The varimax criterion for analytic rotation in factor analysis

An analytic criterion for rotation is defined. The scientific advantage of analytic criteria over subjective (graphical) rotational procedures is discussed. Carroll's criterion and the quartimax criterion are briefly reviewed; the varimax criterion is outlined in detail and contrasted both logically and numerically with the quartimax criterion. It is shown that thenormal varimax solution probably coincides closely to the application of the principle of simple structure. However, it is proposed that the ultimate criterion of a rotational procedure is factorial invariance, not simple structure—although the two notions appear to be highly related. The normal varimax criterion is shown to be a two-dimensional generalization of the classic Spearman case, i.e., it shows perfect factorial invariance for two pure clusters. An example is given of the invariance of a normal varimax solution for more than two factors. The oblique normal varimax criterion is stated. A computational outline for the orthogonal normal varimax is appended.Part of the material in this paper is from the writer's Ph.D. thesis. I am indebted to my committee, Professors F. T. Tyler, R. C. Tryon, and H. D. Carter, chairman, for many helpful suggestions and criticisms. Dr. John Caffrey suggested the namevarimax, and wrote the original IBM 602A computer program for this criterion.I am also indebted to the staff of the University of California Computer Center for help in programming the procedures described in the paper for their IBM 701 electronic computer. Since their installation is partially supported by a grant from the National Science Foundation, the assistance of this agency is acknowledged.     

Tandem criteria for analytic rotation in factor analysis

Two related orthogonal analytic rotation criteria for factor analysis are proposed. Criterion I is based upon the principle that variables which appear on the same factor should be correlated. Criterion II is based upon the principle that variables which are uncorrelated should not appear on the same factor. The recommended procedure is to rotate first by criterion I, eliminate the minor factors, and then rerotate the remaining major factors by criterion II. An example is presented in which this procedure produced a rotational solution very close to expectations whereas a varimax solution exhibited certain distortions. A computer program is provided.     

SIMPLE STRUCTURE AND THE NUMBER OF FACTORS PROBLEM

If the varimax criterion values. for rotations of successively greater numbers of principal axes of the image covariance matrix G are listed, they may be examined for two features: the unimodality of the distribution reflects the clarity of latent structure; and the peak value indicates the number of common factors present. The procedure is applied to nine published example problems and compared to other rules of thumb for determining the rank of a factor matrix.     

Varimax solution for primary mental abilities

The varimax solution for Thurstone's classic Primary Mental Abilities study is presented. Comparisons between the factors of Thurstone's original subjectively rotated factor pattern, Zimmerman's subjectively revised solution, Wrigley, Saunders, and Neuhaus' quartimax results, and the present varimax factor matrix are made by finding correlations between factors defined by these four solutions. It is pointed out that any possible ultimate merit of the varimax solution should be based on its psychological meaningfulness and on the rationale of the varimax criterion—not on its relationship to the other studies.The computations for this paper were done on Illiac, an electronic computer of the Digital Computer Laboratory of the University of Illinois. Mr. S. M. Hunka assisted in these computations.     

A matrix formulation of Kaiser's varimax criterion

Kaiser has given the varimax criterion for the solution of the rotation problem in factor analysis as well as a practical computational procedure for maximizing this criterion. In the present paper, the maximization condition is shown as a matrix equation involving only the unknown orthogonal rotation matrix. This matrix equation can be solved iteratively as a sequence of symmetric eigenproblems.This investigation was supported by Public Health Service grant number MH 07285-03 from the National Institute of Mental Health.     

A direct solution for pairwise rotations in Kaiser's varimax method

The present note contains a completing-the-squares type approach to the varimax rotation problem. This approach yields a direct proof of global optimality of a solution for optimal rotation in a plane. Because varimax rotation can be interpreted as diagonalization of a set of symmetric matrices, the present solution also applies to the diagonalization problem.The author is obliged to Jos M. F. ten Berge for helpful comments.     

