In this article we propose a model-free diagnostic for single-peakedness (unimodality) of item responses. Presuming a unidimensional unfolding scale and a given item ordering, we approximate item response functions of all items based on ordered conditional means (OCM). The proposed OCM methodology is based on Thurstone & Chave's (1929)criterion of irrelevance, which is a graphical, exploratory method for evaluating the "relevance" of dichotomous attitude items. We generalized this criterion to graded response items and quantified the relevance by fitting a unimodal smoother. The resulting goodness-of-fit was used to determine item fit and aggregated scale fit. Based on a simulation procedure, cutoff values were proposed for the measures of item fit. These cutoff values showed high power rates and acceptable Type I error rates. We present 2 applications of the OCM method. First, we apply the OCM method to personality data from the Developmental Profile; second, we analyze attitude data collected by Roberts and Laughlin (1996) concerning opinions of capital punishment.,
Multivariate Behavioral Research
Erasmus MC: University Medical Center Rotterdam

Polak, M.G, de Rooij, M, & Heiser, W.J. (2012). A Model-Free Diagnostic for Single-Peakedness of Item Responses Using Ordered Conditional Means. Multivariate Behavioral Research, 47(5), 743–770. doi:10.1080/00273171.2012.715563