Model I ANOVA

ANALYSIS OF VARIANCE (ANOVA) is a method that deals with differences in means of a variable across categories of observations. A classic example would be the comparison of a behavior of interest between an EXPERIMENTAL group and a control group. The question is whether the mean behavior (measured on an INTERVAL scale) of the subjects in the experimental group differs significantly from the mean behavior of the members of the control group. In other words, does the experimental treatment have an effect? ANOVA is a method that employs ratios of variances—hence the name analysis of variance—to conduct a test of significance. So long as we have observations from all categories, as opposed to a subset of categories, ANOVA uses a FIXED-EFFECTS MODEL. In the example at hand, there are only two categories—namely, the experimental condition and the control condition. So, the fixed-effects model would apply. Note, however, that the fixed-effects model is not restricted to comparisons across just two categories.

ANOVA results typically appear in a table with the following layout (Table 1):

The “sum of squares” entries provide measures of variation between the groups and within each of the groups, with the assumption that the within-group variation is constant. To standardize those measures, we divide each sum of squares by the respective degrees of freedom. That gives us two estimates of VARIANCE (MEAN SQUARE). In the event that the between variance is no larger than the within variance, the ratio of those two quantities (F RATIO) should be 1.0. In that case, any difference between the two groups is entirely due to random chance and not to any group characteristic. In the case above, the F ratio only reaches 1.21, which is not significant at the .05 level. Hence, we have to conclude that the experimental treatment has no effect.

The fixed-effects model is not limited to testing for the effect of just one explanatory variable (ONE-WAY ANOVA). It can also accommodate two such variables (two-way ANOVA), in which one is designated as the row variable and the other as the column variable. It is advisable, however, to design the research in such a way that all the resulting combinations (cells) of rows and columns contain the same number of observations. That is possible, of course, only with experiments in which the investigators have that kind of control. With survey or other observational data, combinations of categories will have whatever observations occur. In that case, one cannot obtain unique estimates for the variance due to each of the explanatory factors. That [Page 657]limits the utility of two-way ANOVA to experimental designs.

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