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.     

Nominally and rigorously parallel test forms

When several test forms are used interchangeably, they will not in practice, despite all efforts, be rigorously parallel. A definition of the standard error of measurement appropriate for this type of situation can be provided; however, it will be different from the definition in classical test theory. Appropriate formulas for the standard error of measurement and for other related quantities under both definitions are derived and compared; also sample statistics for estimating these quantities.The writer is indebted to Lee Cronbach and Julian Stanley for many helpful suggestions for revising a draft of this paper. Part of the work was carried out while the writer was Brittingham Visiting Professor at the University of Wisconsin. This work was supported in part by contract Nonr-2752(00) between the Office of Naval Research and Educational Testing Service. Reproduction in whole or in part for any purpose of the United States Government is permitted. The termnominally parallel was suggested by Lee Cronbach.     

A recruitment theory of simple behavior

A statistical theory of choice is developed using a sequential sampling assumption. Response latency distributions for certain simple reaction-time situations are derived and tested. Both response probability and response latency measures are developed for a two-alternative judgment situation and the relationship between the two measures explored. The sampling parameter is proposed as a means of representing incentive conditions in choice situations and ROC curves are obtained by appropriate manipulations of this parameter. A solution to the overlap problem in simple discrimination-learning situations is also derived.This research was supported in part by a grant from the National Science Foundation. The writer has benefited from discussions with his students, James R. Erickson, James G. Greeno, Duane R. Martin, Kirk H. Smith, and Charles F. Stroebel. For generous suggestions with regard to certain mathematical matters, the writer is indebted to Professors Ingram Olkin, Stephen Orey, and Milton Sobel of the Statistics Department at the University of Minnesota.     

Studies in the learning function

From Thurstone's theoretical learning curve, a solution for the difficulty of the problem and the learning constant of the subject has been developed. The curve is an equilateral hyperbola. Therefore the semi-major axis represents the learning situation in one constant. The vertex of the curve is a point where all of the animals are equated, since they are all making errors at the rate of one error per trial.A grant-in-aid from the National Research Council has made the present analysis possible. We wish to express our appreciation, for the aid and encouragement, of Professor Rufus Crane of the mathematics department, Ohio Wesleyan University and of Professor L. L. Thurstone, University of Chicago.     

An empirical study of the factor analysis stability hypothesis

Note is taken of four related sources of confusion as to the usefulness of Thurstone's factor analysis model and of their resolutions. One resolution uses Tucker's distinction between exploratory and confirmatory analyses. Eight analyses of two sets of data demonstrate the procedures and results of a confirmatory study with statistical tests of some, but not all, relevant hypotheses in an investigation of the stability (invariance) hypothesis. The empirical results provide estimates, as substitutes for unavailable sampling formulations, of effects of variation in diagonal values, in method of factoring, and in samples of cases. Implications of these results are discussed.The computational costs of this study were defrayed, in part, by a research small grant M-1922 from the National Institute of Health, and, in part, by support under project 176-0002 by the University of Iowa Computing Center, Dr. J. P. Dolch, Director. The assistance of Dr. Kern Dickman and Mr. Leonard Wevrick of the University of Illinois and of Mr. Norman Luther of the University of Iowa in handling the computing problems is gratefully acknowledged.     

Factorial invariance and significance

It is shown that invariance requirements remove the indeterminacy in factor determination and lead to an integration of factorial studies with promise of considerable reduction in computational labor. The selection of significant primary factors is discussed, with special reference to Thurstone's simple structure criterion.     

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.     

The effect of difficulty and chance success on item-test correlation and on test reliability

An equation is derived for predicting the effect of chance success, relative to item difficulty, on item-test correlation. The values predicted by this equation and by equations derived by Guilford and Carroll for predicting the effect of chance success on item difficulty and test reliability are compared with empirical values in an experiment which used identical test items in multiple-choice and answer-only form.Condensation of a dissertation presented in partial fulfillment of the requirements for the Ph.D. degree to the University of Chicago. Grateful acknowledgment is made to Professor Harold Gulliksen for his guidance as thesis advisor and to Professor L. L. Thurstone and Dr. D. W. Fiske of the University of Chicago who served as members of the thesis committee. The author is also indebted to Professor S. S. Wilks for review of the derivations and development of statistical tests used in the thesis, to Dr. L. R Tucker for technical advice, and to Dr. W. G. Mollenkopf for critical comments on the derivations and interpretations. The writer expresses appreciation to the Educational Testing Service for making available its technical facilities, and to the University of Chicago for the flexible administrative arrangement which made this thesis possible.     

