Coordinate-Free Relational Models

Relational models generalize log-linear models to sample spaces that do not necessarily have a Cartesian product structure. They are defined by a multiplicative assumption on the cell parameters, with the terms entering the product being constant over subsets of the sample space. The models are defined by specifying the subsets that have parameters associated with them. While log-linear models always have an overall effect, relational models may or may not be reparameterized to have the same effect present in every cell. If such an effect is not present, the usual relationship between the MLEs of probabilities and intensities does not hold. Among others, MLEs for the probabilities do not reproduce the observed sums of the relative frequencies in the defining subsets, and MLEs for intensities do not reproduce the total sample size. Relational models may be also defined by setting the values of generalized odds ratios equal to one, but if the model does not have an overall effect, there is a non-homogeneous one among them. A necessary and sufficient condition of the existence of the MLEs is given, and an algorithm is described to compute them.