Item response theory models (IRT) are increasingly becoming established in social science research, particularly in the analysis of performance or attitudinal data in psychology, education, medicine, marketing and other fields where testing is relevant. We propose the R package eRm (extended Rasch modeling) for computing Rasch models and several extensions. A main characteristic of some IRT models, the Rasch model being the most prominent, concerns the separation of two kinds of parameters, one that describes qualities of the subject under investigation, and the other relates to qualities of the situation under which the response of a subject is observed. Using conditional maximum likelihood (CML) estimation both types of parameters may be estimated independently from each other. IRT models are well suited to cope with dichotomous and polytomous responses, where the response categories may be unordered as well as ordered. The incorporation of linear structures allows for modeling the effects of covariates and enables the analysis of repeated categorical measurements. The eRm package fits the following models: the Rasch model, the rating scale model (RSM), and the partial credit model (PCM) as well as linear reparameterizations through covariate structures like the linear logistic test model (LLTM), the linear rating scale model (LRSM), and the linear partial credit model (LPCM). We use an unitary, efficient CML approach to estimate the item parameters and their standard errors. Graphical and numeric tools for assessing goodness-of-fit are provided.

The first part of this paper describes a series of loglinear preference models based on paired comparisons, a method of measurement whose aim is to order a set of objects according to an attribute of interest by asking subjects to compare pairs of objects. Based on the basic Bradley-Terry specification, two types of models, the loglinear Bradley-Terry model and a pattern approach are presented. Both methods are extended to include subject and object-specific covariates and some further structural effects. In addition, models for derived paired comparisons (based on rankings and ratings) are also included. Latent classes and missing values can be included. The second part of the paper describes the package prefmod that implements the above models in R. Illustrational applications are provided in the last part of the paper.

Preference decisions will usually depend on the characteristics of both the judges and the objects being judged. In the analysis of paired comparison data concerning European universities and students' characteristics, it is demonstrated how to incorporate subject-speci®c information into Bradley±Terry-type models. Using this information it is shown that preferences for universities and therefore university rankings are dramatically different for different groups of students. A log-linear representation of a generalized Bradley±Terry model is speci®ed which allows simultaneous modelling of subject-and object-speci®c covariates and interactions between them. A further advantage of this approach is that standard software for ®tting log-linear models, such as GLIM, can be used.

Summary. This paper provides an alternative methodology for the analysis of a set of Likert responses measured on a common attitudinal scale when the primary focus of interest is on the relative importance of items in the set. The method makes fewer assumptions about the distribution of the responses than the more usual approaches such as comparisons of means, MANOVA or ordinal data methods. The approach transforms the Likert responses into paired comparison responses between the items. The complete multivariate pattern of responses thus produced can be analysed by an appropriately reformulated paired comparison model. The dependency structure between item responses can also be modelled flexibly. The advantage of this approach is that sets of Likert responses can be analysed simultaneously within the Generalized Linear Model framework, providing standard likelihood based inference for model selection. This method is applied to a recent international survey on the importance of environmental problems.

This paper introduces the paired comparison model as a suitable approach for the analysis of partially ranked data. For example, the Inglehart index, collected in international social surveys to examine shifts in post-materialistic values, generates such data on a set of attitude items. However, current analysis methods have failed to account for the complex shifts in individual item values, or to incorporate subject covariates. The paired comparison model is thus developed to allow for covariate subject effects at the individual level, and a reparameterization allows the inclusion of smooth non-linear effects of continuous covariates. The Inglehart index collected in the 1993 International Social Science Programme survey is analysed, and complex non-linear changes of item values with age, level of education and religion are identified. The model proposed provides a powerful tool for social scientists. Copyright 2002 Royal Statistical Society.

This paper is motivated by a Eurobarometer survey on science knowledge. As part of the survey, respondents were asked to rank sources of science information in order of importance. The official statistical analysis of these data however failed to use the complete ranking information. We instead propose a method which treats ranked data as a set of paired comparisons which places the problem in the standard framework of generalized linear models and also allows respondent covariates to be incorporated.An extension is proposed to allow for heterogeneity in the ranked responses. The resulting model uses a nonparametric formulation of the random effects structure, fitted using the EM algorithm. Each mass point is multivalued, with a parameter for each item. The resultant model is equivalent to a covariate latent class model, where the latent class profiles are provided by the mass point components and the covariates act on the class profiles. This provides an alternative interpretation of the fitted model. The approach is also suitable for paired comparison data.

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Measurement invariance is not only an important requirement of tests but also a central point in the examination of the Rasch model. Ponocny (2001) suggested quasi-exact tests for small samples which allow for formulating test-statistics based on matrices obtained using Monte Carlo methods. The purpose of the present study was to analyze the type-I error rates and the empirical power of two test-statistics for the assumption of measurement invariance in comparison with Andersen’s likelihood ratio test (1973). Each simulation was based on 10,000 replications and was a function of sample size (n = 30, 50, 100, 200), test length (k = 5, 9, 17), varying number of items exhibiting model violation, magnitude of violation, and different ability distributions. The results indicate that it is possible to detect large model violations on item level with samples of n = 50 or n = 100, and even weak violations with n = 200. Additionally, the results showed that it is possible to investigate very small samples where a parametric approach is not possible, which is one of the most important advantages of quasi-exact tests.