Irrational beliefs and the anxiety disorders

This study investigated the levels of irrationality reported by a clinical sample of anxiety disorder patients, including simple and social phobics, panic disorder patients, agoraphobics, and obssessive compulsive patients. The levels of irrationality were compared between these groups and a group of normal control subjects, using the Rational Behavior Inventory (RBI). Agoraphobic patients were significantly less rational than control subjects on six of the RBI's subfactors and the total score of the Inventory. Patients in all diagnostic categories except simple phobia were significantly different from control subjects on the projected misfortune subfactor, and patients in all categories except simple phobia and panic disorder were significantly different from controls on the total RBI score. There were only four instances where patients in anxiety disorder categories significantly differed from each other in levels of irrationality. The implications of these findings, methodological limitations, and directions for future research are discussed.Joseph A. Himle, ACSW is a Clinical Social Worker at the Anxiety Disorders Program, Department of Psychiatry, University of Michigan. David P. Himle, Ph.D. is an Associate Professor of Social Work at the University of Michigan. Bruce A. Thyer, Ph.D. is an Associate Professor of Social Work, University of Georgia and Faculty Fellow at the University Affiliated Program. He is also an Associate Clinical Professor at the Department of Psychiatry and Health Behavior, Medical College of Georgia.     

Models of health and wholeness

A church-clinic model of the neighborhood health center, developed initially in a low-income area, has now been established over the past three years in middle- and upper-income areas. The church-clinics, called Wholistic Health Centers, are primary care medical clinics located in church buildings that utilize an interdisciplinary team in the planning for patient treatment and health education. The project integrates primary medical care with pastoral counseling services. Evaluation of the two clinics in middle-to-upper-income areas indicates that the patients are middle to upper class, well educated, and are not medically disenfranchised; their presenting problems are only slightly more often medical (58%) than psychosocial (42%).This work was conducted in part when Dr. Holinger was at the Institute for Psychosomatic and Psychiatric Research and Training, Michael Reese Hospital and Medical Center, Chicago, Illinois.He is also the author ofWholistic Health.The authors gratefully acknowledge the assistance of the Rev. Granger E. Westberg, D.D., Clinical Professor, and Edward A. Lichter, Professor and Chairman, Department of Preventive Medicine and Community Health, University of Illinois at the Medical Center, Abraham Lincoln School of Medicine, Chicago, Illinois, in the preparation of this report.     

Sampling fluctuations resulting from the sampling of test items

Sampling fluctuations resulting from the sampling of test items rather than of examinees are discussed. It is shown that the Kuder-Richardson reliability coefficients actually are measures of this type of sampling fluctuation. Formulas for certain standard errors are derived; in particular, a simple formula is given for the standard error of measurement of an individual examinee's score. A common misapplication of the Wilks-Votaw criterion for parallel tests is pointed out. It is shown that the Kuder-Richardson formula-21 reliability coefficient should be used instead of the formula-20 coefficient in certain common practical situations.Most of the work reported here was carried out under contract with the Office of Naval Research. The writer is indebted to Professor S. S. Wilks, who has checked over certain critical portions of a draft of this paper.     

An interview with a child to assess psychiatric disturbance: A note on its reliability and validity

Forty-six children with enuresis were given a psychiatric interview. The two writers made independent ratings of 10 behavioral variables. Seven of these showed satisfactory interrater reliability. No relationship was established between child psychiatric disturbance assessed in this way and estimates of disorder obtained from information given by mothers and teachers.Professor Max Hamilton, Department of Psychiatry, University of Leeds, kindly provided computing facilities. The work was supported by a grant from the Yorkshire Regional Health Authority.     

A significance test for the hypothesis that two variables measure the same trait except for errors of measurement

The likelihood-ratio significance test is derived for the hypothesis that after correction for attenuation two variables have a perfect correlation in the population from which the sample is drawn.The writer is indebted to Professor John W. Tukey for his valuable suggestions on an earlier draft.     

Probabilistic foundations for coefficients of generalizability

This paper explores certain problems which arise within the context of the theory of generalizability put forward by Cornbach, Rajaratnam, and Gleser. In particular, a formal explication of their theory for the single observation is given, and the various coefflcients of generalizability which they define are related to the estimation of universe scores.This work is based on a Master's thesis submitted to the University of Illinois in 1963. The author is particularly indebted to Dr. Lee J. Cornbach and Dr. Ledyard R Tucker for their unstinted advice and help.     

A simple procedure for rearranging matrices

Guttman's scalogram board technique for reordering the columns and rows of a matrix is described and its disadvantages are pointed out. A simple and inexpensive procedure for doing the same job without these disadvantages is outlined.I am grateful to Professor Jozef Cohen of the University of Illinois for a five-minute conversation which greatly simplified the procedure described here.     

Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations

Thurstone's Case V of the method of paired comparisons assumes equal standard deviations of sensations corresponding to stimuli and zero correlations between pairs of stimuli sensations. It is shown that the assumption of zero correlations can be relaxed to an assumption of equal correlations between pairs with no change in method. Further the usual approach to the method of paired comparisons Case V is shown to lead to a least squares estimate of the stimulus positions on the sensation scale.This research was performed in the Laboratory of Social Relations under a grant made available to Harvard University by the RAND Corporation under the Department of the Air Force, Project RAND.     

