Frank Rijmen

A nonlinear mixed model framework for item response theory

Mixed models take the dependency between observations based on the same cluster into account by introducing 1 or more random effects. Common item response theory (IRT) models introduce latent person variables to model the dependence between responses of the same participant. Assuming a distribution for the latent variables, these IRT models are formally equivalent with nonlinear mixed models. It is shown how a variety of IRT models can be formulated as particular instances of nonlinear mixed models. The unifying framework offers the advantage that relations between different IRT models become explicit and that it is rather straightforward to see how existing IRT models can be adapted and extended. The approach is illustrated with a self-report study on anger.

Statistical inference in generalized linear mixed models: A review

We present a review of statistical inference in generalized linear mixed models (GLMMs). GLMMs are an extension of generalized linear models and are suitable for the analysis of non-normal data with a clustered structure. A GLMM contains parameters common to all clusters (fixed regression effects and variance components) and cluster-specific parameters. The latter parameters are assumed to be randomly drawn from a population distribution. The parameters of this population distribution (the variance components) have to be estimated together with the fixed effects. We focus on the case in which the cluster-specific parameters are normally distributed. The cluster-specific effects are integrated out of the likelihood so that the fixed effects and variance components can be estimated. Unfortunately, the integral over the cluster-specific effects is intractable for most GLMMs with a normal mixing distribution. Within a classical statistical framework, we distinguish between two broad classes of methods to handle this intractable integral: methods that rely on a numerical approximation to the integral and methods that use an analytical approximation to the integrand. Finally, we present an overview of available methods for testing hypotheses about the parameters of GLMMs.

The Random Weights Linear Logistic Test Model.

A generalization of the linear logistic test model of G. H. Fischer (1973), the random weights linear logistic test model, is presented. The generalization consists of a random coef.cient contribution of item stimulus features to the item dif.culties, with the coef.cients varying over persons. Whereas in the common linear logistic test model, only the intercept (ability) is considered random over persons, in the random weights linear logistic test model, also some or all of the item stimulus features are considered as having random coef.cients. It turns out that the random weights linear logistic test model is a special case of the multidimensional random coef.cient multinomial logit model of Adams, Wilson, and Wang (1997). The model is applied to a deductive reasoning task.

Toward Disentangling Sources of Individual Differences in Appraisal and Anger

A theoretical framework is presented to explain individual differences in situation-specific emotional experience in terms of three different sources of variance: (a) individual differences in how one appraises ones circumstances, (b) individual differences in how appraisals are related to the experience of emotion, and (c) individual differences independent from situation and appraisal. The relative contribution and nature of these sources was examined empirically for the experience of anger based on data from two directed imagery studies (total N=1,192). Consistent results across the two studies demonstrated that variability in anger experience primarily stems from variability in how a situation is appraised and to a smaller extent from individual differences in the relations between the appraisals and anger and individual differences independent of appraisal. The findings further identified frustration as the central appraisal involved in anger. Implications for emotion theories and anger management programs are discussed.

Latent Class Models for Diary Method Data: Parameter Estimation by Local Computations

Abstract The increasing use of diary methods calls for the development of appropriate statistical methods. For the resulting panel data, latent Markov models can be used to model both individual differences and temporal dynamics. The computational burden associated with these models can be overcome by exploiting the conditional independence relations implied by the model. This is done by associating a probabilistic model with a directed acyclic graph, and applying transformations to the graph. The structure of the transformed graph provides a factorization of the joint probability function of the manifest and latent variables, which is the basis of a modified and more efficient E-step of the EM algorithm. The usefulness of the approach is illustrated by estimating a latent Markov model involving a large number of measurement occasions and, subsequently, a hierarchical extension of the latent Markov model that allows for transitions at different levels. Furthermore, logistic regression techniques are used to incorporate restrictions on the conditional probabilities and to account for the effect of covariates. Throughout, models are illustrated with an experience sampling methodology study on the course of emotions among anorectic patients.

Estimation and software

The aim of this last chapter is threefold. First, we want to give the reader further insights into the estimation methods for the models presented in this volume. Second, we want to discuss the available software for the models presented in this volume. We will not sketch all possibilities of the software, but only those directly relevant to item response modeling as seen in this volume. Third, we want to illustrate the use of various programs for the estimation of a basic model, the Rasch model, for the verbal aggression data.

Multiple person dimensions and latent item predictors

In this chapter, we discuss two extensions to the item response models presented in the first two parts of this book: more than one random effect for persons (multidimensionality) and latent item predictors. We only consider models with random person weights (following a normal distribution), and with no inclusion of person predictors (except for the constant). The extensions can be applied in much the same way to the other models that were discussed in the first two parts of this book.

Propositional reasoning: the differential contribution of "rules" to the difficulty of complex reasoning problems.

In Experiment 1, complex propositional reasoning problems were constructed as a combination of several types of logical inferences: modus ponens, modus tollens, disjunctive modus ponens, disjunctive syllogism, and conjunction. Rule theories of propositional reasoning can account for how one combines these inferences, but the difficulty of the problems can be accounted for only if a differential psychological cost is allowed for different basic rules. Experiment 2 ruled out some alternative explanations for these differences that did not refer to the intrinsic difficulty of the basic rules. It was also found that part of the results could be accounted for by the notion of representational cost, as it is used in the mental model theory of propositional reasoning. However, the number of models as a measure of representational cost seems to be too coarsely defined to capture all of the observed effects. Frank Rijmen was supported by the Fund for Scientific Research, Flanders (FWO).

Asymptotic and Sampling-Based Standard Errors for Two Population Invariance Measures in the Linear Equating Case

This article describes two methods for obtaining the standard errors of two commonly used population invariance measures of equating functions: the root mean square difference of the subpopulation equating functions from the overall equating function and the root expected mean square difference. The delta method relies on an analytical approximation and provides asymptotic standard errors, whereas the grouped jackknife method is a sampling-based method. The expressions for the delta method are presented for the linear equating function. Both methods were applied to a real data application. The results indicate little difference between the standard errors found by the two methods.

Hypothesis Testing of Equating Differences in the Kernel Equating Framework

Test equating methods are used to produce scores that are interchangeable across different test forms (Kolen & Brennan, 2004). In practice, often more than one equating method is applied to the data stemming from a particular test administration. If differences in estimated equating functions are observed, the question arises as to whether these differences reflect real differences in the underlying “true” equating functions or merely reflect sampling error. That is, are observed differences in equating functions statistically significant?