Goodness of Fit Measure Based on Sample Isotone Regression of Mokken Double Monotonicity Model

Abstract

Based on concepts of Mokken Double Monotonicity model (1971, 1997) and Sample Isotone Regression (Barlow, Bartholomew, Bremmer & Brunk, 1972), a model goodness of fit measure is defined. It permits interpretation of the global deviation from Double Monotonicity in a set of dichotomous response items. To this end, based on the order induced by the difficulty of the items, the disparity function associated with the proportion of positive and negative responses to pairs of items – given in P11 and P 00 matrices – is defined. In each matrix the global deviation from Double Monotonicity is obtained as the sum of discrepancies between the proportions of responses observed on pairs of items and the disparities associated with these proportions.