Course Analysis of Variance Topic AxBxC Factorial ANOVA 1 A x B x C FACTORIAL ANOVA As 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 advisable to include only those factors that directly address research hypotheses The strategy for analyzing a higher order factorial design i e a design with more than 2 crossed factors is a generalization of the strategy for analyzing a two way factorial design EFFECTS IN A HIGHER ORDER FACTORIAL DESIGN To 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 relationshipstatus 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 Status Uninvolved Pill Sex Placebo Drug Male 1 2 3 X 2 6 7 8 X 7 Female 1 2 3 X 2 6 7 8 X 7 Involved Pill Sex Placebo Male 1 2 3 X 2 Female 1 2 3 X 2 Drug 1 2 3 X 2 1 2 3 X 2 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 main effect for each factor However the three way factorial also produces 2 way interactions for each pair 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 twoway 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 Factorial The 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 and relationship 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 Factorial A 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 Pill Sex Placebo Drug Male 2 00 4 50 Female 2 00 4 50 Relationship x Pill Pill Relationship Placebo Drug Uninvolved 2 00 7 00 Involved 2 00 2 00 Sex x Relationship Relationship Sex Uninvolved Involved Male 4 50 2 00 Female 4 50 2 00 The pattern of means for the Sex x Pill interaction was obtained by averaging the samples of 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 the placebo 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 the samples 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 samples of persons who ingested the placebo or drug and shared membership in the same combination of sex and relationship If we eyeball the pattern of means it appears as if there is not a Sex x Pill interaction The difference between uninvolved and involved persons i e 2 50 Course Analysis of Variance Topic AxBxC Factorial ANOVA 3 points is the
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