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Beyond One-Way ANOVA The approach described in the preceding section is called one-way ANOVA. This scenario is easily generalized to accommodate more than one independent variable. These independent variables are either discrete (called factors) or continuous (called covariates). These approaches are called n-way ANOVA or ANCOVA (the “C” indicates the presence of covariates). Two way ANOVA, for example, allows for testing of the effect of two different independent variables on the dependent variable, as well as the interaction of these two independent variables. An interaction effect between two variables describes the way that variables “work together” to have an effect on the dependent variable. This is perhaps best illustrated by an example. Suppose that an analyst wants to know whether the number of health care information workshops attended, as well as a person’s education, are associated with healthy lifestyle behaviors. Although we can surely theorize how attending health care information workshops and a person’s education can each affect an individual’s healthy lifestyle behaviors, it is also easy to see that the level of education can affect a person’s propensity for attending health care information workshops, as well. Hence, an interaction effect could also exist between these two independent variables (factors). The effects of each independent variable on the dependent variable are called main effects (as distinct from interaction effects). To continue the earlier example, suppose that in addition to population, an analyst also wants to consider a measure of the watershed’s preexisting condition, such as the number of plant and animal species at risk in the watershed. Two-way ANOVA produces the results shown in Table 13.4, using the transformed variable mentioned earlier. The first row, labeled “model,” refers to the combined effects of all main and interaction effects in the model on the dependent variable. This is the global F-test. The “model” row shows that the two main effects and the single interaction effect, when considered together, are significantly associated with changes in the dependent variable (p < .000). However, the results also show a reduced significance level of “population” (now, p = .064), which seems related to the interaction effect (p = .076). Although neither effect is significant at conventional levels, the results do suggest that an interaction effect is present between population and watershed condition (of which the number of at-risk species is an indicator) on watershed wetland loss. Post-hoc tests are only provided separately for each of the independent variables (factors), and the results show the same homogeneous grouping for both of the independent variables. Table 13.4 Two-Way ANOVA Results As we noted earlier, ANOVA is a family of statistical techniques that allow for a broad range of rather complex experimental designs. Complete coverage of these techniques is well beyond the scope of this book, but in general, many of these techniques aim to discern the effect of variables in the presence of other (control) variables. ANOVA is but one approach for addressing control variables. A far more common approach in public policy, economics, political science, and public administration (as well as in many others fields) is multiple regression (see Chapter 15). Many analysts feel that ANOVA and regression are largely equivalent. Historically, the preference for ANOVA stems from its uses in medical and agricultural research, with applications in education and psychology. Finally, the ANOVA approach can be generalized to allow for testing on two or more dependent variables. This approach is called multiple analysis of variance, or MANOVA. Regression-based analysis can also be used for dealing with multiple dependent variables, as mentioned in Chapter 17.
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Evan M. Berman (Essential Statistics for Public Managers and Policy Analysts)