A geometric power analysis for general log-linear models

Log-linear models express the association in multivariate frequency data on contingency tables. When the null hypothesis states a log-linear model, a class of distributions obtained by assigning a small offset to the log-linear equation provides a natural local alternative for the power analysis. In this case, a discrepancy from the null can be expressed in terms of interaction parameters and has a data-relevant interpretation. A log-linear model is represented by a smooth surface in the probability simplex and, therefore, the power analysis can be considered from a geometric viewpoint. The proposed concept of geometric power describes the ability of a goodness-of-fit statistic to distinguish between two surfaces in the probability simplex. An extension of this concept to frequency data is also introduced and applied in the context of multinomial sampling. The Monte-Carlo algorithms for the estimation of geometric power and its extension are proposed. An iterative scaling procedure for constructing distributions from a log-linear model and its alternative is described and its convergence is proved. The geometric power analysis is carried out for data from a clinical study.