One-Way ANOVA

Abstract

AbstractIn this and the following chapters, we apply the general theory of linear models to various special cases. This chapter considers the analysis of one-way ANOVA models. A one-way ANOVA model can be written $$y_{ij} = \mu + \alpha_{i} + e_{ij}, \quad i = 1, \cdots, t, \quad j = 1, \cdots, N_i,$$ where \({\rm E}(e_{ij}) = 0, {\rm Var}(e_{ij}) = \sigma^2, {\rm and \ Cov}(e_{ij}, e_{j^{\prime}}, e_{{i^\prime j^\prime}}) = 0 {\rm when} (i, j) \neq (j^\prime, j^\prime)\). For finding tests and confidence intervals, the e ij s are assumed to have a multivariate normal distribution. Here α i is an effect for y ij belonging to the ith group of observations. Group effects are often called treatment effects because one-way ANOVA models are used to analyze completely randomized experimental designs.