A note on the discovery of ag factor by means of Thurstone's centroid method of analysis

A fictitious factor matrix including 16 tests and 3 factors, one of which was ag factor, was prescribed. From it two typical factor problems, including errors of sampling, were derived. Students in training, without awareness of the factor patterns, arrived at essentially correct solutions by the use of Thurstone's centroid method with rotation of axes. Errors in the calculated factor matrix were very close in size to the sampling errors in the correlation coefficients. It is concluded that ag factor need not escape detection by Thurstone's procedures if the criteria of complete simple structure are not demanded.     

Factorial analysis of Thurstone's seven primary abilities

Intercorrelation coefficients among Thurstone's seven primary mental abilities scores were obtained from scores of 170 freshmen engineering students on the experimental edition of thePrimary Mental Abilities Tests. These correlation coefficients were factored to four factors, interpreted as a general factor, a reasoning factor, a verbal factor, and, tentatively, a specific memory factor. The finding of a general factor for a college population corroborates the Thurstones' finding of a general factor for eighth-grade children.     

On Simple Structure and the Solution to Thurstone's Invariant Box Problem

Thurstone's invariant box problem has 2 features which prevent the finding of the obvious simple structure by any of the usual analytic methods of rotation.     

On using multiple imputation for exploratory factor analysis of incomplete data

A simple multiple imputation-based method is proposed to deal with missing data in exploratory factor analysis. Confidence intervals are obtained for the proportion of explained variance. Simulations and real data analysis are used to investigate and illustrate the use and performance of our proposal.     

A factorial analysis of Thurstone's sixteen primary mental abilities tests

This article deals with a factorial analysis of Thurstone'sPrimary Mental Abilities Tests. The analysis was made in order to determine whether the tests designated to measure a particular ability would be found upon completion of the analysis to be clustered together with significant loadings and be isolated from the tests of the other abilities. The results show that most of the tests specifically designed to measure six of the abilities were isolated with loadings varying in size. The tests for the remaining ability were not isolated. It is also shown that the tests are complex in nature, measuring more than any one single ability.     

On the investigation of local identifiability: A counterexample

An assertion that the parameters of a covariance structure are locally identified at a certain point only if the rank of the Jacobian matrix at that point equals the number of parameters, is shown to be false by means of a counterexample.     

Remarks on the identifiability of thurstonian paired comparison models under multiple judgment

It is well-known that the representations of the Thurstonian models for difference judgment data are not unique. It has been shown that equivalence classes can be formed to provide a more meaningful partition of the covariance structures of the Thurstonian ranking models. In this paper, we examine the equivalence relations between Thurstonian covariance structure models for paired comparison data obtained under multiple judgment and discuss their implications on the general identification constraints and methods to check for parameter identifiability in restricted models.The author is indebted to Ulf Böckenholt and Albert Maydeu-Olivares for their significant comments and suggestions which led to considerable improvement in this article.     

Exploratory longitudinal factor analysis in multiple populations

For multiple populatios, a longtidinal factor analytic model which is entirely exploratory, that is, no explicit identification constraints, is proposed. Factorial collapse and period/practice effects are allowed. An invariant and/or stationary factor pattern is permitted. This model is formulated stochastically. To implement this model a stagewise EM algorithm is developed. Finally a numerical illustration utilizing Nesselroade and Baltes' data is presented.The authors wish to thank Barbara Mellers and Henry Kaiser for their helpful comments and John Nesselroade for providing us the data for our illustration. This research wwa supported in part by a grant (No. AG03164) from the National Institute on Aging to William Meredith. Details of the derivations and a copy of the PROC MATRIX program are available upon request from the first author.     

The use of factor analysis in the statistical analysis of multiple time series

Factor analysis has been proposed and used as a method of statistical analysis of several measurements made on one individual repeatedly over time. This paper discusses some difficulties in applying factor analysis to multiple time series and attempts to indicate to what extent such methods can accomplish the goals of time series analysis. Some other methods are suggested.Research sponsored by contract AF41(657)-214 between the USAF School of Aerospace Medicine and Teachers College, Columbia University.     

Matrix analysis of identifiability of some finite markov models

Methods developed by Bernbach [1966] and Millward [1969] permit increased generality in analyses of identifiability. Matrix equations are presented that solve part of the identifiability problem for a class of Markov models. Results of several earlier analyses are shown to involve special cases of the equations developed here. And it is shown that a general four-state chain has the same parameter space as an all-or-none model if and only if its representation with an observable absorbing state is lumpable into a Markov chain with three states.This research was supported by the U.S. Public Health Service under Grant MH-12717 to Indiana University and Grant GM-1231 to the University of Michigan.Now at the University of Texas, Austin.     

On the statistical treatment of residuals in factor analysis

A method for estimation in factor analysis is presented. The method is based on the assumption that the residual (specific and error) variances are proportional to the reciprocal values of the diagonal elements of the inverted covariance (correlation) matrix. The estimation is performed by a modification of Whittle's least squares technique. The method is independent of the unit of scoring in the tests. Applications are given in the form of nine reanalyses of data of various kinds found in earlier literature.The writer wishes to thank Prof. H. Wold, Dr. E. Lyttkens, and Dr. P. Whittle for valuable comments and suggestions.     

