Testing the fit of relational models

Relational models generalize log-linear models to arbitrary discrete sample spaces by specifying effects associated with any subsets of their cells. A relational model may include an overall effect, pertaining to every cell after a reparameterization, and in this case, the properties of the maximum likelihood estimates (MLEs) are analogous to those computed under traditional log-linear models, and the goodness-of-fit tests are also the same. If an overall effect is not present in any reparameterization, the properties of the MLEs are considerably different, and the Poisson and multinomial MLEs are not equivalent. In the Poisson case, if the overall effect is not present, the observed total is not always preserved by the MLE, and the likelihood ratio statistic has a form which can be expressed using the Bregman divergence but does not reduce to its Kullback–Leibler version. The asymptotic chi-squared distribution of the Pearson and likelihood ratio statistics holds, but the generality considered here requires extended proofs.