Spiking Interest in Computational Modeling: A Unified Approach to Cognitive Neuroscience. Review of Computational Explorations in Cognitive Neuroscience: Understanding the Mind by Simulating the Brain, by O'Reilly, R. C., & Munakata, Y.

Randall C. O'Reilly is an assistant professor in the Department of Psychology and Institute of Cognitive Science at the University of Colorado at Boulder. In 1996, he received his Ph.D. in psychology from Carnegie Mellon University under the supervision of Professor James L. McClelland. From 1996 to 1997, Randall O'Reilly was awarded a McDonnell-Pew Cognitive Neuroscience Postdoctoral Fellowship to study at the Department of Brain and Cognitive Sciences at the Massachusetts Institute of Technology. He has served on several National Institutes of Health grant peer review panels and is currently an associate editor of the journal Cognitive Science. Professor O'Reilly's primary research interests are concerned with understanding the biological basis of cognitive processes through a variety of methodologies including computational and formal models of the biological bases of cognition.Yuko Munakata is an associate professor in the Department of Psychology at the University of Denver. In 1996, she received her Ph.D. in psychology from Carnegie Mellon University under the supervision of Professor James L. McClelland. Professor Munakata is a panel member of the NIH Biobehavioral and Behavioral Processes Study Section, a recent recipient of the Boyd R. McCandless Young Scientist Award (American Psychological Association), and Co-Editor (with M. Johnson and R. O. Gilmore) of the book Brain Development and Cognition: A Reader (2nd ed.). From 1996 to 1997, Yuko Munakata was awarded a McDonnell-Pew Cognitive Neuroscience Postdoctoral Fellowship to study at the Department of Brain and Cognitive Sciences at the Massachusetts Institute of Technology. Professor Munakata's primary research interests are concerned with the development and evaluation of neural network models and other representational systems for the purposes of understanding human cognitive development.Richard M. Golden is associate professor of psychology, cognitive science, and electrical engineering at the University of Texas at Dallas. Professor Golden is currently a member of the editorial boards of the journals Neural Networks, Neural Processing Letters, and the Journal of Mathematical Psychology, which focus upon computational and mathematical analyses of neurally inspired mathematical models. He is also a member of the Governing Board of the Society for Text and Discourse and the author of the book Mathematical Methods for Neural Network Analysis and Design. Professor Golden's primary research interests are concerned with the development and evaluation of formal models of higher level cognitive processes through a variety of methodologies from fields such as dynamical systems theory, optimization theory, statistical pattern recognition, and computational cognitive neuroscience.     

Factorial equations for tests of attention

Tests designed to measure what was conceived to be attention were found by factor analysis to involve a factor which is independent of the factors of rote memory, visual-space, number, or perception. To a large extent, at least, this attention factor is independent of content and of mode of presentation of test material. The tests in which the variance is mainly dependent upon this factor are those involving a high degree of sustained or relatively continuous mental effort.An abstract of a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Psychology in the Graduate School of the University of Illinois, 1942.The writer wishes to express his gratitude to Professor Herbert Woodrow who suggested the present problem and under whose guidance the investigation was conducted. A further debt of gratitude is acknowledged to the enlisted men at the Air Corps Technical School at Chanute Field, Illinois, who served as subjects and to Capt. T. W. Harrell and Capt. Richard Faubion.     

Test skewness and kurtosis as functions of item parameters

Indexes of skewness and kurtosis for a test-score distribution are expressed in terms of item parameters. Both are shown to depend, in part, on item means, variances, and covariances. The index of skewness depends also on trivariances. A trivariance is a product moment involving first powers of deviation scores for three items. The index of kurtosis depends on quadrivariances, as well as trivariances. A quadrivariance is a product moment involving first powers of deviation scores for four items. Empirical data are presented for responses of groups of subjects to 25 triads and 25 tetrads of items from five tests.Certain parts of this article represent the results of doctoral research conducted by Hundleby and Goldstein under the direction of Ray in the Department of Psychology at Pennsylvania State University. The authors are indebted to Professor Lester Guest and Professor William Lepley for their supervisory assistance in the final stages of the two dissertations during the absence of the senior author.     

A factor analysis of mechanical ability tests

The intercorrelations of thirty-seven variables, including the Minnesota battery of mechanical ability tests, the seven MacQuarrie tests of mechanical ability, O'Connor's Wiggly blocks, and the Stenquist picture-matching test, were analyzed by Thurstone's centroid method. Five factors, Perceptual, Verbal, Youth, Manual Agility, and Spatial, were taken out. Factors prominent in so-called mechanical ability tests are the Spatial and Perceptual ones with Mac-Quarrie's dotting test significantly high in the Manual Agility factor. Each of the factors can be measured with group pencil-and-paper tests.Acknowledgment is gratefully made to the State Engineering Experiment Station at the Georgia School of Technology for sponsoring and financially supporting the studies; to the Graduate Research Committee of the University of Illinois for providing funds for the purchase of tests and the tabulation of data; and to Dr. E. L. Welker of the University of Illinois Mathematics Department for assistance with statistical problems.