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

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 x₈₉, i = 1, 2, n, j = 1, 2, 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 -) ², where 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 = ² pmatrix 1 & & & \\ & & & \\ & & & \\ & & & 1pmatrix. 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 k, Hotelling's T² 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 (k - 1), (k - 1) (n - 1) where 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 (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 th group has n_ individuals, = 1, 2, g, and ᵍ = ₁ n_ = N. We will show that the treatment mean square and the treatment group interaction can be tested in the same approximate fashion by using the Box procedure.