Properties of the correlational agreement coefficient: A comment to Ünlü and Albert (2004)

The correlational agreement coefficient CA(≤, D) [van Leeuwe, J.F.J., 1974. Item tree analysis. Nederlands Tijdschrift voor de Psychologie 29, 475-484.] is a descriptive measure for the fit of a quasi-order ≤ on an item set to a binary data set D. The coefficient is based on the comparison between the empirical correlations of the items to their assumed theoretical correlations. These theoretical correlations are derived from the assumption that the quasi-order is a correct representation of the data. In a recent paper Ünlü and Albert [Ünlü, A., Albert, D., 2004. The correlational agreement coefficient CA(≤, D) - A mathematical analysis of a descriptive goodness-of-fit measure. Mathematical Social Sciences 48, 281-314.] presented a detailed mathematical investigation of CA(≤, D). They describe a number of problems of this coefficient which show in their opinion that its use to compare quasi-orders is questionable. We do not agree with some of the statements in Ünlü and Albert [Ünlü, A., Albert, D., 2004. The correlational agreement coefficient CA(≤, D) - A mathematical analysis of a descriptive goodness-of-fit measure. Mathematical Social Sciences 48, 281-314.]. Especially we try to show that some of the problems of CA(≤, D) mentioned in Ünlü and Albert [Ünlü, A., Albert, D., 2004. The correlational agreement coefficient CA(≤, D) - A mathematical analysis of a descriptive goodness-of-fit measure. Mathematical Social Sciences 48, 281-314.] are in fact properties which a good measure of fit for a quasi-order should have.