main effect & interaction

http://www.psychstat.missouristate.edu/multibook/mlt09.htmThe ANOVA Summary Table

The results of the analysis are presented in the ANOVA summary table, presented below for the example data.

The items of primary interest in this table are the effects listed under the “Source” column and the values under the “Sig.” column. As in the previous hypothesis test, if the value of “Sig” is less than the value of a as set by the experimenter, then that effect is significant. If a =.05, then the Ability main effect and the Ability BY Method interaction would be significant in this table.

Main effects are differences in means over levels of one factor collapsed over levels of the other factor. This is actually much easier than it sounds. For example, the main effect of Method is simply the difference between the means of final exam score for the two levels of Method, ignoring or collapsing over experience. As seen in the second method of presenting a table of means, the main effect of Method is whether the two marginal means associated with the Method factor are different. In the example case these means were 30.33 and 30.56 and the differences between these means was not statistically significant.

As can be seen from the summary table, the main effect of Ability is significant. This effect refers to the differences between the three marginal means associated with Ability. In this case the values for these means were 27.33, 33.83, and 30.17 and the differences between them may be attributed to a real effect.

Simple Main Effects

A simple main effect is a main effect of one factor at a given level of a second factor. In the example data it would be possible to talk about the simple main effect of Ability atMethod equal blue-book. That effect would be the difference between the three cell means at level a1 (26.67, 31.00, and 33.33). One could also talk about the simple main effect of Method at Ability equal lots (33.33 and 27.00). Simple main effects are not directly tested in this analysis. They are, however, necessary to understand an interaction.

Interaction Effects

An interaction effect is a change in the simple main effect of one variable over levels of the second. An A X B or A BY B interaction is a change in the simple main effect of B over levels of A or the change in the simple main effect of A over levels of B. In either case the cell means cannot be modeled simply by knowing the size of the main effects. An additional set of parameters must be used to explain the differences between the cell means. These parameters are collectively called an interaction.

The change in the simple main effect of one variable over levels of the other is most easily seen in the graph of the interaction. If the lines describing the simple main effects are not parallel, then a possibility of an interaction exists. As can be seen from the graph of the example data, the possibility of a significant interaction exists because the lines are not parallel. The presence of an interaction was confirmed by the significant interaction in the summary table. The following graph overlays the main effect of Ability on the graph of the interaction.

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