 017 II LD SD 688 023 710* 023 454 026 036 047 000 044 693 023 347 030 379 032 404 028 165 029 204 041 568 020 Replication 2 I II LD SD LD SD •335 823 708 425 316 025 015 015 024 039 836 563 •242 615 827 779 495 012 025 032 015 014 019 031 •856 155 573 740 000 Oil 026 017 016 040 341 251 916 631 254 257 096 020 036 012 017 022 027 039 Replication 3 I LD SD 636 511 -024 112 145 708 197 •040 529 ■251 660 316 031 017 023 017 045 019 034 034 023 027 018 046 II LD SD -026 762 891 975 150 430 617 694 669 818 595 000 036 012 009 004 042 025 026 025 018 016 019 039 N> 1 2 3 4 * S 6 7 8 9 10 11 12 Communality = .1 N = 2500 Replication 1 LD SD 142 012 274 024 215 021 ■282* 013 456 016 518 013 413 012 628 Oil 647 013 •253 018 119 019 316 019 II LD SD 892 004 196 019 084 023 925* 006 268 018 471 014 719 009 500 014 328 014 418 017 -510 016 000 019 Replication 2 II LD SD LD SD 125 026 250 022 004 020 844* 010 316 024 315 019 133 026 584 015 402 019 375 015 321 016 271 023 339 023 299 024 295 022 205 013 000 024 490 016 087 023 ■009 019 141 024 637 013 661* 014 052 026 Replication 3 I LD SD 574 015 514 015 314 025 316 025 412 023 303 025 478 017 477 023 614* 017 II LD SD 260 021 422 022 663 010 617 Oil 392 025 288 027 436 019 617 014 005 027 000 025 327 025 ■009 027 610 015 379 022 111* 020 - '4 04 Communality = .3 N = 100 Replication 1 Replication 2 Replication 3 I II I II I II LD SD LD SD LD SD LD SD LD SD LD SD 1 548 082 000 110 -674 052 672 053 090 124 023 121 2 852 037 -138 070 940 016 334 043 -147 052 968 019 3 810 044 182 072 135 091 371 096 573 073 390 082 4 456 086 326 101 172 086 576 070 548 076 000 081 5 779 049 085 069 697 048 108 079 222 068 620 066 6 -078 104 455 094 243 065 804 039 301 079 574 067 7 253 085 598 078 081 094 293 109 -013 093 629 061 8 167 104 414 105 124 110 409 098 633* 055 608 051 9 063 078 646 065 397 072 511 074 028 135 183 103 10 286 078 024 110 155 101 297 094 -003 132 158 119 11 545 077 201 102 548 077 000 097 -155 062 876 030 12 007 108 127 135 879 028 -021 065 316 101 165 095 Communality = .3 N = 150 Replication 1 Replication 2 Replication 3 I II I II I II LD SD LD SD LD SD LD SD LD SD LD SD 1 600 037 695 030 727* 035 568 042 603 048 080 073 2 558 056 551 056 548 059 000 077 860 019 402 037 3 -073 048 889 019 081 103 097 083 548 060 000 079 4 681 050 -072 072 -148 082 371 074 053 071 668 050 5 -043 088 190 088 264 080 352 081 -084 064 800 031 6 136 032 971 009 -219 064 645 054 370 039 779 034 7 249 042 876 018 054 066 707 048 512 055 428 051 8 397 071 246 085 151 099 101 089 736 034 291 055 9 -021 071 635 047 759 042 016 061 -052 055 876 023 10 -239 055 790 034 199 092 031 094 666 052 351 056 11 548 066 000 071 , 247 077 459 066 959 012 074 046 12 258 084 324 071 -051 056 870 033 213 083 273 081 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .3 N = 300 Replication 1 LD SD 548 043 350 135 753 797 796 706 787 •568 593 785 260 041 048 017 016 025 017 023 045 039 030 057 II LD SD 000 055 569 453 606 501 056 673 189 ■315 158 072 149 039 046 021 026 043 019 042 045 054 042 068 Replication 2 II LD SD LD SD 570 462 766 548 378 655 848 958 399 189 ■014 038 040 029 025 039 044 029 019 010 043 049 065 043 306 839 298 000 323 288 ■274 ■061 476 774 078 634 044 017 038 057 056 039 040 033 042 024 058 037 Replication 3 I LD SD 026 467 548 503 817 859 625 232 131 182 709 158 042 040 043 043 016 015 036 055 059 062 032 051 II LD SD 916 