Generalized marginal homogeneity model and its relation to marginal equimoments for square contingency tables with ordered categories

For square contingency tables with ordered categories, Tomizawa (Calcutta Stat Assoc Bull 43:123–125, 1993a; Sankhyā Ser B 60:293–300, 1998) gave theorems that the marginal homogeneity (MH) model is equivalent to certain two or three models holding simultaneously. This paper proposes a generalized MH model, which describes a structure of the odds that an observation will fall in row category i or below and column category i + 1 or above, instead of in column category i or below and row category i + 1 or above. In addition, this paper gives the theorems that the MH model is equivalent to the generalized MH model and some models holding simultaneously whose each indicates: (1) the equality of m-order moment of row and column variables, (2) the equality of skewness of them and (3) the equality of kurtosis of them. When the MH model fits the data poorly, these may be useful for seeing the reason for the poor fit; for instance, the poor fit of the MH model is caused by the poor fit of the equality of row and column means rather than the generalized MH model. Examples are given.