Invited Response to Dallal, G. E. (1992), “The Computer Analysis of Factorial Experiments with Nested Factors,”The American Statistician, 46, 240

Abstract

Dallal (1992) questioned whether the analysis of the unbalanced data should remain unchanged with the specification of an additional factor C, nested within the combinations of A and B. BMDP’s response to his problem is that the inclusion of the third factor affects the formulation of the design variables and that the sums of squares may change. The analysis depends on the definition of the parameters. I use BMDP 4V (Davidson and Toporek), 1991; Dixon, 1990 to analyze the data; the calculated sums of squares are given in Table 1. I present the two models discussed by Dallal(l992) with two weighting options. The BMDP command files for these four models are provided in the Appendix. The first column of Table 1 presents the sums of squares computed from the two-factor model; the second is from the model with Factor C nested within A and B; the third is based on the two-factor model but weighted by the cell sizes; and the last is the weighted analysis of the second model, with cell weights defined such that the cells formed by the combination of A and B are given equal weights. The sums of squares of the first model agree with the results of Dallal(l992). In this model, the six cells defined by the combination of Factors A and B are given equal weights. The second model, on the other hand, creates 27 cells with the equal weights, which changes the parameter estimation procedure for the main effects and interaction effect. If we change the weighting scheme for Model I such that the cell weights are proportional to cell sizes, we then have the third model equivalent to the second model. The fourth model adjusts the cell weights of Model I11 such that the estimation procedure will be equivalent to that of Model I, and thus we get equivalent results. Specifying an additional third factor into the two-factor model changes the model formulation and parameter estimation. Weighting allows the analyst control over parameter definition and estimation particularly for unbalanced data.