A geometric power analysis for general log-linear models

Log-linear models are widely used to express the association in multivariate frequency data on contingency tables. The paper focuses on the power analysis for testing the goodness-of-fit hypothesis for this model type. Conventionally, for the power-related sample size calculations a deviation from the null hypothesis (effect size) is specified by means of the chi-square goodness-of-fit index. It is argued that the odds ratio is a more natural measure of effect size, with the advantage of having a data-relevant interpretation. Therefore, a class of log-affine models that are specified by odds ratios whose values deviate from those of the null by a small amount can be chosen as an alternative. Being expressed as sets of constraints on odds ratios, both hypotheses are represented by smooth surfaces in the probability simplex, and thus, the power analysis can be given a geometric interpretation as well. A concept of geometric power is introduced and a Monte-Carlo algorithm for its estimation is proposed. The framework is applied to the power analysis of goodness-of-fit in the context of multinomial sampling. An iterative scaling procedure for generating distributions from a log-affine model is described and its convergence is proved. To illustrate, the geometric power analysis is carried out for data from a clinical study.