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Higher-Order Factorial Analysis of Variance

PSY 4603 Research Methods. Higher-Order Factorial Analysis of Variance. Three-Way ANOVA. All of the principles concerning a two-way factorial design apply equally well to a three-way or higher-order design.

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Higher-Order Factorial Analysis of Variance

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  1. PSY 4603 Research Methods Higher-Order Factorial Analysis of Variance

  2. Three-Way ANOVA • All of the principles concerning a two-way factorial design apply equally well to a three-way or higher-order design. • With one additional piece of information, you should have no difficulty running an analysis of variance on any factorial design imaginable. • We’ll peek at a simple three-way factorial as an example, since it is the easiest to use. • The only major way in which the three-way differs from the two-way is in the presence of more than one interaction term. • To see this, we must first look at the underlying structural model for a factorial design with three variables:

  3. In this model we have not only main effects, symbolized by i, βJ, and Yk but also two kinds of interaction terms. • The two-variable or first-order interactions are βiJ, Yik, and βYjk which refer to the interaction of variables A and B, A and C, and B and C, respectively. • We also have a second-order interaction term,  βYijk, which refers to the joint effect of all three variables. • The lower-order interactions we have already examined in discussing the two-way.

  4. The higher-order interaction can be viewed in several ways. • Probably the easiest way to view the ABC interaction is to think of the AB interaction itself interacting variable C. • Suppose that we had two levels of each variable and plotted the AB interaction separately for each level of C. • We might have the result shown in this figure. • Notice that for C1 we have one AB interaction, whereas for C2 we have a different one, Thus, AB depends on C, producing an ABC interaction. This same kind of reasoning could be invoked using the AC interaction at different levels of B, or the BC interaction at different levels of A. The result would be the same.

  5. An Example … Variables Affecting Driving Performance • Consider an experiment concerning the driving ability of two different types of drivers - inexperienced (A1) and experienced (A2). • These people will drive on one of three types of roads - first class (B1), second class (B2), or dirt (B3), • under one of two different driving conditions - day (C1) and night (C2). • Thus we have a 2 X 3 X 2 factorial. • The experiment will include four subjects per condition (for a total of 48 subjects), and the dependent variable will be the number of steering corrections in a one-mile section of roadway. The raw data are presented next.

  6. Group (Cell) Means and Ns

  7. The ANOVA Summary Table

  8. Estimated Marginal Means

  9. Estimated Marginal Means (continued)

  10. Multiple Comparisons (Tukey) Tests for the Main Effect of Type of Road

  11. Main Effects

  12. Simple Effects • Since we have a significant interaction, the main effects of A (Experience) and C (Type of Road) should be interpreted with caution. • To this end, the AC interaction has been plotted in the Figure.

  13. When plotted, the data show that for the inexperienced driver night conditions produce considerably more steering corrections than do day conditions, whereas for the experienced driver the difference in the number of corrections made under the two conditions is relatively slight. • Although the data do give us confidence in reporting a significant effect for A (the difference between experienced and inexperienced drivers), they should leave us a bit suspicious about differences due to variable C. • At a quick glance, it would appear that there is a significant C effect for the inexperienced drivers, but possibly not for the experienced drivers.

  14. To examine this question more closely, we must consider the simple effects of C under A1 (Inexperienced) and A2 (Experienced) separately. This analysis is presented in the two tables, from which we can see that there is a significant effect between day and night condition for the inexperienced drivers, but not for the experienced drivers.

  15. So… • From this experiment, we could conclude that there are significant differences as a function of type of roadway, • Between experienced and inexperienced drivers, and • A significant difference between day and night conditions, although only for inexperienced drivers ( p =.001).

  16. Simple Interaction Effects • With higher order factorials, not only can we look at individual levels of some other variable (simple effects), but we can also look at the interaction of two variables at individual levels of some third variable. • These are called a simple interaction effect. • Our 3-Way interaction was not significant but we could still do so for theoretical reasons.

  17. From the analysis of simple interaction effects, it is apparent that the Experience X Road interaction is not significant for the Day data but it is for the night data . When night conditions and dirt roads occur together, differences between Experienced and Inexperienced drivers are magnified.

  18. Two-Way (Lower-Order) Interactions

  19. Time of Day * Level of Driving Experience * Type of Road

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