Contemporary Extensions of the Rasch Model

This paper will introduce, discuss and illustrate two contemporary extensions of theRasch model: the one parameter logistic model (Verhelst and Glas, 1995) and theMultidimensional Rasch model (Hoijtink et al., 1999). Using data with respect tothe measurement of schizotypy (Vollema and Hoijtink, 2000) the most importantfeatures of both models will be illustrated. For the one parameter logistic modelthese include: a (discrete) discrimination parameter for each item; a test for itembias; and, estimation of the location of a person on the (latent) trait that is beingmeasured. For the multidimensional Rasch model these include: specification ofthe model; and, model selection. All analyses presented in this paper can be executedusing either OPLM (Verhelst et al., 1995), TESTFACT (Wilson et al.,1984) or ConQuest (Wu et al., 1998). At the end of the paper some features ofmodels and software that have not been discussed will be summarized.     

Comparison of Hypothesis Testing and Bayesian Model Selection

The main goal of both Bayesian model selection and classical hypotheses testing is to make inferences with respect to the state of affairs in a population of interest. The main differences between both approaches are the explicit use of prior information by Bayesians, and the explicit use of null distributions by the classicists. Formalization of prior information in prior distributions is often difficult. In this paper two practical approaches (encompassing priors and training data) to specify prior distributions will be presented. The computation of null distributions is relatively easy. However, as will be illustrated, a straightforward interpretation of the resulting p-values is not always easy. Bayesian model selection can be used to compute posterior probabilities for each of a number of competing models. This provides an alternative for the currently prevalent testing of hypotheses using p-values. Both approaches will be compared and illustrated using case studies. Each case study fits in the framework of the normal linear model, that is, analysis of variance and multiple regression.     

Bayesian model selection using encompassing priors

This paper deals with Bayesian selection of models that can be specified using inequality constraints among the model parameters. The concept of encompassing priors is introduced, that is, a prior distribution for an unconstrained model from which the prior distributions of the constrained models can be derived. It is shown that the Bayes factor for the encompassing and a constrained model has a very nice interpretation: it is the ratio of the proportion of the prior and posterior distribution of the encompassing model in agreement with the constrained model. It is also shown that, for a specific class of models, selection based on encompassing priors will render a virtually objective selection procedure. The paper concludes with three illustrative examples: an analysis of variance with ordered means; a contingency table analysis with ordered odds-ratios; and a multilevel model with ordered slopes.     

Application of Multilevel Models to Structured Repeated Measurements

The linear mixed-effects model has been widely used for the analysis of continuous longitudinal data. This paper demonstrates that the linear mixed model can be adapted and used for the analysis of structured repeated measurements. A computational advantage of the proposed methodology is that there is no extra burden on the analyst since any software for linear mixed-effects models can be used to fit the proposed models. Two data sets from clinical psychology are used as motivating examples and to illustrate the methods.     

Use of hypotheses for analysis of variance models: challenging the current practice

In social science research, hypotheses about group means are commonly tested using analysis of variance. While deemed to be formulated as specifically as possible to test social science theory, they are often defined in general terms. In this article we use two studies to explore the current practice concerning group mean hypotheses. The first study consists of a content analysis of published articles where the reconstructed reality of hypotheses use is explored. The second study is a qualitative interview study with researchers, adding information about daily practice. We argue that, at present, hypotheses are not used to their utmost potential and that progress can be made by using informative hypotheses instead of the current non-informative hypotheses. Informative hypotheses capitalize on knowledge that researchers already possess and enable them to focus in their proceeding projects. The substantive focus of our work is the case of applied psychology.     

An item response model with single peaked item characteristic curves: The PARELLA model

In this paper an item response model (the PARELLA model) designed specifically for the measurement of attitudes and preferences will be introduced. In contrast with the item response models currently used (e.g. the Rasch model and, the two and three parameter logistic model) the item characteristic curve is single peaked instead of monotonically increasing. The model and its properties will be introduced. After this the consequences of these properties for the item writing stage, the evaluation of the concordance of data and model, and the applicability of the model will be discussed. The paper is finished with an example concerning the measurement of change in the attitude in the car-environment issue.     

How to handle missing data in regression models using information criteria*

An important application of multiple regression is predictor selection. When there are no missing values in the data, information criteria can be used to select predictors. For example, one could apply the small‐sample‐size corrected version of the Akaike information criterion (AIC), the (AICC). In this article, we discuss how information criteria should be calculated when the dependent variable and/or the predictors contain missing values. Therewith, we extensively discuss and evaluate three models that can be employed to deal with the missing data, that is, to predict the missing values. The most complex model, that is, the model with all available predictors, outperforms the other models. These results also apply to more general hypotheses than predictor selection and also to structural equation modeling (SEM) models.     

Testing homogeneity in a random intercept model using asymptotic, posterior predictive and plug-in p-values

In this paper three statistics and three discrepancy measures with which homogeneity in the random intercept model can be investigated will be evaluated. The first two can be used to test the homogeneity of level one residual variances across level two units and the third can be used to test whether effects should be fixed or random. Each statistic and discrepancy measure will be evaluated using asymptotic (if available), posterior predictive and plug in p -values. A simulation study will be used to investigate the frequency properties of these p -values. In the discussion it will be indicated how the results obtained for the random intercept model with one explanatory variable can be useful during the construction of general two level models.     

Objective Bayes Factors for Inequality Constrained Hypotheses

This paper will present a Bayes factor for the comparison of an inequality constrained hypothesis with its complement or an unconstrained hypothesis. Equivalent sets of hypotheses form the basis for the quantification of the complexity of an inequality constrained hypothesis. It will be shown that the prior distribution can be chosen such that one of the terms in the Bayes factor is the quantification of the complexity of the hypothesis of interest. The other term in the Bayes factor represents a measure of the fit of the hypothesis. Using a vague prior distribution this fit value is essentially determined by the data. The result is an objective Bayes factor.