Analysis of Nonnegative Observations using Gamma Model with 2 Factors (ANOGaM-2): Theory, Method and Applications with Real-life Data (including R code)

Two-factor ANOVA is widely used in experimental studies but relies on additivity, normality, independence, and homoscedasticity. These assumptions are often violated for nonnegative, positively skewed observations. Although Box–Cox-type transformations are commonly used, they may reduce interpretability and require a subjective choice of transformation. We propose an alternative framework in which nonnegative observations affected by two factors are modeled by gamma distributions with unknown shape and scale parameters that may depend on factor levels. We develop likelihood ratio tests (LRTs) for main and interaction effects. The asymptotic LRT (ALRT) uses the asymptotic chi-square distribution, which may be inaccurate for small to moderate samples. We therefore propose a parametric bootstrap LRT (PBLRT) that determines critical values by simulation. Extensive simulations show that the PBLRT maintains the nominal significance level well. Real-data examples demonstrate its applicability and show that its inferences can differ from those of traditional ANOVA.