The influence of communality,factor strength,and loading size on the sampling characteristics of factor loadings

A Monte Carlo approach is employed in determining whether or not certain variables produce systematic effects on the sampling variability of individual factor loadings. A number of sample correlation matrices were generated from a specified population, factored, and transformed to a least-squares fit to the population values. Influences of factor strength, communality and loading size are discussed in relation to the statistics summarizing the results of the above procedures. Influences producing biased estimators of the population values are also discussed.This study was supported in part by NSF Grant GB 4230. Computing assistance was obtained from the Western Data Processing Center and Health Sciences Computing Facility, UCLA, sponsored by NIH Grant FR-3.     

On the sampling errors of factor loadings

The results of three empirical studies on the sampling fluctuation of centroid factor loadings are reported. The first study is based on data which happened to be available on 8 variables for 700 cases and which were factored to three factors for subsamples. The second study is based on fictitious data for 2500 cases which provided separate analyses on 25 samples for each of three situations: 5 variables, one factor; 5 variables, two factors; and 6 variables, three factors. The third study, based on real data for 9 variables and 7000 cases, involves separate factorization for 35 samples of 200 cases. The three studies agree in showing that the sampling behavior of first centroid factor loadings is much like that of correlation coefficients, whereas the sampling fluctuations for loadings beyond the first are disturbingly large.     

The determination of the factor loadings of a given test from the known factor loadings of other tests

A technique is indicated by which approximations to the factor loadings of a new test may be obtained if factor loadings of a given group of tests and the correlations of the new test with the other tests are known. The technique is applicable to any orthogonal system and is especially adapted to cases in which a _ji a _jk = 0 wheni k. Application is also made to the simultaneous determination of the factor weights of a group of tests in which no additional common factor is present. The technique is useful in adding tests to a completed factorial solution and in using factorial solutions involving errors to give results which are approximately correct.     

Some properties of the communality in multiple factor theory

Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established. The following theorems are applicable to a battery ofn tests which are describable in terms ofr common factors, with orthogonal reference vectors.1. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors.2. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors and then—1 remaining tests. Corollary: The square of the multiple correlation of a testj with then—1 remaining tests is equal to or less than the communality of testj. It cannot exceed the communality.3. The square of the multiple correlation of a testj with then—1 remaining tests equals the communality of testj if the group of tests containsr statistically independent ests teach with a communality of unity.4. With correlation coefficients corrected for attenuation, when the number of tests increases indefinitely while the rank of the correlational matrix remains unchanged, the communality of a testj equals the square of the multiple correlation of testj with then—1 remaining tests.5. With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a testj with then—1 remaining tests approaches the communality of testj as a limit when the number of tests increases indefinitely while the rank of correlational matrix remains the same. This has not yet been proved for the general case.The author wishes to express his appreciation of the encouragement and assistance given him by Dr. L. L. Thurstone.     

Stability of factor loadings

Each of 17 tests was embedded in six different eight-test matrices. Stability of general factor loadings were determined over the six contexts. Correlations of factor loadings ranged from 0.52 to 0.94 with an average value of 0.83.     

The sampling distributions of Gaussian ROC statistics

For a discrimination experiment, a plot of the hit rate against the false-alarm rate--the ROC curve--summarizes performance across a range of confidence levels. In many content areas, ROCs are well described by a normal-distribution model and the z-transformed hit and false-alarm rates are approximately linearly related. We examined the sampling distributions of three parameters of this model when applied to a ratings procedure: the area under the ROC (Az), the normalized difference between the means of the underlying signal and noise distributions (da), and the slope of the ROC on z-coordinates (s). Statistical bias (the degree to which the mean of the sampling distribution differed from the true value) was trivial for Az, small but noticeable for da, and substantial for s. Variability of the sampling distributions decreased with the number of trials and was also affected by the number of response categories available to the participant and by the overall sensitivity level. Figures in the article and tables available on line can be used to construct confidence intervals around ROC statistics and to test statistical hypotheses.     

Subjective sampling distributions and the additivity of estimates

Abstract.— Previous studies of sampling distributions have been conducted almost exclusively under the assumption that persons behave in accordance with the "fundamental convention" of probability, i.e. that the sum of all probability estimates will equal 1. When this assumption was tested by asking subjects to give "unrestricted" probability estimates of all possible outcomes of samples from a given population, a general tendency of overestimation made the sum of all probabilities exceed 1 to a considerable extent. The subjective sampling distributions appeared to be unaffected by sample size ( N=5 or 10) and number of outcomes, and were flatter than the corresponding "objective" sampling distributions.     

The relation between factor loadings and biserial correlations in item analysis

The first centroid factor loadings obtained from various interitem relations are compared with item discrimination indices commonly used in item analysis. Depending upon what type of matrix is factored, the factor loadings are shown to be related to point biserial and biserial correlations.     

Estimating the standard errors of rotated factor loadings by jackknifing

The jackknife by groups and modifications of the jackknife by groups are used to estimate standard errors of rotated factor loadings for selected populations in common factor model maximum likelihood factor analysis. Simulations are performed in whicht-statistics based upon these jackknife estimates of the standard errors are computed. The validity of thet-statistics and their associated confidence intervals is assessed. Methods are given through which the computational efficiency of the jackknife may be greatly enhanced in the factor analysis model.Computing assistance was obtained from the Health Sciences Computing Facility, UCLA, sponsored by NIH Special Research Resources Grant RR-3.The author wishes to thank his doctoral committee co-chairmen, Drs James W. Frane and Robert I. Jennrich, UCLA, for their contributions to this research.     

