Testing the fit of relational models

TL;DR

This paper explores the properties of relational models, especially regarding maximum likelihood estimates and goodness-of-fit tests, highlighting differences when an overall effect is absent and providing theoretical insights into their statistical behavior.

Contribution

It characterizes the properties of MLEs and goodness-of-fit tests in relational models, especially when the overall effect is not included, extending existing theoretical frameworks.

Findings

MLE properties differ without overall effect

Likelihood ratio statistic relates to Bregman divergence

Asymptotic equivalence of chi-squared and likelihood ratio tests

Abstract

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 thus, the likelihood ratio statistic is not identical with twice the Kullback-Leibler divergence. However, as demonstrated, its general form may be obtained from the Bregman divergence. The asymptotic equivalence of the Pearson chi-squared and likelihood ratio statistics holds, but the generality considered here requires extended proofs.