Controlling item exposure and test overlap on the fly in computerized adaptive testing

This paper proposes an on‐line version of the Sympson and Hetter procedure with test overlap control (SHT) that can provide item exposure control at both the item and test levels on the fly without iterative simulations. The on‐line procedure is similar to the SHT procedure in that exposure parameters are used for simultaneous control of item exposure rates and test overlap rate. The exposure parameters for the on‐line procedure, however, are updated sequentially on the fly, rather than through iterative simulations conducted prior to operational computerized adaptive tests (CATs). Unlike the SHT procedure, the on‐line version can control item exposure rate and test overlap rate without time‐consuming iterative simulations even when item pools or examinee populations have been changed. Moreover, the on‐line procedure was found to perform better than the SHT procedure in controlling item exposure and test overlap for examinees who take tests earlier. Compared with two other on‐line alternatives, this proposed on‐line method provided the best all‐around test security control. Thus, it would be an efficient procedure for controlling item exposure and test overlap in CATs.     

The weighted varimax rotation and the promax rotation

Kaiser's iterative algorithm for the varimax rotation fails when (a) there is a substantial cluster of test vectors near the middle of each bounding hyperplane, leading to non-bounding hyperplanes more heavily overdetermined than those at the boundaries of the configuration of test vectors, and/or (b) there are appreciably more thanm (m factors) tests whose loadings on one of the factors of the initialF-matrix, usually the first, are near-zero, leading to overdetermination of the hyperplane orthogonal to this initialF-axis before rotation. These difficulties are overcome by weighting the test vectors, giving maximum weights to those likely to be near the primary axes, intermediate weights to those likely to be near hyperplanes but not near primary axes, and near-zero weights to those almost collinear with or almost orthogonal to the first initialF-axis. Applications to the Promax rotation are discussed, and it is shown that these procedures solve Thurstone's hitherto intractable “invariant” box problem as well as other more common problems based on real data.     

Factor comparison of two Minnesota Multiphasic Personality Inventory short forms

A sample of 315 valid Minnesota Multiphasic Personality Inventory (MMPI) protocols were selected from inpatient files and scored for both the MMPI-168 and Faschingbauer Abbreviated MMPI short forms. Each short form was then factor-analyzed by a principal axis strategy with varimax rotation. The six factors extracted from each short form were then compared as to their similarity by use of the s index. This procedure showed five of the six factors in each short form as having a significant relationship of the pattern of salient variables they had in common with the complimentary factors of the other form. These data suggest that both short forms, though based on different construction methodologies, share the same underlying personality attributes. Future research is suggested to replicate these results and extend them to the full MMPI.     

Die‐by Framing both Lengthens and Shortens Life: Further Evidence on Constructed Beliefs in Life Expectancy

Decisions regarding consumption over the lifespan require some estimate of how long that lifespan is likely to be. Payne et al. (2013) found that respondents' estimates of their own life expectancy are on average 8.6 years shorter when elicited using a die‐by frame than when elicited by a live‐to frame. If decision makers act on these life expectancies, then an arbitrary detail of framing will lead to drastically different choices. We propose that the framing effect is sensitive to the iterative probabilistic elicitation procedure employed by the previous literature. Study 1 compares the framing effect across the iterative probabilistic procedure and a point estimate procedure that simply asks respondents the age they will live to/die by. The iterative probabilistic procedure implies a life expectancy six years shorter in the die‐by frame than in the live‐to frame, replicating the results of Payne et al. (2013). With the point estimate procedure, however, the framing effect reverses: the die‐by frame increases life expectancy by three years. In Study 2, we test for the framing effect using a point estimate procedure on a representative sample of 2000 Britons. Again, and in contrast with the previous literature, we find that the die‐by frame implies longer life. Our results reinforce the previous literature that beliefs around life expectancy are constructed. We recommend caution when attempting to debias life expectancy estimates or using life expectancies in choice architecture. Copyright © 2017 John Wiley & Sons, Ltd.     