The influence of temporal distributions on causal induction from tabular data

Two experiments investigated the joint influence of statistical and temporal information on causal inference from tabular data. Participants were presented with unambiguous data sets containing information about relative effect frequencies in cause-present and cause-absent situations. In addition to contingency information, the stimuli also revealed information about the temporal distribution of effects. The participants took this information into account when making causal judgments, so that the mere advancing or postponing of the effect in time was attached with causal significance, even when the cause did not increase the overall probability of the effect. These results cannot be reconciled with standard contingency accounts of causal induction.     

An investigation of the sampling distributions of item discrimination indices

The sampling properties of four item discrimination indices (biserialr, Cook's indexB, theU–L 27 per cent index, and DeltaP) were investigated in order to ascertain their sampling properties when small samples drawn from actual test data rather than constructed data were employed. The empirical results indicated that the mean index values approximated the population values and that values of the standard deviations computed from large sample formulas were good approximations to the standard deviations of the observed distributions based on samples of size 120 or less. Goodness of fit tests comparing the observed distributions with the corresponding distribution of the product-moment correlation coefficient based upon a bivariate normal population indicated that this correlational model was inappropriate for the data. The lack of adequate mathematical models for the sampling distributions of item discrimination indices suggests that one should avoid indices whose only real reason for existence was the simplification of computational procedures.This research reported herein was performed pursuant to a contract (OE-2-10-071) with the United States Office of Education, Department of Health, Education and Welfare.     

Note on the squared multiple correlation as a lower bound to communality

A simple proof that the squared multiple correlation of a variable with the remaining variables in the set of variables is a lower bound to the communality of that variable is presented.     

The effect of agency and communality on self-esteem: Gender differences in longitudinal data

Prior studies have noted a positive relationship between masculinity (M) and self-esteem (S-E) for both males and females. However, during adolescence, males and females have been described as experiencing a divergence into two domains: agency for males and communality for females. The present study hypothesized that, contrary to the cross-sectional positive correlational relationship between M and S-E for both males and females, an agentic orientation during adolescence would predict heightened S-E at young adulthood for males only. Conversely, a communal orientation was hypothesized to predict heightened S-E for females only at young adulthood. Using structural equation models with latent variables to analyze data from a community sample of 654 subjects (M=192, F=462; White=432; Black=97; Hispanic=64; Asian=61), these hypotheses were confirmed. In addition, as found in typical masculinity-femininity (M-F) studies, both males and females evidenced a positive relationship between S-E and M, as measured on the Personal Attributes Questionnaire and essentially no relationship between F and S-E. Possible reasons for this complexity and implications for future research are discussed.     

Asymptotic comparison of missing data procedures for estimating factor loadings

Large sample properties of four methods of handling multivariate missing data are compared. The criterion for comparison is how well the loadings from a single factor model can be estimated. It is shown that efficiencies of the methods depend on the pattern or arrangement of missing data, and an evaluation study is used to generate predictive efficiency equations to guide one's choice of an estimating procedure. A simple regression-type estimator is introduced which shows high efficiency relative to the maximum likelihood method over a large range of patterns and covariance matrices.     

Recovery of weak factor loadings in confirmatory factor analysis under conditions of model misspecification

This article presents the results of two Monte Carlo simulation studies of the recovery of weak factor loadings, in the context of confirmatory factor analysis, for models that do not exactly hold in the population. This issue has not been examined in previous research. Model error was introduced using a procedure that allows for specifying a covariance structure with a specified discrepancy in the population. The effects of sample size, estimation method (maximum likelihood vs. unweighted least squares), and factor correlation were also considered. The first simulation study examined recovery for models correctly specified with the known number of factors, and the second investigated recovery for models incorrectly specified by underfactoring. The results showed that recovery was not affected by model discrepancy for the correctly specified models but was affected for the incorrectly specified models. Recovery improved in both studies when factors were correlated, and unweighted least squares performed better than maximum likelihood in recovering the weak factor loadings.     

Note on the squared multiple correlation as a lower bound to communality

It is demonstrated that the squared multiple correlation of a variable with the remaining variables in a set of variables is a function of the communalities and the squared canonical correlations between the observed variables and common factors. This equation is shown to imply a strict inequality between the squared multiple correlation and communality.     

Standard errors for the class of orthomax-rotated factor loadings: Some matrix results

The partial derivative matrices of the class of orthomax-rotated factor loadings with respect to the unrotated maximum likelihood factor loadings are derived. The reported results are useful for obtaining standard errors of the orthomax-rotated factor loadings, with or without row normalization (standardization) of the initial factor loading matrix for rotation. Using a numerical example, we verify our analytic formulas by comparing the obtained standard error estimates with that from some existing methods. Some advantages of the current approach are discussed.Authorship is determined by alphabetical order. The authors contributed equally to the research. Kentaro Hayashi is now at the Department of Mathematics, Bucknell University, Lewisburg, PA 17837 (email: khayashi@Bucknell.edu). Yiu-Fai Yung is now at the SAS Institute, Inc., SAS Campus Drive, Cary, NC 27513 (email: yiyung@wnt.sas.com).Part of the research was completed while Yiu-Fai Yung was a visiting scholar at the Department of Psychology, the Ohio State University. The visit was supported in part by grant N4856118101 from the NIMH and the Mason and Linda Stephenson Travel Award from the Department of Psychology, University of North Carolina at Chapel Hill. The authors are grateful to Michael Browne who suggested some relevant references and provided valuable comments on the research, and to Robert Cudeck who provided the FAS program for the numerical comparison. The expert comments by the reviewers are deeply appreciated.