On admissible communalities in factor analysis

A set of inequalities on communalities developed by Darroch [1] is used to provide some rules for determining admissible sets of communalities in factor analysis.     

On Bayesian estimation in unrestricted factor analysis

It is shown that the common and unique variance estimates produced by Martin & McDonald's Bayesian estimation procedure for the unrestricted common factor model have a predictable sum which is always greater than the maximum likelihood estimate of the total variance. This fact is used to justify a suggested simple alternative method of specifying the Bayesian parameters required by the procedure.     

Revisiting Thurstone's and Coombs' scales on the seriousness of crimes and offences

The classical studies by Thurstone (1927b) and Coombs (1967) on the seriousness of crimes and offences are replicated here for a sample of male and female students. Eighty subjects used both forced choice and graded dominance ratios to assess all pairs of 10 crimes/offences. The direct ratio estimates correlate very highly with pair-comparison probabilities, thus making it possible to interpret the former as direct estimates of dominance probabilities. These data can be scaled with excellent fit in Thurstone's sense. The resulting scales are quite similar to those obtained by Thurstone and Coombs. An analysis of the individual data shows, however, that 69 per cent of the average data are aggregated over a variety of distinct subgroups of individuals who partition the crimes/offences into subsets that are incomparable in their seriousness. Thurstone's and Coombs' seriousness scales must therefore be considered a population lawfulness.     

The square root method and multiple group methods of factor analysis

The square root method for the solution of a set of simultaneous linear equations or the reduction of a matrix has been known for some time under a variety of names. Because of its usefulness in statistical work, especially in factor analysis, the square root method is presented in general terms and an example given. Several independently developed multiple group methods for factor analysis are compared and synthesized. Their fundamental concepts are set forth and an appropriate system of notation developed. Detailed computational procedures are outlined, and the square root method is emphasized as a computing aid in multiple group analysis.The opinions expressed in this paper are those of the author and do not necessarily reflect official Department of the Army policy. The author is now with The RAND Corporation, Santa Monica, Calif.     

Application of the quartimax method of rotation to Thurstone's primary mental abilities study

This study compares a quartimax rotation of the centroid factor loadings for Thurstone's Primary Mental Abilities Test Battery with factorings of the same correlation matrix by Thurstone (simple structure), Zimmerman (revised simple structure), Holzinger and Harman (bi-factor analysis), and Eysenck (group factor analysis). The quartimax results agree very closely with the solutions of Holzinger and Harman and of Eysenck, and reasonably well with the two simple structure analyses. The principal difference is the general factor provided by the quartimax solution. Reproduction of the factorial structure is sufficiently good to justify its use at least as the first stage of rotation. More extensive trial of the method will be needed with more varied data before it will be possible to decide whether quartimax factors meet psychological requirements sufficiently well without further rotation.We wish to thank Professor L. G. Henyey and the University of California Computer Center for making the IBM 701 electronic computer available for this study, and the National Science Foundation for its support of the work of the Computer Center. Professor H. F. Kaiser of the University of Illinois has made helpful criticisms of the paper, and Mr. Louis S. Davis of the University of California has assisted with preparation of the tables. The research was supported in part by the United States Air Force under Contract No. AF 33 (038)-25726 monitored by the Air Force Personnel and Training Research Center. Permission is granted for reproduction, translation, publication, use and disposal in whole and in part by or for the United States Government.A 701 program for calculation of the quartimax and varimax loadings, prepared by Professor H. F. Kaiser, is available in the library of computer programs held by the Computer Center at the University of California (Program No. 464). Mr. J. O. Neuhaus and Mr. K. W. Dickman have prepared a quartimax program for Illiac at the University of Illinois. This Illiac program will be usable on three other computers recently built or under construction: Mistic (Michigan State University), Silliac (University of Sydney), and the machine being constructed by Iowa State College.     

Assessing the local identifiability of probabilistic knowledge structures

Given a collection Q of problems, in knowledge space theory Doignon & Falmagne, (International Journal of Man–Machine Studies 23:175–196, 1985) the knowledge state of a student is the collection K ? Q of all problems that this student is capable of solving. A knowledge structure is a pair (Q, ), where is a collection of knowledge states that contains at least the empty set and Q. A probabilistic knowledge structure (PKS) is a knowledge structure (Q, , π), where π is a probability distribution on the knowledge states. The PKS that has received the most attention is the basic local independence model BLIM; Falmagne & Doignon, (British Journal of Mathematical and Statistical Psychology 41:1–23, 1988a, Journal of Mathematical Psychology 32:232–258, 1988b). To the best of our knowledge, systematic investigations in the literature concerning the identifiability of the BLIM are totally missing. Based on the theoretical work of Bamber and van Santen (Journal of Mathematical Psychology 29:443–473, 1985), the present article is aimed to present a method and a corresponding computerized procedure for assessing the local identifiability of the BLIM, which is applicable to any finite knowledge structure of moderate size.     

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.