On component analyses

Principal components analysis can be redefined in terms of the regression of observed variables upon component variables. Two criteria for the adequacy of a component representation in this context are developed and are shown to lead to different component solutions. Both criteria are generalized to allow weighting, the choice of weights determining the scale invariance properties of the resulting solution. A theorem is presented giving necessary and sufficient conditions for equivalent component solutions under different choices of weighting. Applications of the theorem are discussed that involve the components analysis of linearly derived variables and of external variables.Preparation of this article was supported in part by National Institute of Aging Grant NIA-AG03164-03 to William Meredith.     

Testing equality of correlated proportions with incomplete data on both responses

Two test statistics are proposed for testing the equality of two correlated proportions when some observations are missing on both responses. The performance of these tests in terms of size and power is compared with other tests by means of Monte Carlo simulations. The proposed tests are easily computed and compare favorably with other tests.     

Tests for equality of several alpha coefficients when their sample estimates are dependent

In a variety of measurement situations, the researcher may wish to compare the reliabilities of several instruments administered to the same sample of subjects. This paper presents eleven statistical procedures which test the equality ofm coefficient alphas when the sample alpha coefficients are dependent. Several of the procedures are derived in detail, and numerical examples are given for two. Since all of the procedures depend on approximate asymptotic results, Monte Carlo methods are used to assess the accuracy of the procedures for sample sizes of 50, 100, and 200. Both control of Type I error and power are evaluated by computer simulation. Two of the procedures are unable to control Type I errors satisfactorily. The remaining nine procedures perform properly, but three are somewhat superior in power and Type I error control.A more detailed version of this paper is also available.     

Describing the elephant: Structure and function in multivariate data

There is a unity underlying the diversity of models for the analysis of multivariate data. Essentially, they constitute a family models, most generally nonlinear, for structural/functional relations between variables drawn from a behavior domain.     

Rasch-representable reaction time distributions

This article investigates properties of a representation based on the Rasch test model for reaction times (RT) that was proposed by Micko. Necessary and sufficient conditions for a set of RT distributions to be Rasch-representable are derived. It is shown that independent serial and independent parallel processing models cannot be reconciled with the representation. However, random extreme models compatible with the Reasch-representation exist that assume RT is determined by the longest or he shortest processing time of a random number of independent paraloel channels. Nonparametric properties of Rasch-representable distributions are derived that can be used for testing the model and for estimating its parameters. Conditions are presented for Rasch-representable distributions to form a scale family. Finally, Rasch-represent-able distributions are characterized interms of their hazard functions.For helpful discussions, we are grateful to Hans Irtel, Christoph Micko, Hartmann Scheiblechner, and Hans-Henning Schultz.     

The derivation of some tests for the rasch model from the multinomial distribution

The present paper is concerned with testing the fit of the Rasch model. It is shown that this can be achieved by constructing functions of the data, on which model tests can be based that have power against specific model violations. It is shown that the asymptotic distribution of these tests can be derived by using the theoretical framework of testing model fit in general multinomial and product-multinomial models. The model tests are presented in two versions: one that can be used in the context of marginal maximum likelihood estimation and one that can be applied in the context of conditional maximum likelihood estimation.I am indebted to Norman Verhelst and Niels Veldhuijzen for their helpful comments. Requests for reprints should be sent to Cees A. W. Glas, Cito, PO Box 1034, 6801 MG Arnhem, THE NETHERLANDS.     

An alternative test for multisample sphericity

A test for multisample sphericity based on the efficient scores criterion is obtained as an alternative to the likelihood ratio test developed by Mendoza.     

Univariate repeated measures techniques applied to multivariate data

Repeated measures designs in psychology have traditionally been analyzed by the univariate mixed model approach, in which the repeated measures effect is tested against an error term based on the subject by treatment interaction. This paper considers the extension of this analysis to designs in which the individual repeated measures are multivariate. Sufficient conditions for a valid multivariate mixed model analysis are given, and a test is described to determine whether or not given data satisfy these conditions.     

Best linear prediction of composite universe scores

The problem of predicting universe scores for samples of examinees based on their responses to samples of items is treated. A general measurement procedure is described in which multiple test forms are developed from a table of specifications and each form is administered to a different sample of examinees. The measurement model categorizes items according to the cells of such a table, and the linear function derived for minimizing error variance in prediction uses responses to these categories. In addition, some distinctions are drawn between aspects of the approach taken here and the familiar regressed score estimates.The author thanks Robert L. Brennan, Michael J. Kolen, and Richard Sawyer for helpful comments and corrections, and anonymous reviewers for suggested improvements.