Computing the Analysis of Variance Table for Experiments Involving Qualitative Factors and Zero Amounts of Quantitative Factors

Abstract

In a factorial experiment involving two factors, one quantitative and one qualitative, one may simply treat both factors as qualitative and obtain an analysis of variance table by straightforward methods. The sum of squares due to the quantitative factor can then be partitioned into sums of squares due to the linear, quadratic, etc. orthogonal contrasts, while the sum of squares due to the qualitative factor can be partitioned into meaningful orthogonal contrasts. If, however, one of the levels of the quantitative factor is zero, the analysis is complicated by the fact that a zero amount of each level of the qualitative factor is the same treatment. It is clear, for example, that if one applies a zero amount of six different fertilizers to six experimental units, one treats each experimental unit in exactly the same way. The foregoing situation arises frequently in biological assay and detailed analyses have been derived under the assumption that the quantitative factor follows a particular response law for each level of the qualitative factor. A short summary of this type of analysis is given in Kempthorne [1, Sec. 18.8]. I have been unable to find in the literature any description of how to compute the analysis of variance table for the situation where one of the levels of the quantitative factor is zero. The computations for the analysis of variance table which is appropriate for experiments involving quantitative and qualitative factors including zero amounts are presented in the remainder of this paper. Suppose that a factorial experiment involves one quantitative factor, A, having a levels and one qualitative factor, B, having b levels. When none of the levels of the quantitative factor A is a zero amount, the usual analysis of variance table for a two-way classification with one observation per experimental unit applies. If yij denotes the yield of the experimental unit that receives level i of factor A and level j of factor B, the sum of squares due to factor B,