A Model-Free Diagnostic for Single-Peakedness of Item Responses Using Ordered Conditional Means

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) Thurstone, L. L. and Chave, E. J. 1929. The measurement of attitude: A psychophysical method and some experiments with a scale for measuring attitude toward the church. Chicago, IL: University of Chicago Press.. Crossref] , Google Scholar] 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) Roberts, J. S. and Laughlin, J. E. 1996. A unidimensional item response model for unfolding responses from a graded disagree-agree response scale. Applied Psychological Measurement, 20: 231–255. Crossref], Web of Science ®] , Google Scholar] concerning opinions of capital punishment.