Bayesian modeling of measurement error in predictor variables using item response theory

It is shown that measurement error in predictor variables can be modeled using item response theory (IRT). The predictor variables, that may be defined at any level of an hierarchical regression model, are treated as latent variables. The normal ogive model is used to describe the relation between the latent variables and dichotomous observed variables, which may be responses to tests or questionnaires. It will be shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a Bayesian framework using Gibbs sampling. In this article, handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model. Examples using real data are given.This paper is part of the dissertation by Fox (2001) that won the 2002 Psychometric Society Dissertation Award.     

The robustness of estimates of total indirect effects in covariance structure models estimated by maximum

The large sample distribution of total indirect effects in covariance structure models in well known. Using Monte Carlo methods, this study examines the applicability of the large sample theory to maximum likelihood estimates oftotal indirect effects in sample sizes of 50, 100, 200, 400, and 800. Two models are studied. Model 1 is a recursive model with observable variables and Model 2 is a nonrecursive model with latent variables. For the large sample theory to apply, the results suggest that sample szes of 200 or more and 400 or more are required for models such as Model 1 and Model 2, respectively.For helpful comments on a previous draft of this paper, we are grateful to Gerhard Arminger, Clifford C. Clogg, and several anonymous reviewers.     

A Monte Carlo investigation of conjoint analysis index-of-fit: Goodness of fit,significance and power

Researchers in the field of conjoint analysis know the index-of-fit values worsen as the judgmental error of evaluation increases. This simulation study provides guidelines on the goodness of fit based on distribution of index-of-fit for different conjoint analysis designs. The study design included the following factors: number of profiles, number of attributes, algorithm used and judgmental model used. Critical values are provided for deciding the statistical significance of conjoint analysis results. Using these cumulative distributions, the power of the test used to reject the null hypothesis of random ranking is calculated. The test is found to be quite powerful except for the case of very small residual degrees of freedom.The authors thank the editor, the three reviewers and Ellen Foxman for helpful comments on the paper. Sanjay Mishra was a doctoral student at Washington State University at the time this research was completed. He is currently in the Department of Marketing at the University of Kansas.     

Fungible Correlation Matrices: A Method for Generating Nonsingular,Singular, and Improper Correlation Matrices for Monte Carlo Research

For a fixed set of standardized regression coefficients and a fixed coefficient of determination (R-squared), an infinite number of predictor correlation matrices will satisfy the implied quadratic form. I call such matrices fungible correlation matrices. In this article, I describe an algorithm for generating positive definite (PD), positive semidefinite (PSD), or indefinite (ID) fungible correlation matrices that have a random or fixed smallest eigenvalue. The underlying equations of this algorithm are reviewed from both algebraic and geometric perspectives. Two simulation studies illustrate that fungible correlation matrices can be profitably used in Monte Carlo research. The first study uses PD fungible correlation matrices to compare penalized regression algorithms. The second study uses ID fungible correlation matrices to compare matrix-smoothing algorithms. R code for generating fungible correlation matrices is presented in the supplemental materials.     

A Comparison of ML,WLSMV, and Bayesian Methods for Multilevel Structural Equation Models in Small Samples: A Simulation Study

Multilevel structural equation models are increasingly applied in psychological research. With increasing model complexity, estimation becomes computationally demanding, and small sample sizes pose further challenges on estimation methods relying on asymptotic theory. Recent developments of Bayesian estimation techniques may help to overcome the shortcomings of classical estimation techniques. The use of potentially inaccurate prior information may, however, have detrimental effects, especially in small samples. The present Monte Carlo simulation study compares the statistical performance of classical estimation techniques with Bayesian estimation using different prior specifications for a two-level SEM with either continuous or ordinal indicators. Using two software programs (Mplus and Stan), differential effects of between- and within-level sample sizes on estimation accuracy were investigated. Moreover, it was tested to which extent inaccurate priors may have detrimental effects on parameter estimates in categorical indicator models. For continuous indicators, Bayesian estimation did not show performance advantages over ML. For categorical indicators, Bayesian estimation outperformed WLSMV solely in case of strongly informative accurate priors. Weakly informative inaccurate priors did not deteriorate performance of the Bayesian approach, while strong informative inaccurate priors led to severely biased estimates even with large sample sizes. With diffuse priors, Stan yielded better results than Mplus in terms of parameter estimates.     

Relationships between the Vocational Preference Inventory and the Edwards Personal Preference Schedule

The Vocational Preference Inventory and the Edwards Personal Preference Schedule were administered to 372 undergraduates. The two instruments were compared using canonical analysis. The analysis revealed three significant relationships between components of the two instruments. The relationships were viewed as supportive of Holland's theory of personality types.     

Asymptotic distributions of the estimators of communalities in factor analysis

Asymptotic distributions of the estimators of communalities are derived for the maximum likelihood method in factor analysis. It is shown that the common practice of equating the asymptotic standard error of the communality estimate to the unique variance estimate is correct for standardized communality but not correct for unstandardized communality. In a Monte Carlo simulation the accuracy of the normal approximation to the distributions of the estimators are assessed when the sample size is 150 or 300. This study was carried out in part under the ISM Cooperative Research Program (90-ISM-CRP-9).     

Random versus rational strategies for initial configurations in nonmetric multidimensional scaling

An examination is made concerning the utility and design of studies comparing nonmetric scaling algorithms and their initial configurations, as well as the agreement between the results of such studies. Various practical details of nonmetric scaling are also considered.This research was supported by NSF Grants SOC 76-24512 and SOC 76-24394.     

Graph Theory Approach to Detect Examinees Involved in Test Collusion

Test collusion (TC) is sharing of test materials or answers to test questions before or during the test (important special case of TC is item preknowledge). Because of potentially large advantages for examinees involved, TC poses a serious threat to the validity of score interpretations. The proposed approach applies graph theory methodology to response similarity analyses for identifying groups of examinees involved in TC without using any knowledge about parts of test that were affected by TC. The approach supports different response similarity indices (specific to a particular type of TC) and different types of groups (connected components, cliques, or near-cliques). A comparison with an up-to-date method using real and simulated data is presented. Possible extensions and practical recommendations are given.