Modeling Responses and Response Times in Tests With the Hierarchical Model and the Three-Parameter Lognormal Distribution

Abstract

The hierarchical model of van der Linden is the most popular model for responses and response times in tests. It is composed of two separate submodels—one for the responses and one for the response times—that are joined at a higher level. The submodel for the response times is based on the lognormal distribution. The lognormal distribution is a skew distribution with a support from zero to infinity. Such a support is unrealistic as the solution process demands a minimal processing time that sets a response time threshold. Ignoring this response time threshold misspecifies the model and threatens the validity of model-based inferences. In this article, the response time model of van der Linden is replaced by a model that is based on the three-parameter lognormal distribution. The three-parameter lognormal distribution extends the lognormal distribution by an additional location parameter that bounds the support away from zero. Two different approaches to model fitting are proposed and evaluated with regard to parameter recovery in a simulation study. The extended model is applied to two data sets. In both data sets, the extension improves the fit of the hierarchical model.