Observed-score equating using the marginal distributions of two tests is not necessarily the universally best approach it has been claimed to be. On the other hand, equating using the conditional distributions given the ability level of the examinee is theoretically ideal. Possible ways of dealing with the requirement of known ability are discussed, including such methods as conditional observed-score equating at point estimates or posterior expected conditional equating. The methods are generalized to the problem of observed-score equating with a multivariate ability structure underlying the scores.

It is very important to choose appropriate variables to be analyzed in multivariate analysis when there are many observed variables such as those in a questionnaire. What is actually done in scale construction with factor analysis is nothing but variable selection.

In this paper, we take several goodness-of-fit statistics as measures of variable selection and develop backward elimination and forward selection procedures in exploratory factor analysis. Once factor analysis is done for a certain number p of observed variables (the p-variable model is labeled the current model), simple formulas for predicted fit measures such as chi-square, GFI, CFI, IFI and RMSEA, developed in the field of the structural equation modeling, are provided for all models obtained by adding an external variable (so that the number of variables is p + 1) and for those by deleting an internal variable (so that the number is p − 1), provided that the number of factors is held constant.

A program SEFA (Stepwise variable selection in Exploratory Factor Analysis) is developed to actually obtain a list of the fit measures for all such models. The list is very useful in determining which variable should be dropped from the current model to improve the fit of the current model. It is also useful in finding a suitable variable that may be added to the current model. A model with more appropriate variables makes more stable inference in general.

The criteria traditionally often used for variable selection is magnitude of communalities. This criteria gives a different choice of variables and does not improve fit of the model in most cases.

The asymptotic normal distribution of the maximum likelihood estimator of Cronbach's alpha (under normality) is derived for the case when no assumptions are made about the covariances among items. The asymptotic distribution is also considered for the special case of compound symmetry and compared to the exact distribution.

In the context of structural equation modeling, a general interaction model with multiple latent interaction effects is introduced. A stochastic analysis represents the nonnormal distribution of the joint indicator vector as a finite mixture of normal distributions. The Latent Moderated Structural Equations (LMS) approach is a new method developed for the analysis of the general interaction model that utilizes the mixture distribution and provides a ML estimation of model parameters by adapting the EM algorithm. The finite sample properties and the robustness of LMS are discussed. Finally, the applicability of the new method is illustrated by an empirical example.

Item response theory posits “local independence,” or conditional independence of item responses given item parameters and examinee proficiency parameters. The usual definition of local independence, however, addresses the context of fixed tests, and initially appears to yield incorrect response-pattern probabilities in the context of adaptive testing. The paradox is resolved by introducing additional notation to deal with the item selection mechanism.

In this paper, a class of probability models for ranking data, the order-statistics models, is investigated. We extend the usual normal order-statistics model into one where the underlying random variables follow a multivariate normal distribution. Bayesian approach and the Gibbs sampling technique are used for parameter estimation. In addition, methods to assess the adequacy of model fit are introduced. Robustness of the model is studied by considering a multivariate-t distribution. The proposed method is applied to analyze the presidential election data of the American Psychological Association (APA).

Interval estimates of the Pearson, Kendall tau-a and Spearman correlations are reviewed and an improved standard error for the Spearman correlation is proposed. The sample size required to yield a confidence interval having the desired width is examined. A two-stage approximation to the sample size requirement is shown to give accurate results.

Cureton & Mulaik (1975) proposed the Weighted Varimax rotation so that Varimax (Kaiser, 1958) could reach simple solutions when the complexities of the variables in the solution are larger than one. In the present paper the weighting procedure proposed by Cureton & Mulaik (1975) is applied to Direct Oblimin (Clarkson & Jennrich, 1988), and the rotation method obtained is called Weighted Oblimin. It has been tested on artificial complex data and real data, and the results seem to indicate that, even though Direct Oblimin rotation fails when applied to complex data, Weighted Oblimin gives good results if a variable with complexity one can be found for each factor in the pattern. Although the weighting procedure proposed by Cureton & Mulaik is based on Landahl's (1938) expression for orthogonal factors, Weighted Oblimin seems to be adequate even with highly oblique factors. The new rotation method was compared to other rotation methods based on the same weighting procedure and, whenever a variable with complexity one could be found for each factor in the pattern, Weighted Oblimin gave the best results. When rotating a simple empirical loading matrix, Weighted Oblimin seemed to slightly increase the performance of Direct Oblimin.

A new algorithm for obtaining exact person fit indexes for the Rasch model is introduced which realizes most powerful tests for a very general family of alternative hypotheses, including tests concerning DIF as well as model-deviating item correlations. The method is also used as a goodness-of-fit test for whole data sets where the item parameters are assumed to be known. For tests with 30 items at most, exact values are obtained, for longer tests a Monte Carlo-algorithm is proposed. Simulated examples and an empirical investigation demonstrate test power and applicability to item elimination.

Multilevel covariance structure models have become increasingly popular in the psychometric literature in the past few years to account for population heterogeneity and complex study designs. We develop practical simulation based procedures for Bayesian inference of multilevel binary factor analysis models. We illustrate how Markov Chain Monte Carlo procedures such as Gibbs sampling and Metropolis-Hastings methods can be used to perform Bayesian inference, model checking and model comparison without the need for multidimensional numerical integration. We illustrate the proposed estimation methods using three simulation studies and an application involving student's achievement results in different areas of mathematics.

Correspondence analysis leads to a graphical representation of the associations between categories of the row and column variables of a contingency table. Greenacre's (1988) formulation of joint correspondence analysis is a multivariate extension which finds the optimal joint display of contingency tables between all pairs of variables in a set. Greenacre presented a discrepancy function and an alternating least squares algorithm for its minimization. Boik (1996) presented an alternative algorithm, also of the alternating least squares type, for minimizing the same discrepancy function. In this paper, a noniterative procedure, not based on the minimization of any discrepancy function, is described.

