In health studies, questionnaire items are often scored on an ordinal scale, for example on a Likert scale. For such questionnaires, item response theory (IRT) models provide a useful approach for obtaining summary scores for subjects (i.e., the model's random subject effect) and characteristics of the items (e.g., item difficulty and discrimination). In this article, we describe a model that allows the items to additionally exhibit different within-subject variance, and also includes a subject-level random effect to the within-subject variance specification. This permits subjects to be characterized in terms of their mean level, or location, and their variability, or scale, and the model allows item difficulty and discrimination in terms of both random subject effects (location and scale). We illustrate application of this location-scale mixed model using data from the Social Subscale of the Drinking Motives Questionnaire (SS-DMQ) assessed in an adolescent study. We show that the proposed model fits the data significantly better than simpler IRT models, and is able to identify items and subjects that are not well-fit by the simpler models. The proposed model has useful applications in many areas where questionnaires are often rated on an ordinal scale, and there is interest in characterizing subjects in terms of both their mean and variability.
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