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.

翻译：对数线性模型广泛用于表达列联表多元频数数据中的关联性。本文聚焦于检验此类模型拟合优度假设的功效分析。传统上，在基于功效的样本量计算中，通过卡方拟合优度指数来指定与零假设的偏离（效应量）。本文论证了比值比作为效应量更具自然性，因其具有数据相关解释的优势。因此，可选择一类以比值比偏离零假设较小值来指定的对数仿射模型作为备择假设。由于两种假设均被表达为对比值比的约束集合，它们在概率单纯形上被表示为光滑曲面，从而功效分析也可获得几何解释。本文引入几何功效概念，并提出用于其估计的蒙特卡洛算法。该框架被应用于多项抽样背景下拟合优度的功效分析。描述了一种从对数仿射模型生成分布的迭代缩放过程，并证明了其收敛性。为阐明方法，本文对一项临床研究数据进行了几何功效分析。