Linear structural equations with latent variables

An interdependent multivariate linear relations model based on manifest, measured variables as well as unmeasured and unmeasurable latent variables is developed. The latent variables include primary or residual common factors of any order as well as unique factors. The model has a simpler parametric structure than previous models, but it is designed to accommodate a wider range of applications via its structural equations, mean structure, covariance structure, and constraints on parameters. The parameters of the model may be estimated by gradient and quasi-Newton methods, or a Gauss-Newton algorithm that obtains least-squares, generalized least-squares, or maximum likelihood estimates. Large sample standard errors and goodness of fit tests are provided. The approach is illustrated by a test theory model and a longitudinal study of intelligence.This investigation was supported in part by a Research Scientist Development Award (KO2-DA00017) and a research grant (DA01070) from the U. S. Public Health Service.     

Constructing a covariance matrix that yields a specified minimizer and a specified minimum discrepancy function value

A method is presented for constructing a covariance matrix Σ*₀ that is the sum of a matrix Σ(γ₀) that satisfies a specified model and a perturbation matrix,E, such that Σ*₀=Σ(γ0) +E. The perturbation matrix is chosen in such a manner that a class of discrepancy functionsF(Σ*₀, Σ(γ₀)), which includes normal theory maximum likelihood as a special case, has the prespecified parameter value γ₀ as minimizer and a prespecified minimum δ A matrix constructed in this way seems particularly valuable for Monte Carlo experiments as the covariance matrix for a population in which the model does not hold exactly. This may be a more realistic conceptualization in many instances. An example is presented in which this procedure is employed to generate a covariance matrix among nonnormal, ordered categorical variables which is then used to study the performance of a factor analysis estimator. We are grateful to Alexander Shapiro for suggesting the proof of the solution in section 2.     

Representationalism and indeterminate perceptual content

Representationalists currently cannot explain counter-examples that involve indeterminate perceptual content, but a double content (DC) view is more promising. Four related cases of perceptual imprecision are used to outline the DC view, which also applies to imprecise photographic content. Next, inadequacies in the more standard single content (SC) view are demonstrated. The results are then generalized so as to apply to the content of any kinds of non-conventional representation. The paper continues with evidence that a DC account provides a moderate rather than extreme realist account of perception, and it concludes with an initial analysis of the failure of nomic covariance accounts of information in indeterminacy cases.

Latent variable modeling in heterogeneous populations

Common applications of latent variable analysis fail to recognize that data may be obtained from several populations with different sets of parameter values. This article describes the problem and gives an overview of methodology that can address heterogeneity. Artificial examples of mixtures are given, where if the mixture is not recognized, strongly distorted results occur. MIMIC structural modeling is shown to be a useful method for detecting and describing heterogeneity that cannot be handled in regular multiple-group analysis. Other useful methods instead take a random effects approach, describing heterogeneity in terms of random parameter variation across groups. These random effects models connect with emerging methodology for multilevel structural equation modeling of hierarchical data. Examples are drawn from educational achievement testing, psychopathology, and sociology of education. Estimation is carried out by the LISCOMP program.Presidential address delivered at the Psychometric Society meetings in Los Angeles, USA and Leuven, Belgium, July 1989. The research was supported by Grant No. SES-8821668 from the National Science Foundation and by Grant No. OERI-G-86-003 from the Office for Educational Research and Improvement, Department of Education. I thank Leigh Burstein, Mike Hollis, Linda Muthén, and Albert Satorra for helpful discussions and Tammy Tam, Jin-Wen Yang, Suk-Woo Kim, and Lynn Short for computational assistance. Designs were created by Arlette Collier, Rita Ling and Jennifer Edic-Bryant.     

Inferential theory for partially disattenuated correlation coefficients

Four measurement designs are presented for use with correlation coefficients corrected, in one variable, for attenuation due to unreliability—coefficients that we term partially disattenuated correlation coefficients. Asymptotic expressions are derived for the variances and covariances of the estimates accompanying each design. Empirical simulation results that bear on the preceding mathematical developments are then presented. In addition to providing insights into the distributions of the estimates, the empirical results demonstrate satisfactory Type I error control for typical inferential applications. Power is shown to be equal to or greater than that of corresponding product-moment correlations in three of the four designs. Implications for practice are discussed.Support for the research reported in this article was provided by the Natural Sciences and Engineering Research Council of Canada. The authors acknowledge with thanks the contributions of Nancy E. Heckman to some of the theoretical aspects of the study.     

On the use of ridit analysis

Ridit analysis is statistical method for comparing ordinal-scale responses. In this paper, the extact variance and asymptotic distribution of the average ridit is developed, including the cases in which the reference group is sampled or the comparison group is finite. The appropriate use and interpretation of ridit analysis is also discussed.The authors wish to thank Andrew Klugh for this support, and the references and David Feigenbaum for their very helpful comments.     

The analysis of structural equation models by means of derivative free nonlinear least squares

It is shown that the PAR Derivative-Free Nonlinear Regression program in BMDP can be used to fit structural equation models, producing generalized least squares estimates, standard errors, and goodness-of-fit test statistics. Covariance structure models more general than LISREL can be analyzed. The approach is particularly useful for dealing with new non-standard models and experimenting with alternate methods of estimation. The research of the second author was supported by the NSF grant MCS 83-01587. We wish to thank our referees for some very valuable suggestions.     

Probability matrix decomposition models

In this paper, we consider a class of models for two-way matrices with binary entries of 0 and 1. First, we considerBoolean matrix decomposition, conceptualize it as alatent response model (LRM) and, by making use of this conceptualization, generalize it to a larger class of matrix decomposition models. Second,probability matrix decomposition (PMD) models are introduced as a probabilistic version of this larger class of deterministic matrix decomposition models. Third, an algorithm for the computation of the maximum likelihood (ML) and the maximum a posteriori (MAP) estimates of the parameters of PMD models is presented. This algorithm is an EM-algorithm, and is a special case of a more general algorithm that can be used for the whole class of LRMs. And fourth, as an example, a PMD model is applied to data on decision making in psychiatric diagnosis. This paper is based on a chapter of the first author's doctoral dissertation, written at the University of Leuven and supervised by Paul De Boeck.     

ANOVA and ANCOVA of pre- and post-test,ordinal data

With random assignment to treatments and standard assumptions, either a one-way ANOVA of post-test scores or a two-way, repeated measures ANOVA of pre- and post-test scores provides a legitimate test of the equal treatment effect null hypothesis for latent variable . In an ANCOVA for pre- and post-test variablesX andY which are ordinal measures of and , respectively, random assignment and standard assumptions ensure the legitimacy of inferences about the equality of treatment effects on latent variable . Sample estimates of adjustedY treatment means are ordinal estimators of adjusted post-test means on latent variable .     

On the estimation of polychoric correlations and their asymptotic covariance matrix

A general theory for parametric inference in contingency tables is outlined. Estimation of polychoric correlations is seen as a special case of this theory. The asymptotic covariance matrix of the estimated polychoric correlations is derived for the case when the thresholds are estimated from the univariate marginals and the polychoric correlations are estimated from the bivariate marginals for given thresholds. Computational aspects are also discussed.The research was supported by the Swedish Council for Research in the Humanities and Social Sciences (HSFR) under the programMultivariate Statistical Analysis. The author thanks a reviewer for pointing out an error in the original version of the paper.