On the Analysis of Variance of a Two-Way Classification with Unequal Sub-Class Numbers

Abstract

In many avenues of research it is necessary to analyse the variance of data which are classified in two ways with unequal numbers of observations falling into each sub-class of the classification. For data of this kind special. methods of analysis are required because the inequality of the sub-class numbers causes lack of orthogonality among the main effects and interaction comparisons. Table I below gives the basic notation for dealing with an analysis of a two-way classification with unequal sub-class numbers. Several writers have dealt with the analysis of data of this form and various methods have been put forward. Some of the more prominent articles and discussions are cited below [1-13]. A simple preliminary step common to all methods is to separate the variance within sub-classes from the variance between sub-classes. Table II gives the analysis of variance for this preliminary step. The problem of extending the analysis to the main effects and to the interaction between the main effects now arises. The (pq 1) degrees of freedom for between sub-classes can be partitioned in the usual way into (p 1) degrees of freedom for between A classes, (q 1) degrees of freedom for between B classes and (p 1) (q 1) degrees of freedom for the interaction between the two classifications. The main difficulties arise in determining the correct sums of squares to be associated with each of these. One difficulty is that the addition theorem for sums of squares does not apply unless the sub-class numbers are proportional, and thus the interaction sum of squares cannot be computed by the usual method of differences. In fact, situations may occur where this procedure would give a negative result for the sum of squares for interaction. Frequently we assume, from the nature of the data or from previous information or experience, that interaction is absent or if present, negligible. Making this assumption, we are interested in testing if there are any significant differences between the A-classes and between the B-classes.