An Extension of Box's Results on the Use of the $F$ Distribution in Multivariate Analysis

Abstract

The mixed model in a 2-way analysis of variance is characterized by a fixed classification, e.g., treatments, and a random classification, e.g., plots or individuals. If we consider $k$ different treatments each applied to everyone of $n$ individuals, and assume the usual analysis of variance assumptions of uncorrelated errors, equal variances and normality, an appropriate analysis for the set of $nk$ observations $x_{ij}, i = 1, 2, \\cdots n, j = 1, 2, \\cdots k$, is ???? where the $F$ ratio under the null hypothesis has the $F$ distribution with $(k - 1)$ and $(k - 1)(n - 1)$ degrees of freedom. As is well known, if we extend the situation so that the errors have equal correlations instead of being uncorrelated, the $F$ ratio has the same distribution. Under the null hypothesis, the numerator estimates the same quantity as the denominator, namely, $(1 - \\rho)\\sigma^2$, where $\\rho$ is the constant correlation coefficient among the treatments. This case can also be considered as a sampling of $n$ vectors (individuals) from a $k$-variate normal population with variance-covariance matrix $$V = \\sigma^2 \\begin{pmatrix} 1 & \\rho & \\cdots & \\rho \\\\ \\rho & & & \\vdots \\\\ \\vdots & & & \\rho \\\\ \\rho & \\cdots & \\rho & 1\\end{pmatrix}.$$ If we consider this type of formulation and suppose the $k$ treatment errors to have a multivariate normal distribution with unknown variance-covariance matrix (the same for each individual), then the usual test described above is valid for $k = 2$. For $k > 2$, and $n \\geqq k$, Hotelling's $T^2$ is the appropriate test for the homogeneity of the treatment means. However, the working statistician is sometimes confronted with the case where $k > n$, or he does not have the adequate means for computing large order inverse matrices and would therefore like to use the original test ratio which in general does not have the requisite $F$ distribution. Box [1] and [2] has given an approximate distribution of the test ratio to be $F\\lbrack(k - 1)\\epsilon, (k - 1)(n - 1)\\epsilon\\rbrack$ where $\\epsilon$ is a function of the population variances and covariances and may further be approximated by the sample variances and covariances. We show in Section 3 that $\\epsilon \\geqq (k - 1)^{-1}$, and therefore a conservative test would be $F(1, n - 1)$. Box referred only to one group of $n$ individuals. We shall extend his results to a frequently occurring case, namely, the analysis of $g$ groups where the $\\alpha$th group has $n_\\alpha$ individuals, $\\alpha = 1, 2, \\cdots g$, and $\\Sigma^g_{\\alpha = 1} n_\\alpha = N$. We will show that the treatment mean square and the treatment $\\times$ group interaction can be tested in the same approximate fashion by using the Box procedure.