• categorical data;
  • contingency tables;
  • homogeneous constraints;
  • maximum likelihood;
  • non-log-linear models;
  • sampling plan [non]invariance

Abstract.  This study gives a generalization of Birch's log-linear model numerical invariance result. The generalization is given in the form of a sufficient condition for numerical invariance that is simple to verify in practice and is applicable for a much broader class of models than log-linear models. Unlike Birch's log-linear result, the generalization herein does not rely on any relationship between sufficient statistics and maximum likelihood estimates. Indeed the generalization does not rely on the existence of a reduced set of sufficient statistics. Instead, the concept of homogeneity takes centre stage. Several examples illustrate the utility of non-log-linear models, the invariance (and non-invariance) of fitted values, and the invariance (and non-invariance) of certain approximating distributions.