Latent class models for nonmonotone dichotomous items

Starting from perfectly discriminating nonmonotone dichotomous items, a class of probabilistic models with or without response errors and with or without intrinsically unscalable respondents is described. All these models can be understood as simply restricted latent class analysis. Thus, the estimation and identifiability of the parameters (class sizes and item latent probabilities) as well as the chi-squared goodness-of-fit tests (Pearson and likelihood-ratio) are free of the problems. The applicability of the proposed variants of latent class models is demonstrated on real attitudinal data.This research was supported by the Kulturamt der Stadt Wien, Magistratsabteilung 7.The author wishes to thank the editor, Ivo W. Molenaar, as well as Clifford C. Clogg and the anonymous reviewers for their valuable comments on the earlier drafts of this paper.     

Determining the number of components from the matrix of partial correlations

A common problem for both principal component analysis and image component analysis is determining how many components to retain. A number of solutions have been proposed, none of which is totally satisfactory. An alternative solution which employs a matrix of partial correlations is considered. No components are extracted after the average squared partial correlation reaches a minimum. This approach gives an exact stopping point, has a direct operational interpretation, and can be applied to any type of component analysis. The method is most appropriate when component analysis is employed as an alternative to, or a first-stage solution for, factor analysis.     

Some properties of binormamin

Three properties of the binormamin criterion for analytic transformation in factor analysis are discussed. Particular reference is made to Carroll's oblimin class of criteria.The research reported in this note was supported in part by the Program in Applied Mathematics and Statistics. National Science Foundation, and by the National Institute of Education, Department of Health, Education, and Welfare.     

Testing the assumptions underlying tetrachoric correlations

A method is proposed for empirically testing the appropriateness of using tetrachoric correlations for a set of dichotomous variables. Trivariate marginal information is used to get a set of one-degree of freedom chi-square tests of the underlying normality. It is argued that such tests should preferrably preceed further modeling of tetrachorics, for example, modeling by factor analysis. The assumptions are tested in some real and simulated data.Presented at the Psychometric Society meeting in Santa Barbara, California, June 25–26, 1984. The research of the first author was supported by Grant No. SES-8312583 from the National Science Foundation.     

Retrieving the correlation matrix from a truncated PCA solution: The inverse principal component problem

Whenr Principal Components are available fork variables, the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose. The approximation is generally not perfect unlessr =k. In the present paper it is shown that, whenr is at or above the Ledermann bound,r principal components are enough to perfectly reconstruct the correlation matrix, albeit in a way more involved than taking the loading matrix times its transpose. In certain cases just below the Ledermann bound, recovery of the correlation matrix is still possible when the set of all eigenvalues of the correlation matrix is available as additional information.     

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.     

Reliability of Summed Item Scores Using Structural Equation Modeling: An Alternative to Coefficient Alpha

A method is presented for estimating reliability using structural equation modeling (SEM) that allows for nonlinearity between factors and item scores. Assuming the focus is on consistency of summed item scores, this method for estimating reliability is preferred to those based on linear SEM models and to the most commonly reported estimate of reliability, coefficient alpha.     

Latent roots of random data correlation matrices with squared multiple correlations on the diagonal: A monte carlo study

In order to make the parallel analysis criterion for determining the number of factors easy to use, regression equations for predicting the logarithms of the latent roots of random correlation matrices, with squared multiple correlations on the diagonal, are presented. The correlation matrices were derived from distributions of normally distributed random numbers. The independent variables are log (N–1) and log {[n(n–1)/2]–[(i–1)n]}, whereN is the number of observations;n, the number of variables; andi, the ordinal position of the eigenvalue. The results were excellent, with multiple correlation coefficients ranging from .9948 to .9992.This research was supported by the Office of Naval Research under Contract N00014-67-A-0305-0012, Lloyd G. Humphreys, principal investigator, and by the Department of Computer Science of which Richard G. Montanelli, Jr., is a member.     

Factor invariance of the Millon Clinical Multiaxial Inventory

Several factor analyses of the Millon Clinical Multiaxial Inventory (MCMI) have resulted in very similar solutions. Interpretation of this consistency is hampered by the fact that the 20 scales of the inventory share items. Overlapping items cause the scales to be linearly dependent and may create structure in the interscale correlation matrix which is separate from the subject response patterns. A factor analysis was performed on the matrix of item-overlap coefficients which describes the underlying artifactual structure of the instrument. Data from two new subject samples were factor analyzed and compared to previously published studies. Similarity coefficients among factors across studies were calculated.     

中学生学习成功感量表的初步研究

在访谈、开放式问卷等的基础上,自编了中学生学习成功感量表,并对上海市随机抽取的642名中学生进行了测量。结果表明：该量表的49个项目均有较好的区分度;四个分量表之间的相关表明该量表的建构是正确的;因素分析表明成功感分量表可以分为：与学习本身有关的积极情感、与他人有关的积极情感、满意感和学业自我效能感四个因子;该量表有较高的信度和效度,可以作为评鉴中学生学习成功感的有效工具。     

A numerical approach to the approximate and the exact minimum rank of a covariance matrix

A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.The authors are obliged to Paul Bekker for stimulating and helpful comments.     

