Extended Marginal Homogeneity Model Based on Complementary Log-Log Transform for Square Tables

Abstract

For square contingency tables with the same ordinal row and column classifications, McCullagh (1977) gave the marginal cumulative logistic model, which is an extension of the marginal homogeneity (MH) model using the logit transform. The present paper proposes a different extension of the MH model using the complementary log-log transform. In addition, the present paper gives the theorem that the MH model is equivalent to the proposed model and the equality of row and column marginal means holding simultaneously. In data analysis, if the MH model fits the data poorly, the theorem may be useful for seeing the reason for the poor fit. As example, the occupational status data for British father-son pairs are analyzed.

Extension of Marginal Complementary Log–Log Model and Separations of Marginal Homogeneity for Ordinal Categorical Data

“…Saigusa et al [13] and Shinoda et al [14] gave the separation of marginal homogeneity. We would like to note that Theorems 1 and 2 include these results.…”

“…The ML model with = 0 is the MH model, namely the ML model is the extension of the MH model. Moreover, Miyamoto et al [12] proposed the conditional ML (CML) model, and Saigusa et al [13] proposed the marginal complementary log-log (MCLL) model. The ML (CML) model states that one (conditional) marginal distribution is a location shift of the other (conditional) marginal distribution on a logistic scale.…”