505 000 193 570 357 ■497 203 224 301 095 664 017 040 050 054 021 026 040 051 062 056 051 034 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .3 N = 600 Replication 1 LD SD 348 082 455 262 717 595 285 548 040 023 030 038 018 028 042 030 726 776 342 422 012 013 041 016 II LD SD 139 035 987* 006 427 030 357 041 334 027 356 028 113 042 000 031 603 014 507 014 063 042 873 007 Replication 2 I II LD SD LD SD 607 259 818 245 227 946 658 548 025 033 016 040 041 009 023 031 074 639 234 083 038 023 036 042 551 693 ■189 219 154 039 389 000 024 023 030 047 045 022 026 039 558 376 563 010 033 033 033 051 Replication 3 LD SD •021 887 222 681 868 548 037 010 043 023 013 029 339 526 701 031 344 638 038 033 018 049 025 014 II LD SD 676 300 188 218 074 000 025 018 042 033 031 032 123 010 398 107 830 753 041 043 024 049 013 Oil "J "j Communality = .3 N = 2500 Replication 1 Replication 2 Replication 3 I II I II I II LD SD LD SD LD SD LD SD LD SD LD SD 1 -097 015 760 009 619 009 581 009 399 019 022 019 2 -329 016 636 012 -196 014 892* 007 -319 018 -455 014 3 306 010 862 005 548 013 000 018 356 012 751 008 4 633 008 721 007 389 014 500 015 548 017 000 017 5 784 009 069 015 158 013 790 008 141 019 634 013 6 1 121 020 314 018 755 005 611 007 622 010 630 009 7 236 022 -004 019 811 004 584 006 492 009 775 006 8 643 013 290 016 702 010 -256 013 450 017 205 016 9 247 014 654 012 123 018 465 015 073 024 087 023 10 594 014 210 015 211 017 573 015 153 014 833 008 11 548 015 000 020 421 020 221 019 565 015 139 016 12 518 018 -182 019 197 020 236 021 631 014 125 017 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .5 N = 100 Replication 1 I LD SD 191 889 898 ■266 049 695 415 225 710 707 616 149 097 025 025 063 110 055 056 060 051 056 061 077 II LD SD 043 230 -018 928 335 107 755 856 252 000 257 609 118 050 056 024 092 081 040 029 067 070 072 061 Replication 2 I LD SD 609 075 554 707 825 827 567 307 ■434 263 648 582 050 072 070 054 040 036 073 083 067 075 046 073 II LD SD 491 747 208 000 036 308 ■183 103 630 762 555 037 054 044 088 075 066 051 085 115 061 049 050 094 Replication 3 LD SD 562 540 538 425 732 491 291 707 737 688 227 119 071 066 068 100 036 048 096 071 051 032 081 113 II LD SD -027 600 551 258 584 770 502 000 324 655 727 468 095 055 066 112 044 032 087 088 067 034 057 104 " 4 <£> 1 2 r r j 4 5 6 7 8 9 10 11 12 Communality = .5 N = 150 Replication 1 LD SD 375 433 117 707 387 473 ■110 570 302 734 072 522 072 064 063 053 075 067 071 051 074 032 057 066 II LD SD 477 410 829 000 052 191 663 ■311 374 600 740 074 073 074 037 069 077 073 054 067 077 037 039 078 Replication 2 LD SD 717 372 382 398 707 805 651 689 320 170 127. 098 039 050 079 068 050 034 053 039 072 083 046 061 II LD SD 391 608 -065 535 000 131 ■021 419 563 326 905 817 047 056 074 053 066 057 067 046 058 080 024 032 Replication 3 I LD SD 831 651 -071 039 880 ■097 ■232 118 787 707 466 530 025 040 077 083 020 077 071 050 030 042 051 063 II LD SD 484 -420 442 005 ■366 659 382 •966 241 000 671 135 0.38 053 063 088 044 047 076 015 053 066 041 068 00 o 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .5 N = 300 Replication 1 LD SD 504 931 -114 436 596 628 ■010 249 102 707 240 386 032 018 046 051 038 039 054 043 050 027 044 054 II LD SD 666 034 764 429 225 116 317 785 467 000 776 120 026 037 030 045 043 047 050 023 051 047 024 053 Replication 2 LD SD 849 613 538 289 707 •334 650 363 504 271 465 068 023 033 025 