MATRIX METHODS IN COMPONENTS OF VARIANCE AND COVARIANCE ANALYSIS
Summary. The sampling variance of the least squares estimates of the components of variance in an unbalanced (non-orthogonal) one-way classification and the large sample variances of the maximum likelihood estimates of these quantities are summarized in a paper by Crump [1]. The present paper outlines a method of obtaining these results by the use of matrix algebra, and extends them to the sampling variances of estimates of components of covariance when two variables are considered. The methods are also used to obtain the large sample variance-covariance matrix of the maximum likelihood estimates of the components of variance and covariance. PART I. COMPONENTS OF VARIANCE 1. Model and analysis of variance. We are concerned with data in a 1-way classification with unequal numbers of observations in the classes. The linear model is taken as Xij = + ai + eii, where xij is the jth observation in the ith class. We will assume that there are c such classes (i = 1, -, c), the ith class containing ni observations (j = 1 - , n,); and let Eini = N. is a general mean, and {ai} and {eij} are random samples of size c and N from two normally distributed populations having zero means and variances o2 and _ , respectively. This is Eisenhart's Model II [3] and it is to this model that the discussion confines itself. The problem is to find the sampling variances of the estimates of o_2 and o_2 based on the usual analysis of variance of between and within classes. These estimates are