Simple Interactions with the Welch-Satterthwaite Adjustment for Heterogeneous VariancesTesting the Omnibus EffectsExplaining the 3-way Interaction in SASExplaining Lower Order Effects in SASCourse: Analysis of Variance Topic: AxBxC Factorial ANOVA 1 A x B x C FACTORIAL ANOVAAs we discussed during our introduction to factorial ANOVA, factorial designs enable researchers to examine the main and interactive effects of variables. Factorial designs, however, are not limited to only two variables. As may variables as desired can be added to and crossed in a factorial design. As we’ll discuss momentarily, however, the number of main effects and interactions dramatically increase as factors are added to the design. Consequently, it is advisableto include only those factors that directly address research hypotheses. The strategy for analyzinga higher order factorial design (i.e., a design with more than 2-crossed factors) is a generalizationof the strategy for analyzing a two-way factorial design.EFFECTS IN A HIGHER ORDER FACTORIAL DESIGNTo keep things simple, we’ll begin our exploration of higher order designs with a 2 x 2 x 2 factorial. This design involves three fully crossed variables each of which has 2 levels. Consequently, we have 8 samples. To facilitate the discussion we’ll return to Dr. Infomercial’s love-potion experiment. If you recall, Dr. Infomercial developed a drug that increases feelings of love toward a target person. Initial tests of the drug indicated that the drug increased feelings of love relative to a placebo for males but had no effect for females. Follow-up interviews revealed that all of the female participants were engaged in romantic relationships while all of the male participants were uninvolved (i.e., single). Weary of this confound between sex and relationship-status, Dr. Infomercial conducted a second study in which males and females who were either involved or uninvolved in a relationship ingested either the love-drug or a placebo. As in the previous study, participants subsequently rated their feelings-of-love for an opposite sex target person (actually multiple targets were used to avoid confounding sex with idiosyncratic characteristics of the target and participants were randomly assigned to different target person – such a factor could be added to the design). The data are as follows:Relationship StatusUninvolved InvolvedPill PillSex Placebo Drug Sex Placebo DrugMale 1,2,3 (2X) 6, 7, 8 (7X) Male 1,2,3 (2X) 1,2,3 (2X)Female 1,2,3 (2X) 6, 7, 8 (7X) Female 1,2,3 (2X) 1,2,3 (2X)Recall that two-way factorial designs produce main effects for each factor and a 2-way interaction for the combination of both factors. A three-way factorial design also produces a maineffect for each factor. However, the three-way factorial also produces 2-way interactions for eachpair of factors and a 3-way interaction for the combination of all three factors. In the love-drug example, there is the potential for three main effects (sex, pill, and relationship status), three two-way interactions (Sex x Pill, Sex x Relationship, and Relationship x Pill), and one three-way interaction (Sex x Relationship x Pill). Let’s examine the meaning of main effects, 2-way interactions, and a 3-way interaction in a higher order factorial.Course: Analysis of Variance Topic: AxBxC Factorial ANOVA 2 Main Effects in a Higher Order FactorialThe meaning and interpretation of a main effect in a higher order factorial is the same as in a two-way factorial. A main effect is the effect of one variable averaging across all other variables. The main effect of sex, for example, compares that average love-score of males and females averaging across whether they are uninvolved or involved in a relationship and ingested the placebo or drug. In other words, the marginal mean for males is an average of the four samples of males (M = 3.25) and the marginal mean for females is an average of the four samples of females (M = 3.25). Likewise, the main effect for pill compares those persons who ingested the placebo with those persons who ingested the drug averaging across levels of sex andrelationship status. The marginal mean for placebo (M = 2.00) is an average of the four samples of persons who ingested the placebo and the marginal mean for drug is an average of the four samples of persons who ingested the drug (M = 4.5).Two-Way Interactions in a Higher Order FactorialA 2-way interaction in a higher order factorial occurs when the effect of one variable changes (in magnitude or direction) across levels of a second variable while averaging across all other variables. The following tables display the means for each of the potential 2-way interactions in the love drug example.Sex x Pill Relationship x Pill Sex x RelationshipPill Pill RelationshipSex Placebo Drug Relationship Placebo Drug Sex Uninvolved InvolvedMale 2.00 4.50 Uninvolved 2.00 7.00 Male 4.50 2.00Female 2.00 4.50 Involved 2.00 2.00 Female 4.50 2.00The pattern of means for the Sex x Pill interaction was obtained by averaging the samplesof persons who are uninvolved or involved in a relationship and shared membership in the same combination of sex and pill. If we eyeball the pattern of means, it appears as if there is not a Sex x Pill interaction. The effect of pill for males appears to be the same as the effect of pill for females. For both males and females, the drug increases love scores by 2.50 points relative to theplacebo (while averaging across relationship status. Likewise, the effect of sex (difference between males and females) is uniform across levels of pill (while averaging across relationship status).The pattern of means for the Relationship x Pill interaction was obtained by averaging thesamples of males and females who shared membership in the same combination of relationship and pill. If we eyeball the pattern of means, it appears as if there is a Relationship x Pill interaction. The effect of pill appears to change across levels of relationship. While the drug increases love scores by 5 points relative to the placebo for persons uninvolved in a relationship, the drug increases love scores by 0 points relative to the placebo for persons involved in a relationship (while averaging across sex). Consequently, it appears as if the pill is effective only for persons not involved in a relationship!The pattern of means for the Sex x Relationship interaction was obtained by averaging the
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