052 034 047 037 052 045 036 027 065 II LD SD -018 275 813 353 000 661 268 256 •002 739 852 093 039 042 017 051 046 036 046 056 057 026 016 053 Replication 3 I LD SD 273 407 452 599 742 330 546 659 707 236 672 288 048 038 051 040 025 062 039 026 039 055 021 077 II LD SD 602 680 076 043 383 174 314 605 000 491 649 051 035 032 061 051 034 076 049 029 048 045 023 075 CO 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .5 N = 600 Replication 1 II LD SD LD SD 549 112 707 032 040 022 285 659 411 611 866 •031 382 105 507 039 029 031 025 016 030 030 037 034 179 032 420 037 000 028 247 041 •051 033 341 040 470 029 ■064 028 836* 021 608 025 171 051 235 036 Replication 2 I II LD SD LD SD 440 ■925 810 •191 571 358 262 055 707 026 007 009 031 023 041 042 040 026 143 331 007 041 023 037 641 022 -163 025 577 014 921* 012 363 026 154 047 051 048 439 037 000 034 386 037 836 012 513 034 Replication 3 LD SD 619 •447 542 383 511 665 •648 707 028 031 029 023 031 020 019 026 445 533 080 7 47 024 027 025 Oil II LD SD 181 •443 •065 754 287 463 ■485 000 034 029 039 016 034 023 024 030 666 022 471 025 974* 009 600 012 00 N) 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .5 N = 2500 Replication 1 LD SD -286 026 234 707 179 514 111 599 ■371 689 820 926 012 013 019 010 016 012 021 013 018 007 006 004 II LD SD 853 871 175 000 752 •468 242 154 097 645 532 116 007 006 020 013 010 014 018 016 020 008 008 012 Replication 2 II LD SD LD SD 639 492 196 •363 849 651 661 494 087 707 811 773 011 014 014 016 006 009 013 015 016 012 006 008 477 -052 684 678 077 533 ■076 202 815 000 525 002 012 020 010 Oil 013 010 015 020 009 016 008 014 Replication 3 I LD SD 707 013 456 044 321 915 560 155 191 170 311 465 927 018 021 017 005 012 020 019 021 018 014 004 II LD SD 000 017 -030 240 481 147 511 138 353 023 341 580 219 025 023 014 Oil 013 027 021 032 023 013 010 00 w 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .7 N = 100 Replication 1 Replication 2 Replication * 7 <5 1 1 I II I II I II LD SD LD SD LD SD LD SD LD SD LD SD 544 084 -022 099 722 029 631 032 873 024 385 040 352 084 381 096 796 040 112 061 039 116 192 150 393 093 376 096 720 045 408 054 212 106 154 136 134 116 132 117 156 090 317 096 530 084 326 103 019 132 137 134 315 090 393 085 837 039 000 069 572 063 542 067 837 029 000 064 615 045 748 050 366 052 817* 042 014 102 458 091 777 032 523 049 092 112 413 114 577 058 376 072 315 095 330 115 572 066 -065 093 897 027 -108 057 211 113 Oil 142 837* 047 000 078 439 062 -817* 047 533 068 347 086 009 108 386 106 282 090 299 103 576 070 336 088 127 088 593 066 507 065 536 062 754 048 -267 076 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .7 N = 150 Replication 1 LD SD 902 745 217 837 244 944 •030 336 •126 547 109 251 017 031 054 027 057 013 085 071 058 052 083 080 II LD SD 323 449 733 000 814 103 261 105 807 300 197 021 038 042 044 059 031 042 088 094 039 061 089 093 Replication 2 I LD SD 394 837 482 115 729 686 420 824 740 649 048 854 056 028 035 057 025 038 049 032 039 053 070 025 II LD SD 597 000 863 714 593 515 628 ■176 •133 ■Oil 698 167 046 054 020 043 025 044 043 056 063 066 041 053 Replication 3 LD SD 123 619 047 800 159 ■087 041 116 205 163 •278 837 089 029 097 027 058 057 087 075 088 063 051 027 II LD SD 214 742 130 448 778 686 301 535 221 657 906 000 086 026 097 044 030 046 077 063 078 051 027 067 CO Ul 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .7 N = 300 Replication 1 Replication 2 Replication 3 I II | I II I II LD SD LD SD LD SD LD SD LD SD LD SD 00 cn 372 049 315 055 -152 034 -885 014 344 047 214 057 837 020 000 042 -073 044 777 023 343 044 607 037 715 018 621 022 397 028 826 018 612 036 -287 042 839 023 072 038 419 036 554 037 033 352 033 779 021 709 023 500 029 785 023 324 037 713 015 653 016 837 022 000 046 501 048 159 052 558 024 733 018 559 020 764 014 -012 037 951 012 035 051 599 040 771 014 568 017 301 058 114 059 704 026 346 033 379 032“ 846 018 r I 914 010 205 023 -500 036 795 026 -961 006 -091 026 768 019 448 028 300 047 346 052 314 046 359 053 499 033 539 036 837 016 000 034 385 038 617 033 739 028 -103 039 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .7 N = 600 Replication 1 I II LD SD LD SD 348 041 139 039 082 025 987 005 455 030 427 033 262 037 357 035 717 019 334 021 595 026 356 026 285 041 113 041 726 012 603 012 776 012 507 015 342 038 063 039 837 015 000 027 422 016 873 007 Repli I LD SD 607 021 259 036 818 014 245 037 227 042 ?946 007 658 022 074 029 639 023 234 033 837 012 083 044 ation 2 II LD SD 551 022 693 022 -189 024 219 046 154 054 039 021 389 024 558 030 376 027 563 031 000 028 010 046 I LD SD 837 016 -021 032 887 009 222 039 681 020 868 Oil 339 037 526 030 701 019 031 035 344 023 638 013 :ion 3 II LD SD OOP 025 676 024 300 015 188 046 218 029 074 024 123 041 010 035 398 025 107 048 830 013 753 Oil 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .7 N = 2500 Replication 1 II LD SD LD SD 742 395 229 067 614 711 601 616 413 837 010 016 018 018 014 010 013 012 017 008 699 655 008 013 134 518 370 636 •180 226 356 328 403 000 015 014 017 012 018 016 017 016 017 014 598 102 008 015 Replication 2 II LD SD LD SD 837 007 445 952 238 944 308 497 ■054 •139 488 917 547 Oil 002 020 003 019 017 019 014 014 003 009 000 012 635 163 208 ■245 387 •064 509 842 528 381 725 Oil 008 019 Oil 017 020 016 008 Oil 006 007 Replication 3 LD SD 375 579 -046 456 666 494 696 ■398 148 837 022 015 020 016 007 016 010 014 022 008 558 428 Oil 013 II LD SD 276 017 149 017 717 Oil 096 021 629 008 056 021 404 012 721* 012 139 019 000 013 617 010 652 010 00 oo Communality = .9 N = 100 Replication 1 Replication 2 Replication 3 I II I II I II LD SD LD SD LD SD LD SD LD SD LD SD 1 925 020 -113 057 453 072 506 066 949 019 000 057 2 077 109 405 084 949 Oil 000 055 481 084 006 122 3 544 069 326 076 417 073 -509 057 430 080 -042 140 4 061 079 747 048 998 004 -046 048 823 035 -091 074 5 356 ( 081 418 074 035 088 335 086 627 054 669 053 6 800 1 032 332 057 -026 079 652 060 341 088 156 126 7 949 014 000 048 691 045 474 064 921 017 242 043 8 114 068 907* 030 -341 066 774 043 729 051 459 061 9 767 039 300 062 044 071 860 035 658 055 071 112 10 602 053 254 079 464 075 486 066 571 073 451 084 11 721 043 459 060 897 018 342 049 399 088 458 090 12 577 071 094 098 892 021 -424 041 682 049 431 062 Communality = .9 N = 150 Replication 1 Replication 2 Replication $ I II I II I II LD SD LD SD LD SD LD SD LD SD LD SD 1 572 057 409 061 312 063 441 074 207 084 312 103 2 190 057 844 030 949 015 000 049 525 047 631 055 3 145 095 111 110 499 059 248 073 381 098 188 117 4 949* 028 000* 055 015 049 939 015 949 031 000 068 5 667 038 136 069 463 053 479 062 813 018 538 028 6 285 079 263 101 904 019 -222 051 575 052 512 061 7 584 036 629 036 316 087 144 083 836 027 345 051 8 402 066 589 055 434 048 736 032 364 076 431 088 