Testing Hypotheses in Unbalanced Variance Components Models for Two-Way Layouts

Abstract

Consider the model equation $y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + e_{ijk} (i = 1,2,\cdots, r; j = 1,2,\cdots, s; k = 1,2,\cdots, n_{ij})$, where $\mu$ is a constant and $\alpha_i, \beta_j, \gamma_{ij}, e_{ijk}$ are distributed independently and normally with zero means and variances $\sigma_A^2, \sigma_B^2, \sigma^2_{AB}, \sigma^2$, re spectively. In this paper procedures for testing hypotheses on $\sigma_A^2/\sigma_B^2, \sigma^2/\sigma^2$, and $\sigma^2_{AB}/\sigma^2$ are given. The test procedure for $\sigma^2_{AB}/\sigma^2$ is compared with the corresponding test procedures when $\alpha_i, \beta_j$, and $\gamma_{ij}$ are fixed effects instead of being random.