Standard errors for rotated factor loadings

Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented.This research was supported in part by NIH Grant RR-3. The authors are grateful to Dorothy T. Thayer who implemented the algorithms discussed here as well as those of Lawley and Maxwell. We are particularly indebted to Michael Browne for convincing us of the significance of this work and for helping to guide its development and to Harry H. Harman who many years ago pointed out the need for standard errors of estimate.     

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.     

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.     

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.     

Influence curves for factor loadings

Influence curves for the initial and rotated loadings are derived for the maximum likelihood factor analysis (MLFA) model. Cook's distances based on the empirical influence curves of factor loadings are proposed for the identification of influential observations. The distances are shown to be invariant under scale transformation and factor rotation. We find that an observation with a very large Cook's distance based on the sample influence curve may not necessarily exert an excessive influence on the factor loadings pattern but may change the ordering of the factors. The issue of the switching of factors is also studied by means of the empirical influence curve and factor scores.     

Standard errors for procrustes solutions

The asymptotic standard errors for the procrustes solutions are derived for orthogonal rotation, direct oblique rotation and indirect oblique rotation. The standard errors for the first two rotations are obtained using the augmented information matrices. For the indirect oblique solution, the standard errors of rotated parameters are derived from the information matrix of unrotated loadings using the chain rule for information matrices. For all three types of rotation, the standard errors of rotated parameters are presented for unstandardized and standardized manifest variables. Numerical examples show the similarity of theoretical and simulated values.     

A look,by simulation,at the validity of some asymptotic distribution results for rotated loadings

In the last few years, a number of asymptotic results for the distribution of unrotated and rotated factor loadings have been given. This paper investigates the validity of some of these results based on simulation techniques. In particular, it looks at principal component extraction and quartimax rotation on a problem with 13 variables. The indication is that the asymptotic results are quite good.     

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.     

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.     

Standard errors of prediction for the vineland adaptive behavior scales

When a scale has been administered to the same person more than once, the question arises whether variation in obtained scores is due to measurement error or whether it reflects actual change in the examinee's functional level. This article offers standard errors of prediction and confidence intervals for the Vineland Adaptive Behavior Scales (VABS) that can help in making such decisions where these scales are concerned. The values presented here were derived from the normative data provided in the VABS manuals.     

Rotation for simple loadings

Existing analytic oblique rotation schemes proceed by optimizing a simplicity function applied to the reference structure. This article suggests optimizing a simplicity function applied to primary loadings directly. The feasibility of the suggestion is demonstrated using the quartimin criterion. An algorithm to implement the optimization is derived and the existence of an admissible solution proved. Practical comparisons with the biquartimin method are made using Thurstone's Box Problem and Holzinger and Swineford's Twenty-Four Psychological Tests Problem.     

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.     

Scanning for reversed and rotated targets

Children and adults scanned arrays of identical shapes for mirror reversed or 180° rotated targets. The decomposition of scanning time into ‘target identification time’ and ‘context scanning time’ indicated that it takes longer to identify reversed than rotated targets but that it takes equally long in both conditions to scan context. This confirms the notion of a specific difficulty in coding mirror reversed shapes.When the shapes used contained vertical lines, ‘scanning time’ increased while ‘target identification time’ was unaffected. This is probably due to a tendency to fixate lines at right angles to the direction of scan and does not reflect a coding difficulty.     

Compensating for distortion in viewing pictures obliquely

Gombrich (1972) suggested that interpretation of obliquely viewed pictures depends largely on the foreshortened image projected to the eye, the attendant distortion rarely being important. Sixteen university students judged whether 128 drawings viewed at 41 and 26 deg to the page pictured rectangular or nonrectangular boxes. Projective geometry predicted Ss’ classifications; for many pictures, predictions for normal and oblique viewing differed. Nevertheless, at 41 deg, Ss classified 77% of the pictures where predictions differed according to the normal prediction. Furthermore. performance on cases where the predictions agreed was no better. Even at 26 deg, normal classifications still dominated. Ss demonstrated substantial capacity to compensate for oblique viewing and small tendency to judge according to the foreshortened image.     

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.     

Standard errors in covariance structure models: Asymptotics versus bootstrap

Commonly used formulae for standard error (SE) estimates in covariance structure analysis are derived under the assumption of a correctly specified model. In practice, a model is at best only an approximation to the real world. It is important to know whether the estimates of SEs as provided by standard software are consistent when a model is misspecified, and to understand why if not. Bootstrap procedures provide nonparametric estimates of SEs that automatically account for distribution violation. It is also necessary to know whether bootstrap estimates of SEs are consistent. This paper studies the relationship between the bootstrap estimates of SEs and those based on asymptotics. Examples are used to illustrate various versions of asymptotic variance–covariance matrices and their validity. Conditions for the consistency of the bootstrap estimates of SEs are identified and discussed. Numerical examples are provided to illustrate the relationship of different estimates of SEs and covariance matrices.