Which Sums of Squares Are Best in Unbalanced Analysis of Varia

Three fundamental concepts of science and statistics are entities, variables (which are formal representations of properties of entities), and relationships between variables. These concepts help to distinguish between two uses of the statistical tests in analysis of variance (ANOVA), namely • to test for relationships between the response variable and the predictor variables in an experiment • to test for relationships among the parameters of the model equation in an experiment. Two methods of computing ANOVA sums of squares are: • Higher-level Terms are Omitted from the generating model equations (HTO = SPSS ANOVA EXPERIMENTAL ≈ SAS Type II ≈ BMDP4V with Weights are Sizes) • Higher-level Terms are Included in the generating model equations (HTI = SPSS ANOVA UNIQUE = SPSS MANOVA UNIQUE = SAS Type III = BMDP4V with Weights are Equal = BMDP2V = MINITAB GLM = SYSTAT MGLH = Data Desk Type 3). This paper evaluates the HTO and HTI methods of computing ANOVA sums for squares for fulfilling the two uses of the ANOVA statistical tests. Evaluation is in terms of the hypotheses being tested and relative power. It is concluded that (contrary to current practice) the HTO method is generally preferable when a researcher wishes to test the results of an experiment for evidence of relationships between variables.