The asymptotic correlations between the estimates of factor and component loadings are obtained for the exploratory factor analysis model with the assumption of a multivariate normal distribution for manifest variables. The asymptotic correlations are derived for the cases of unstandardized and standardized manifest variables with orthogonal and oblique rotations. Based on the above results, the asymptotic standard errors for estimated correlations between factors and components are derived. Further, the asymptotic standard error of the mean squared canonical correlation for factors and components, which is an overall index for the closeness of factors and components, is derived. The results of a Monte Carlo simulation are presented to show the usefulness of the asymptotic results in the data with a finite sample size.

The paper addresses and discusses whether the tradition of accepting point-symmetric item characteristic curves is justified by uncovering the inconsistent relationship between the difficulties of items and the order of maximum likelihood estimates of ability. This inconsistency is intrinsic in models that provide point-symmetric item characteristic curves, and in this paper focus is put on the normal ogive model for observation. It is also questioned if in the logistic model the sufficient statistic has forfeited the rationale that is appropriate to the psychological reality. It is observed that the logistic model can be interpreted as the case in which the inconsistency in ordering the maximum likelihood estimates is degenerated.

The paper proposes a family of models, called the logistic positive exponent family, which provides asymmetric item chacteristic curves. A model in this family has a consistent principle in ordering the maximum likelihood estimates of ability. The family is divided into two subsets each of which has its own principle, and includes the logistic model as a transition from one principle to the other. Rationale and some illustrative examples are given.

Robust schemes in regression are adapted to mean and covariance structure analysis, providing an iteratively reweighted least squares approach to robust structural equation modeling. Each case is properly weighted according to its distance, based on first and second order moments, from the structural model. A simple weighting function is adopted because of its flexibility with changing dimensions. The weight matrix is obtained from an adaptive way of using residuals. Test statistic and standard error estimators are given, based on iteratively reweighted least squares. The method reduces to a standard distribution-free methodology if all cases are equally weighted. Examples demonstrate the value of the robust procedure.

When modeling the relationship between two nominal categorical variables, it is often desirable to include covariates to understand how individuals differ in their response behavior. Typically, however, not all the relevant covariates are available, with the result that the measured variables cannot fully account for the associations between the nominal variables. Under the assumption that the observed and unobserved variables follow a homogeneous conditional Gaussian distribution, this paper proposes RC(M) regression models to decompose the residual associations between the polytomous variables. Based on Goodman's (1979, 1985)RC(M) association model, a distinctive feature of RC(M) regression models is that they facilitate the joint estimation of effects due to manifest and omitted (continuous) variables without requiring numerical integration. The RC(M) regression models are illustrated using data from the High School and Beyond study (Tatsuoka & Lohnes, 1988).

In the literature on the measurement of change, reliable change is usually determined by means of a confidence interval around an observed value of a statistic that estimates the true change. In recent literature on the efficacy of psychotherapies, attention has been particularly directed at the improvement of the estimation of the true change. Reliable Change Indices, incorporating the reliability-weighted measure of individual change, also known as Kelley's formula, have been proposed. According to current practice, these indices are defined as the ratio of such an estimator and an intuitively appealing criterion and then regarded as standard normally distributed statistics. However, because the authors fail to adopt an adequate standard error of the estimator, the statistical properties of their indices are unclear. In this article, it is shown that this can lead to paradoxical conclusions. The adjusted standard error is derived.

We investigate under what conditions the matrix of factor loadings from the factor analysis model with equal unique variances will give a good approximation to the matrix of factor loadings from the regular factor analysis model. We show that the two models will give similar matrices of factor loadings if Schneeweiss' condition, that the difference between the largest and the smallest value of unique variances is small relative to the sizes of the column sums of squared factor loadings, holds. Furthermore, we generalize our results and discus the conditions under which the matrix of factor loadings from the regular factor analysis model will be well approximated by the matrix of factor loadings from Jöreskog's image factor analysis model. Especially, we discuss Guttman's condition (i.e., the number of variables increases without limit) for the two models to agree, in relation with the condition we have shown, and conclude that Schneeweiss' condition is a generalization of Guttman's condition. Some implications for practice are discussed.

In item response models of the Rasch type (Fischer & Molenaar, 1995), item parameters are often estimated by the conditional maximum likelihood (CML) method. This paper addresses the loss of information in CML estimation by using the information concept of F-information (Liang, 1983). This concept makes it possible to specify the conditions for no loss of information and to define a quantification of information loss. For the dichotomous Rasch model, the derivations will be given in detail to show the use of the F-information concept for making comparisons for different estimation methods. It is shown that by using CML for item parameter estimation, some information is almost always lost. But compared to JML (joint maximum likelihood) as well as to MML (marginal maximum likelihood) the loss is very small. The reported efficiency in the use of information of CML to JML and to MML in several comparisons is always larger than 93%, and in tests with a length of 20 items or more, larger than 99%.

An important feature of distance-based principal components analysis, is that the variables can be optimally transformed. For monotone spline transformation, a nonnegative least-squares problem with a length constraint has to be solved in each iteration. As an alternative algorithm to Lawson and Hanson (1974), we propose the Alternating Length-Constrained Non-Negative Least-Squares (ALC-NNLS) algorithm, which minimizes the nonnegative least-squares loss function over the parameters under a length constraint, by alternatingly minimizing over one parameter while keeping the others fixed. Several properties of the new algorithm are discussed. A Monte Carlo study is presented which shows that for most cases in distance-based principal components analysis, ALC-NNLS performs as good as the method of Lawson and Hanson or sometimes even better in terms of the quality of the solution.