Testing the Hypothesis of no Fixed Main-Effects in Scheffe's Mixed Model

Abstract

Various approaches are available for the formulation of linear models for the analysis of variance. The mixed model in which one factor is a fixed-effects factor, and one factor is a random-effects factor can be obtained for instance as a limiting case of the general models of Cornfield and Tukey [3], and of Wilk and Kempthorne [8]. An entirely different approach is that given by Scheffe [6], and is the one considered in the present paper. Whatever the approach used, a hypothesis of interest will usually be the hypothesis of no fixed-effects. Consider a two-way layout in which $A$ denotes a fixed-effects factor and $B$ a random-effects factor. Let $I$ and $J$ be the numbers of levels of factors $A$ and $B$ respectively at which measurements are taken $(I > 1, J > 1)$. Let $K$ be the number of replications performed in each cell $(K > 1)$. In the light of $K$ be the number of replications performed in each cell $(K > 1)$. In the light of Table 1 of [3], Table 3 of [8], and formulas (46) and (54) of [6], an adequate $F$-type statistic to use for testing the hypothesis $H_A$ that all main-effects corresponding to the levels of factor $A$ are zero appears to be \begin{equation*}\tag{1.1}\mathscr{F} = (\mathrm{MS})_A/(\mathrm{MS})_{AB}.\end{equation*} The usual mean squares $(\mathrm{MS})_A$ and $(\mathrm{MS})_{AB}$ corresponding to factor $A$ and to $A \times B$ interactions are explicitly defined below. With the normal theory models which are commonly used in the case of two fixed-effects factors and in the case of two random-effects factors (e.g., in [7]), the criterion (1.1) has under the hypothesis $H_A$ the $F$-distribution with $I - 1$ and $(I - 1)(J - 1)$ d.f. In Scheffe's mixed model [6], this is no longer the case. When $J \geqq I$, a Hotelling $T^2$ statistic can then be constructed for the test of $H_A$. When multiplied by a constant factor, this statistic has the $F$-distribution with $I - 1$ and $J - I + 1$ d.f. While requiring a larger amount of computational work, the $T^2$ test will have little power when $J - I + 1$ is small. It is therefore tempting to construct a test of $H_A$, based on the ratio (1.1), by assuming that the law of $\mathscr{F}$ is not much different under $H_A$ from that of $F$ with $I - 1$ and $(I - 1)(J - 1)$ d.f. In Sub-section 4.1, we investigate the possible ill-effects of this assumption. They can be considerable, and remedies are suggested in Subsections 4.2 and 4.3.