An iterative procedure for extracting skill maps from data

The methodologies for the construction of a knowledge structure mainly refer to the query to experts, the skill maps, and the data-driven approaches. This last method is of growing interest in recent literature. In this paper, an iterative procedure for building a skill map from a set of data is introduced. This procedure is based on the minimization of the distance between the knowledge structure delineated by a given skill map and the data. The accuracy of the proposed method is tested through a number of simulation studies where the amount of noise in the data is manipulated as well as the kind of structure to be reconstructed. Results show that the procedure is accurate and that its performance tends to be sufficiently stable even with high error rates. The procedure is compared to two already-existing methodologies to derive knowledge structures from a set of data. The use of the corrected Akaike Information Criterion (AICc) as a stopping criterion of the iterative reconstruction procedure is tested against the app criterion introduced by Schrepp. Moreover, two empirical applications on clinical data are reported, and their results show the applicability of the procedure.     

The Conservatism Scale: In Search of a Replicable Factor Structure

Abstract

Responses of three independent groups of subjects to the Wilson-Patterson Conservatism Scale were submitted to principal components analyses followed by varimax rotation. Inspection of the unrotated factor loadings provided evidence for the presence of a general factor in all three groups, supporting the contention of the scale's authors (Wilson & Patterson, 1968). Detailed analyses of the present and previously published results revealed a consistent underlying theme of fundamental religious conservatism. Examination of loadings obtained from two-, three-, and four-factor rotations, using the FACTOREP procedure for factor comparison, failed to identify any other consistently replicable factor structure, a result consistent with conclusions based on an examination of previously published research.     

Iterating between lessons on concepts and procedures can improve mathematics knowledge

Background Knowledge of concepts and procedures seems to develop in an iterative fashion, with increases in one type of knowledge leading to increases in the other type of knowledge. This suggests that iterating between lessons on concepts and procedures may improve learning. Aims The purpose of the current study was to evaluate the instructional benefits of an iterative lesson sequence compared to a concepts‐before‐procedures sequence for students learning decimal place‐value concepts and arithmetic procedures. Samples In two classroom experiments, sixth‐grade students from two schools participated (N = 77 and 26). Method Students completed six decimal lessons on an intelligent‐tutoring systems. In the iterative condition, lessons cycled between concept and procedure lessons. In the concepts‐first condition, all concept lessons were presented before introducing the procedure lessons. Results In both experiments, students in the iterative condition gained more knowledge of arithmetic procedures, including ability to transfer the procedures to problems with novel features. Knowledge of concepts was fairly comparable across conditions. Finally, pre‐test knowledge of one type predicted gains in knowledge of the other type across experiments. Conclusions An iterative sequencing of lessons seems to facilitate learning and transfer, particularly of mathematical procedures. The findings support an iterative perspective for the development of knowledge of concepts and procedures.     

Bootstrap confidence intervals for principal covariates regression

Principal covariate regression (PCOVR) is a method for regressing a set of criterion variables with respect to a set of predictor variables when the latter are many in number and/or collinear. This is done by extracting a limited number of components that simultaneously synthesize the predictor variables and predict the criterion ones. So far, no procedure has been offered for estimating statistical uncertainties of the obtained PCOVR parameter estimates. The present paper shows how this goal can be achieved, conditionally on the model specification, by means of the bootstrap approach. Four strategies for estimating bootstrap confidence intervals are derived and their statistical behaviour in terms of coverage is assessed by means of a simulation experiment. Such strategies are distinguished by the use of the varimax and quartimin procedures and by the use of Procrustes rotations of bootstrap solutions towards the sample solution. In general, the four strategies showed appropriate statistical behaviour, with coverage tending to the desired level for increasing sample sizes. The main exception involved strategies based on the quartimin procedure in cases characterized by complex underlying structures of the components. The appropriateness of the statistical behaviour was higher when the proper number of components were extracted.     

