A Simplified Method for Determining Expected Mean Squares and Error Terms in the Analysis of Variance.
This paper illustrates how expected mean squares needed in the analysis of variance can be arrived at via the use of only one rule: the E(MS) for any source of variation for any ANOVA model is the specified effect (main effect or interaction) plus the specified effect in combination with any random effect (including combinations of random effects). Once the procedure has been mastered, it can be quickly applied to ANOVA designs of any complexity. IN DESIGNING AND ANALYZING research studies that use the analysis of variance (ANOVA), it is often necessary to determine the expected mean square, E(MS), of the various sources of variation. These E(MS) com ponents define the proper error term for the F-test for each main effect and interaction. Except with the fixed effects ANOVA model, the within-cell variance is not the correct error term for all effects. For example, the six-factor ANOVA used by Long and Henderson (2) re quired eight different error terms to test the various ef fects for significance. Knowledge of E(MS) is essential in determining variance components and in generalizability studies. Some computer programs such as BMD 08V print out E(MS) values as well as compute the F-values for each effect for which the design yields a proper error term. Other programs, such as the FINN multivariate This content downloaded from 207.46.13.106 on Sun, 26 Feb 2017 18:56:54 UTC All use subject to http://about.jstor.org/terms 14 JOURNAL OF EXPERIMENTAL EDUCATION package, require the user to specify the error terms to be used for the various hypotheses tested. Most canned com puter programs, however, operate as if all factors are fixed. The user must ignore the F-tests on the output and com pute the correct F-ratios using mean square values, i. e., the error terms that are congruent with the inferences he wishes to make. Even when computer programs such as BMD 08V are used, it is useful to know whether or not an effect is going to be testable at the design state of a project. The proper error term is defined by the E(MS). com ponents for any effect. The correct error term includes all components except the one to be tested. The E(MS) values for the various effects in an ANOVA design will indicate whether or not the design will yield the proper error term without resorting to statistical compromises of model simplification (pooling) or quasi-F tests?un fortunately frequently necessary with mixed ANOVA models. Knowledge of the correct error term may affect the design. For example, the researcher may elect to in clude more levels of certain random factors in his design to give the error terms for the interesting effects of degrees of freedom, reduce the critical F-value, and pro vide the associated F-tests with greater power. Perhaps the design can be modified so that the interesting effects are testable without the patchwork of constructed error terms and the related statistical uncertainties of their use. Knowledge of the E(MS) for an effect illuminates the design alternatives and identifies which sources or varia tions are to be involved in possible pooling/model sim plification decisions.