Preparing to Present Your Summary of a Three-Way Factorial ANOVA The factorial ANOVA for your example may have three or more factors (classification variables). Professor Karl wants you to focus in on only three of the factors. Here I shall simply refer to the three factors as “A,” “B,” and “C.” Were any of the three main effects significant? If so, present the marginal means and describe differences between or among the groups. Were any of the three two-way interactions significant? If so, describe the simple main effects. Did the researchers test the simple main effects? Was the triple interaction significant? If so, describe the simple interactions. Did the researchers test the simple interactions? If the statistical presentation in your article is so poor that you are unable to find the means to answer the questions above, find another article. When I call on you, you should be prepared to, with little delay, answer the questions above and explain what A, B, and C are, and what the scores are. You should be prepared to give supporting statistics – F, degrees of freedom, and p. If reported, you should share estimates of magnitude of effect for each of the seven effects. Be prepared to criticize the statistical presentation, if it fell short of adequate. Were estimates of effect size provided? Were they accompanied by confidence intervals? Example of Factorial ANOVA Swami, V., Furnham, A., & Joshi, K. (2008). The influence of skin tone, hair length, and hair colour on ratings of women’s physical attractiveness, health and fertility. Scandinavian Journal of Psychology, 49, 429–437.
Respondents were asked to rate the attractiveness of sketches of women. The sketches differed in skin tone (dark, tan, or light) and hair color (brunette or blonde). The respondents varied in sex (female or male). The authors reported a Skin Tone × Hair Colour × Sex interaction F(1.71, 410.42) = 4.25, p < .05, .
The triple interaction is illustrated in the plots below. The simple interactions do not look very much different from each other, but that is not surprising given the low value of the partial eta-squared.
If we ignore the sex of respondent, the Skin Tone x Hair Colour interaction is more impressive – the authors reported a partial eta-squared of .18 for that effect. As you can see from the plots, the simple main effect of hair color was small when skin tone was light or tan, but when skin tone was dark, the simple main effect of hair color was large. The solid line is brunette, the broken line blonde.
First, note that the omnibus analysis is a four-way mixed factorial ANOVA: 3(skin tone) x 2 (hair length) × 2 (hair colour) × 2(participant sex). One factor (sex) is between subjects, the other three are within-subjects. When a within-subjects factor has three or more levels, the traditional analysis assumes sphericity (more on this later in the semester). When this assumption is violated, one can correct for the violation with the Greenhouse-Geisser correction, which results in non-integer values of degrees of freedom. A superior approach is to use the multivariate approach analysis, which I shall teach to you later.
Second, note that the authors used partial eta-squared as the measure of strength of effect. I think this inappropriate when all of the effects represent constructs that are variable in the real world. If you inspect Table 1, summing the reported values of eta-squared, you will find that the authors claim to have explained over 129% of the variance in physical attractiveness.