A Penalty Approach to Differential Item Functioning in Rasch Models

A new diagnostic tool for the identification of differential item functioning (DIF) is proposed. Classical approaches to DIF allow to consider only few subpopulations like ethnic groups when investigating if the solution of items depends on the membership to a subpopulation. We propose an explicit model for differential item functioning that includes a set of variables, containing metric as well as categorical components, as potential candidates for inducing DIF. The ability to include a set of covariates entails that the model contains a large number of parameters. Regularized estimators, in particular penalized maximum likelihood estimators, are used to solve the estimation problem and to identify the items that induce DIF. It is shown that the method is able to detect items with DIF. Simulations and two applications demonstrate the applicability of the method

Visualization of Categorical Response Models - from Data Glyphs to Parameter Glyphs

The multinomial logit model is the most widely used model for nominal multi-category responses. One problem with the model is that many parameters are involved, another that interpretation of parameters is much harder than for linear models because the model is non-linear. Both problems can profit from graphical representations. We propose to visualize the effect strengths by star plots, where one star collects all the parameters connected to one explanatory variable. In contrast to conventional star plots, which are used to represent data, the plots represent parameters and are considered as parameter glyphs. The set of stars for a fitted model makes the main features of the effects of explanatory variables on the response variable easily accessible. The method is extended to ordinal models and illustrated by several data sets

Extended Ordered Paired Comparison Models with Application to Football Data from German Bundesliga

A general paired comparison model for the evaluation of sports competitions is proposed. It efficiently uses the available information by allowing for ordered response categories and team-specific home advantage effects. Penalized estimation techniques are used to identify clusters of teams that share the same ability. The model is extended to include team-specific explanatory variables. It is shown that regularization techniques allow to identify the contribution of explanatory variables to the success of teams. The usefulness of the methods is demonstrated by investigating the performance and its dependence on the budget for football teams of the German Bundesliga

BTLLasso - A Common Framework and Software Package for the Inclusion and Selection of Covariates in Bradley-Terry Models

In paired comparison models, the inclusion of covariates is a tool to account for the heterogeneity of preferences and to investigate which characteristics determine the preferences. Although methods for the selection of variables have been proposed no coherent framework that combines all possible types of covariates is available. There are three different types of covariates that can occur in paired comparisons, the covariates can either vary over the subjects, the objects or both the subjects and the objects of the paired comparisons. This paper gives an overview over all possible types of covariates in paired comparisons and introduces a general framework to include covariate effects into Bradley-Terry models. For each type of covariate, appropriate penalty terms that allow for sparser models and therefore easier interpretation are proposed. The whole framework is implemented in the R-package BTLLasso. The main functionality and the visualization tools of the package are introduced and illustrated by using real data sets

Uncertainty as Response Style in Latent Trait Models

It is well known that the presence of response styles can affect estimates in item response models. Various approaches to account for response styles have been suggested, in particular the tendency to extreme or middle categories has been included in the modelling of item responses. A response style that has been rarely considered is the noncontingent response style, which occurs if persons have a tendency to respond randomly and nonpurposefully, which might also be a consequence of indecision. A model is proposed that extends the Rasch model and the Partial Credit Model to account for a response style that accounts for subject-specific uncertainty when responding to items. It is demonstrated that ignoring the subject-specific uncertainty may yield biased estimates of model parameters. Uncertainty as well as the underlying trait are linked to explanatory variables. The parameterization allows to identify subgroups that differ in response style and underlying trait. The modeling approach is illustrated by using data on the confidence of citizens in public institutions

Subject-specific modelling of paired comparison data: A lasso-type penalty approach

In traditional paired comparison models heterogeneity in the population is simply ignored and it is assumed that all persons or subjects have the same preference structure. In the models considered here the preference of an object over another object is explicitly modelled as depending on subject-specific covariates, therefore allowing for heterogeneity in the population. Since by construction the models contain a large number of parameters we propose to use penalized estimation procedures to obtain estimates of the parameters. The used regularized estimation approach penalizes the differences between the parameters corresponding to single covariates. It enforces variable selection and allows to find clusters of objects with respect to covariates. We consider simple binary but also ordinal paired comparisons models. The method is applied to data from a pre-election study from Germany. </jats:p

Response Styles in the Partial Credit Model

In the modeling of ordinal responses in psychological measurement and survey-based research, response styles that represent specific answering patterns of respondents are typically ignored. One consequence is that estimates of item parameters can be poor and considerably biased. The focus here is on the modeling of a tendency to extreme or middle categories. An extension of the partial credit model is proposed that explicitly accounts for this specific response style. In contrast to existing approaches, which are based on finite mixtures, explicit person-specific response style parameters are introduced. The resulting model can be estimated within the framework of generalized mixed linear models. It is shown that estimates can be seriously biased if the response style is ignored. In applications, it is demonstrated that a tendency to extreme or middle categories is not uncommon. A software tool is developed that makes the model easy to apply

Subject-specific Bradley-Terry-Luce Models with Implicit Variable Selection

The Bradley-Terry-Luce (BTL) model for paired comparison data is able to obtain a ranking of the objects that are compared pairwise by subjects. The task of each subject is to make preference decisions in favor of one of the objects. This decision is binary when subjects prefer either the first object or the second object, but can also be ordinal when subjects make their decisions on a Likert scale. Since subject-specific covariates, which reflect characteristics of the subject, may affect the preference decision, it is essential to incorporate subject-specific covariates into the model. However, the inclusion of subject-specific covariates yields a model that contains many parameters and thus estimation becomes challenging. To overcome this problem, we propose a procedure that is able to select and estimate only relevant variables