Comparison Of Oblique And Orthogonal Simple Structure Solutions For Personality And Interest Factors

The study was directed to the need to structure in a few variables the domain measured by personality and interest measures commonly employed in educational counseling: Strong, Kuder, EPPS and the Study of Values. Despite initial uncertainty regarding number of factors to be employed, effects of ipsative scores and of mixing test formats, both an oblique and orthogonal rotation yielded nearly identical results. Of the twenty factors identified by both the biquartimin and varimax solutions, Seven linked vocational interest clusters with personality. Two of the remaining factors had only interest loadings, while of the eleven personality factors, only four were scale specific. Definition of the 16 common factors required that extraction proceed beyond the unit latent root criterion. The results offer evidence that over- extracting factors does not confuse the results of rotation. Further, psycho- metric differences between tests had essentially no effect on the factors found. Of three oblimin rotations attempted, only the biquartimin was successful, yielding results essentially like those of the varimax solution. Because of the vast difference in computation time for these two solutions (computer time 20 times greater for biquartimin), however, the orthogonal varimax remains the method of choice.     

Factor structure of items from the personality Adjective Check List in a sample of normal adolescents

We present the results of a principal-components analysis of the Personality Adjective Check List (PACL) at the item level, rather than at the scale level. The PACL was constructed by Strack (1987) to evaluate the personality scales proposed by Millon (1981). Data were obtained from a sample of normal adolescents. A scree test was applied to determine the number of factors to be extracted, and the retained factors were then rotated using a varimax procedure. Using the data from this second analysis, congruence coefficients were calculated by correlating the factor weights from the two analyses of men and women. Results reveal the existence in the whole sample of five unipolar personality factors, which we have called: Aggressive, Conscientious, Inhibited, Impulsive and Gregarious. These are not similar to those obtained from a sample of normal adults.     

Convergence properties of an iterative procedure of ipsatizing and standardizing a data matrix,with applications to parafac/candecomp preprocessing

Centering a matrix row-wise and rescaling it column-wise to a unit sum of squares requires an iterative procedure. It is shown that this procedure converges to a stable solution. This solution need not be centered row-wise if the limiting point of the interations is a matrix of rank one. The results of the present paper bear directly on several types of preprocessing methods in Parafac/Candecomp.     

On the estimation of parameters in latent structure analysis

In this note, we describe the iterative procedure introduced earlier by Goodman to calculate the maximum likelihood estimates of the parameters in latent structure analysis, and we provide here a simple and direct proof of the fact that the parameter estimates obtained with the iterative procedure cannot lie outside the allowed interval. Formann recently stated that Goodman's algorithm can yield parameter estimates that lie outside the allowed interval, and we prove in the present note that Formann's contention is incorrect.This research was supported in part by Research Contract No. NSF SOC 76-80389 from the Division of the Social Sciences of the National Science Foundation. The author is indebted to C. C. Clogg for helpful comments and for the numerical results reported here (see, e.g., Table 1).     

Methodological examination of the PEST (Parametric Estimation by Sequential Testing) procedure

Any psychophysical procedure requires arbitrary decisions on the part of E at the beginning of the experiment. Even within adaptive stimulus programming, e.g., PEST, procedures, which are modified according to the moment-by-moment responses of S, E must initialize the procedure. The PEST procedure was employed in three different auditory tasks and examined for its immunity to initializing conditions. PEST, despite its adaptive features, is not entirely immune to initializing conditions. It is suggested that an iterative approach may assist the initializing of conditions for psychophysical procedures.     

Mood as input and catastrophic worrying

The authors describe 3 experiments investigating a "mood-as-input" approach to understanding catastrophic worrying. Experiment 1 found that induced negative mood increased the number of steps emitted in both a catastrophizing interview procedure and a positive iteration task. Experiment 2 found that the number of items that worriers emitted in an iterative item generation task was dependent on the stop rules specified by the procedure. Experiment 3 found that manipulating the stop rules for catastrophizing had differential effects on worriers and nonworriers, depending on the nature of the stop rules specified. These results suggest that mood provides information about continuing or terminating the catastrophizing process that is interpreted in the context of the stop rules for the task. It is argued that the mood-as-input hypothesis accounts for the facts of exacerbated catastrophizing in worriers better than explanations couched in terms of either mood congruency effects or worriers possessing a generalized perseverative iterative style.