The Imagination of Images and Images of Imagination: How the Imagination Relates to Aging,Health, and Spirituality

ABSTRACT

Psychological literature suggests that religion and spirituality increase in late adulthood. Yet, operational definitions of spirituality and religiosity remain widely debated and inadequate for the concepts they are designed to measure. The empirical studies of religion and spirituality as one ages are of poor design and often measure only limited aspects of religion or spirituality. Few empirical studies exist which have been conceived to only study religiosity and spirituality in late adulthood. The purpose of this study was to determine the defining aspects of religiosity and spirituality using the Allport, Ross Intrinsic, Extrinsic Religiosity Scale, Ellison's Spiritual Well-Being Scale, and Neugarten's Life Satisfaction Instrument. Using a principal component factor analysis, the study examined the factor structure using an older adult sample of 320 individuals 65 years of age and older. Having a purpose in life combines with intrinsic religious questions for the first factor. Life satisfaction questions group together on two factors and extrinsic religiosity is clearly one factor. The scales used hold together well when combined. A new, shortened scale to measure aspects of religiosity and spirituality is proposed.     

Psychometric properties of the Danish MCMI-I translation

A translation of the MCMI-I has been in use in Denmark for some years. An untested assumption in the interpretation of the pattern of test results is that the psychometric characteristics of the Danish and American versions are similar. The purpose of this study was to evaluate the psychometric properties of the questionnaire by using traditional psychometric analysis techniques on the results of a sample consisting of 423 patients and 179 normal controls. Coefficient alpha was calculated for the 20 clinical subscales of the test and the Danish results were strikingly similar to the original coefficients reported by Millon. Furthermore, factor analysis of the subscales showed a factor structure very similar to American findings, and it is concluded that the psychometric properties of the Danish MCMI are not significantly different from the original.     

Psychometric properties of the Obsessive Compulsive Inventory-Revised among Icelandic college students

In the present study, the psychometric properties of the Obsessive Compulsive Inventory-Revised (OCI-R) (Foa, et al., 2002) were investigated. The OCI-R was administered to a total of 816 Icelandic college students. The Maudsley Obsessive Compulsive Inventory (MOCI) and the Penn State Worry Questionnaire (PSWQ) were administered to a subsample of 304 students, and the PI-WSUR (Padua Inventory Washington State University Revision), the PSWQ and the Multidimensional Perfectionism Scale (MPS) to another subsample of 212 students. Also, 89 students filled in the OCI-R twice with two weeks in between. The factor structure of the instrument was replicated in a confirmatory factor analysis. Cronbach's alpha and test-retest reliabilities of the OCI-R total scale and all subscales were satisfactory. Further, the OCI-R total score showed a significantly higher correlation with the PI-WSUR and the MOCI than with the PSWQ or the MPS. Finally, conceptually similar scales of the OCI-R and the MOCI as well as the PI-WSUR showed higher inter-correlations than the correlations that were observed between conceptually more distinct scales. It is concluded that the Icelandic version of the OCI-R has strong psychometric properties in a student population.     

A comparative assessment of the factor structures and psychometric properties of the GHQ-12 and the GHQ-20 based on data from a Finnish population-based sample

The General Health Questionnaire (GHQ) is a self-report questionnaire designed to identify psychological distress. Psychometric properties of two versions of GHQ-12 and GHQ-20 were assessed in a large population-based sample of Finnish twins, ages 22 to 27 (n= 4580). Participants were randomized into two subgroups, namely Twin1 (n= 2294) and Twin2 (n= 2286). The GHQ-12 data were assessed using Confirmatory Factor Analysis (CFA). The factor structure of the GHQ-20 was first assessed with Exploratory Factor Analysis (EFA) in the Twin1 dataset, and the results obtained were then subjected to CFA in Twin1 and Twin2 datasets. The CFA of the GHQ-12 indicated that the best fit and the simplest solution were provided by the three-factor solution in both subpopulations. Analyses of the GHQ-20 suggested that the four-factor structure was superior to the three-factor model. This result is also theoretically justifiable. Compared to the 12-item version, GHQ-20 provides additional fourth factor of anhedonia, suggesting some discriminative power.     

In Spite of Indeterminacy Many Common Factor Score Estimates Yield an Identical Reproduced Covariance Matrix

It was investigated whether commonly used factor score estimates lead to the same reproduced covariance matrix of observed variables. This was achieved by means of Schönemann and Steiger’s (1976) regression component analysis, since it is possible to compute the reproduced covariance matrices of the regression components corresponding to different factor score estimates. It was shown that Thurstone’s, Ledermann’s, Bartlett’s, Anderson-Rubin’s, McDonald’s, Krijnen, Wansbeek, and Ten Berge’s, as well as Takeuchi, Yanai, and Mukherjee’s score estimates reproduce the same covariance matrix. In contrast, Harman’s ideal variables score estimates lead to a different reproduced covariance matrix.     

Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm

Maximum likelihood estimation of item parameters in the marginal distribution, integrating over the distribution of ability, becomes practical when computing procedures based on an EM algorithm are used. By characterizing the ability distribution empirically, arbitrary assumptions about its form are avoided. The Em procedure is shown to apply to general item-response models lacking simple sufficient statistics for ability. This includes models with more than one latent dimension.Supported in part by NSF grant BNS 7912417 to the University of Chicago and by SSRC (UK) grant HR6132 to the University of Lancaster.We are indebted to Mark Reiser and Robert Gibbons for computer programming. David Thissen clarified a number of points in an earlier draft.     

Psychometric properties of the Danish 16PF and EPQ

A Danish translation of the Cattell's 16PF has been used in studies evaluating the effects of prenatal drug exposure. This paper reports a psychometric analysis of the 16PF and Eysenck's EPQ based on a sample of 558 young Danes. Many 16PF scales had unacceptable psychometric properties (as indicated by coefficient alpha and item–total score correlations), but more satisfactory results were obtained with the EPQ N and E scales. A factor analysis of all 16PF and EPQ scales suggested a six factor solution that roughly corresponds to the second-order factor structure obtained by Krug and Johns (1986). It is concluded that the second-order factor structure should be the basis of interpretation of the 16PF in both practical and research contexts.