Analysis of frequency data: The ANOFA framework

Abstract

Analyses of frequencies are commonly done using a chi-square test. This test, derived from a normal approximation, is deemed generally efficient (controlling type-I error rates fairly well and having good statistical power). However, in the case of factorial designs, it is difficult to decompose a total test statistic into additive interaction effects and main effects. Herein, we present an alternative test based on the $G$ statistic. The test has similar type-I error rates and power as the former one. However, it is based on a total statistic that is naturally decomposed additively into interaction effects, main effects, simple effects, contrast effects, etc., mimicking precisely the logic of ANOVAs. We call this set of tools ANOFA (Analysis of Frequency data) to highlight its similarities with ANOVA. We also examine how to render plots of frequencies along with confidence intervals. Finally, quantifying effect sizes and planning statistical power are described under this framework. The ANOFA is a tool that assesses the significance of effects instead of the significance of parameters; as such, it is more intuitive to most researchers than alternative approaches based on generalized linear models.

Keywords

• frequency; contingency table; analyses of frequencies; additive decomposition