Models for Complex Contingency Tables and Polychotomous Dosage Response Curves

Abstract

Probability models, in terms of main effect and interaction parameters, are given for the cell frequencies of complex contingency tables. These models, related to the logistic handling of quantal data, are essentially the same ones ordinarily employed by others who have studied such tables. The identification of specific parameters rather than of parametric conditions is intended to facilitate understanding what has been done in the field and to stimulate further research on contingency tables. Parametrization is given alternatively in arithmetric and logarithmic terms, with arithmetic additivity corresponding to unions of events (categories), logarithmic additivity to intersections. An analogous model is presented for the handling of the polychotomous dosage response situation, one in which drug treatment may elicit any of several discrete responses. Slope and intercept parameters are postulated corresponding to each of the several possible responses. Alternative handlings of the polychotomous response problem are discussed and it is brought out that the proposed model, in contrast, gives a unique result without the need for specifying an ordering of response categories. Further, the need for considering certain correlation problems and certain unrealistic consequences arising in multivariate probit models are avoided. In the polychotomous response situation it is brought out that arbitrary combination of response categories is inappropriate, albeit corresponding combinations are permissible in the complex contingency table situation. The implications of this limitation are discussed. Possible extensions of the model, including multiple regression and non-linear regression forms, are mentioned.