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.

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.

Maximum likelihood parameter estimation of the continuous time linear stochastic state space model is considered on the basis of large N discrete time data using a structural equation modeling (SEM) program. Random subject effects are allowed to be part of the model. The exact discrete model (EDM) is employed which links the discrete time model parameters to the underlying continuous time model parameters by means of nonlinear restrictions. The EDM is generalized to cover not only time-invariant parameters but also the cases of stepwise time-varying (piecewise time-invariant) parameters and parameters varying continuously over time according to a general polynomial scheme. The identification of the continuous time parameters is discussed and an educational example is presented.

When multiple items are clustered around a reading passage, the local independence assumption in item response theory is often violated. The amount of information contained in an item cluster is usually overestimated if violation of local independence is ignored and items are treated as locally independent when in fact they are not. In this article we provide a general method that adjusts for the inflation of information associated with a test containing item clusters. A computational scheme was presented for the evaluation of the factor of adjustment for clusters in the restrictive case of two items per cluster, and the general case of more than two items per cluster. The methodology was motivated by a study of the NAEP Reading Assessment. We present a simulated study along with an analysis of a NAEP data set.

The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank of I × J × K arrays with K = 2 to a particular class of arrays with K ≥ 2. It is shown that the typical rank is I when the array is tall in the sense that JK − J < I < JK. In addition, typical rank results are given for the case where I equals JK − J.

In this paper, we propose a (n−1)2 parameter, multistage ranking model, which represents a generalization of Luce's model. We propose the n×n item-rank relative frequency matrix (p-matrix) as a device for summarizing a set of rankings. As an alternative to the traditional maximum likelihood estimation, for the proposed model we suggest a method which estimates the parameters from the p-matrix. An illustrative numerical example is given. The proposed model and its differences from Luce's model are briefly discussed. We also show some special p-matrix patterns possessed by the Thurstonian models and distance-based models.

We present an hierarchical Bayes approach to modeling parameter heterogeneity in generalized linear models. The model assumes that there are relevant subpopulations and that within each subpopulation the individual-level regression coefficients have a multivariate normal distribution. However, class membership is not known a priori, so the heterogeneity in the regression coefficients becomes a finite mixture of normal distributions. This approach combines the flexibility of semiparametric, latent class models that assume common parameters for each sub-population and the parsimony of random effects models that assume normal distributions for the regression parameters. The number of subpopulations is selected to maximize the posterior probability of the model being true. Simulations are presented which document the performance of the methodology for synthetic data with known heterogeneity and number of sub-populations. An application is presented concerning preferences for various aspects of personal computers.

This note uses the EM-algorithm in an item response model as an illustration of a general method of parameter estimation, which geometrically can be described as an alternating projection method.

Situations sometimes arise in which variables collected in a study are not jointly observed. This typically occurs because of study design. An example is an equating study where distinct groups of subjects are administered different sections of a test. In the normal maximum likelihood function to estimate the covariance matrix among all variables, elements corresponding to those that are not jointly observed are unidentified. If a factor analysis model holds for the variables, however, then all sections of the matrix can be accurately estimated, using the fact that the covariances are a function of the factor loadings. Standard errors of the estimated covariances can be obtained by the delta method. In addition to estimating the covariance matrix in this design, the method can be applied to other problems such as regression factor analysis. Two examples are presented to illustrate the method.

Current practice in factor analysis typically involves analysis of correlation rather than covariance matrices. We study whether the standard z-statistic that evaluates whether a factor loading is statistically necessary is correctly applied in such situations and more generally when the variables being analyzed are arbitrarily rescaled. Effects of rescaling on estimated standard errors of factor loading estimates, and the consequent effect on z-statistics, are studied in three variants of the classical exploratory factor model under canonical, raw varimax, and normal varimax solutions. For models with analytical solutions we find that some of the standard errors as well as their estimates are scale equivariant, while others are invariant. For a model in which an analytical solution does not exist, we use an example to illustrate that neither the factor loading estimates nor the standard error estimates possess scale equivariance or invariance, implying that different conclusions could be obtained with different scalings. Together with the prior findings on parameter estimates, these results provide new guidance for a key statistical aspect of factor analysis.

It is well-known that the representations of the Thurstonian Case III and Case V models for paired comparison data are not unique. Similarly, when analyzing ranking data, other equivalent covariance structures can substitute for those given by Thurstone in these cases. That is, we may more broadly define the family of covariance structures satisfying Case III and Case V conditions. This paper introduces the notion of equivalence classes which defines a more meaningful partition of the covariance structures of the Thurstonian ranking models. In addition, the equivalence classes of Case V and Case III are completely characterized.

The relationship between the EM algorithm and the Bock-Aitkin procedure is described with a continuous distribution of ability (latent trait) from an EM-algorithm perspective. Previous work has been restricted to the discrete case from a probit-analysis perspective.

A new model of confirmatory factor analysis (CFA) for multitrait-multimethod (MTMM) data sets is presented. It is shown that this model can be defined by only three assumptions in the framework of classical psychometric test theory (CTT). All other properties of the model, particularly the uncorrelatedness of the trait with the method factors are logical consequences of the definition of the model. In the model proposed there are as many trait factors as different traits considered, but the number of method factors is one fewer than the number of methods included in an MTMM study. The covariance structure implied by this model is derived, and it is shown that this model is identified even under conditions under which other CFA-MTMM models are not. The model is illustrated by two empirical applications. Furthermore, its advantages and limitations are discussed with respect to previously developed CFA models for MTMM data.