Interpreting Interactions in Factorial ANOVA

Interpreting Interactions in Factorial ANOVA

Problem: Marascuilo & Levin (1970) stated that analyzing simple effects of a significant interaction caused a type 4 error. A type 4 error is a wrong interpretation of a correctly rejected null hypothesis. The authors say this is akin to a physician prescribing incorrect medicine after a correct diagnosis. Marascuilo & Levin (1970) proposed two solutions to this problem: (a) analyzing the interaction contrasts or (b) analyzing the interaction effects. The phia package in R provides solutions for these methods, along with testing simple effects (see page 6). Martinez (2012) suggested that the method chosen must be decided by the questions being asked. Solution: Analysis of interaction contrasts (Solution A) was supported by Rosnow & Rosenthal (1989), but they called it residual or leftover contrasts. This solution works by removing lower-order effects by subtracting marginal and row/column means. This provides a sense of difference beyond what is expected from marginal means (Martinez, 2012; see page 7).

Solution B, analysis of the interaction effects can take one of a few methods. One method utilizes pairwise comparisons to investigate interaction effects. The phia package will conduct both approaches to interaction effects (Martinez, 2012; see page 9).

For more information: Marascuilo, L.A., & Levin, J.R. (1970). Appropriate post hoc comparisons for interaction and nested hypotheses in analysis of variance designs: The elimination of type IV errors. American Educational Research Journal, 7(3), 397-421.

Martinez, H.R. (2012). Analysing interactions of fitted models. Available at:

Rosnow, R.L., & Rosenthal, R. (1989). Definition and interpretation of interaction effectsPsychological Bulletin, 105(1), 143-146.



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