9 643 037 612 038 -193 064 629 049 090 101 101 126 10 614 035 659 034 101 088 310 079 500 064 592 057 11 613 044 552 048 -115 049 900 021 379 075 570 062 12 306 075 248 098 545 065 171 072 416 080 109 105 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .9 N = 300 Replication 1 I LD SD 193 061 712 031 532 044 424 033 320 044 949* 016 196 054 629 028 425 046 031 057 •180 045 670 035 II LD SD 170 196 181 754 541 000 278 443 507 555 782 128 066 041 054 021 040 033 057 036 041 039 025 048 Replication 2 LD SD 486 240 363 628 126 949 816 613 940 800 017 077 040 055 054 034 036 008 024 023 008 013 059 041 II LD SD 017 303 254 •050 852 000 ■212 663, 167 568 164 843 057 054 052 041 020 027 039 022 024 018 061 022 Replication 3 LD SD 457 533 514 610 622 742 292 358 160 369 972 949 046 040 049 031 017 019 046 052 066 032 006 007 II LD SD 330 467 305 296 774 453 571 408 095 891 067 000 046 037 049 044 013 033 038 049 058 014 030 035 < J D 1 2 3 4 5 6 7 8 9 10 11 12 Communality = .9 N = 600 Replication 1 LD SD 949 004 455 687 323 559 145 046 567 831 003 377 996 031 020 030 021 033 032 026 012 038 035 003 II LD SD 000 024 331 337 699 568 811 863 336 303 448 006 029 038 028 020 020 015 014 031 024 037 036 022 Replication 2 II LD SD LD SD 929 731 ■456 602 794 460 949 007 023 031 028 012 025 006 731 308 ■121 466 624 017 037 040 031 025 ■033 ■080 265 150 576 709 000 022 034 037 033 014 019 021 359 550 122 191 067 028 031 051 043 040 Replication 3 LD SD 738 855 005 807 949 018 013 042 012 007 328 -894 ! 481 ' 045 532 451 452 042 010 024 034 030 027 025 II LD SD 115 005 029 464 000 028 023 052 017 017 133 •087 735 798 052 259 712 038 021 017 018 041 035 018 ID t-O 1 2 T 4 5 6 7 8 9 10 11 12 Communality = .9 N = 2500 Replication 1 I LD SD II LD SD Replication 2 II LD SD LD SD Replication 3 LD SD 552 Oil 546 012 503 011 764 008 524 007 810 005 370 013 590 013 666 012 127 013 319 014 644 010 464 016 137 021 271 017 602 014 540 012 465 014 426 017 309 019 469 015 377 015 357 016 175 020 656 007 609 007 330 014 890 006 103 022 203 021 367 010 896 005 096 021 023 021 848 004 430 008 949 003 000 012 778 007 317 011 319 015 624 012 268 016 636 012 949 003 000 010 470 016 143 018 779 009 -030 016 763 009 -235 014 -030 013 863 006 916 003 333 007 956 002 208 008 794 004 562 006 686 008 392 013 922 004 067 010 163 015 692 010 476 012 649 009 299 018 -020 024 949 004 000 014 II LD SD < J D

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Asset Metadata

Creator Pennell, Roger John (author)

Core Title The Influence Of Communality And N On The Sampling Distributions Of Factor Loadings

Degree Doctor of Philosophy

Degree Program Psychology

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

Tag OAI-PMH Harvest,psychology, general

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Cliff, Norman (committee chair), Priest, Robert F. (committee member), Rigby, Gerald (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-142556

Unique identifier UC11360241

Identifier 6713039.pdf (filename),usctheses-c18-142556 (legacy record id)

Legacy Identifier 6713039.pdf

Dmrecord 142556

Document Type Dissertation

Rights Pennell, Roger John

Type texts

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

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

Repository Name University of Southern California Digital Library

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