Course Analysis of Variance Topic AxB Factorial ANOVA 1 A x B FACTORIAL ANOVA Analysis of variance is a powerful tool in part because it can be used to analyze a factorial design that involves the effects of multiple variables Consider the following example Dr Infomercial has been tinkering with a drug Love Potion 9 21 that causes the user to fall in love with the first person s he encounters The doctor tested the drug by measuring participants feelings of love for a target person after ingesting either the drug or a placebo As expected persons who ingested the drug reported stronger feelings of love M 4 5 than did persons who ingested the placebo M 2 Assume that the difference is significant The doctor is about to kick off a media blitz when her research assistant mentions that the data look different when aggregated by sex i e half of the participants were male The data are as follows Placebo Male Drug Male Placebo Female Drug Female 1 2 3 X 2 6 7 8 X 7 1 2 3 X 2 1 2 3 X 2 The above data can be thought of as a one factor design that has four levels Alternatively the four levels could be thought of as having resulted from the crossing of two factors pill drug or placebo and sex male or female We can visually represent this latter conceptualization by altering our presentation of the data as follows Pill Sex Male Female Placebo 1 2 3 X 2 1 2 3 X 2 Drug 6 7 8 X 7 1 2 3 X 2 To the doctor s dismay it appears as if the pill differentially affects males and females In particular it appears as if the love drug is effective only for males EFFECTS IN A FACTORIAL DESIGN The latter conceptualization of the data is referred to as a 2 x 2 factorial design A factorial design occurs when all variables are completely crossed that is when the levels of the variables are combined in all possible combinations The number of numbers in the 2 x 2 notation indicates the number of variables in the design e g there are 2 twos corresponding to sex and pill And the value of each number indicates the number of levels in the corresponding variable e g pill has 2 levels and sex has 2 levels A 2 x 3 design for example has two variables the first of which has two levels and the second of which has 3 levels A 2 x 2 x 2 design involves 3 variables each of which has 2 levels Notice that multiplying across the notation indicates the number of samples involved e g a 2 x 2 has 4 samples For the moment we will deal only with designs that involve 2 factors Factorial designs produce two general types of effects Main effects and interactions Each variable in the design has associated with it a main effect and in the two factor design there is one interaction produced by the combination of the variables as we will discuss later designs with more than two factors have more than one interaction Course Analysis of Variance Topic AxB Factorial ANOVA 2 Main Effects A main effect is the average effect of a variable That is a main effect is the effect of a variable averaged across the other variables Likewise the means associated with a main effect are referred to as marginal means and are computed by averaging across the levels of the other variables The following table for example reveals the marginal means of pill Pill Sex Placebo Drug Male 1 2 3 X 2 6 7 8 X 7 Female 1 2 3 X 2 1 2 3 X 2 Marginal means 2 4 5 The marginal means of pill are 2 for the placebo condition and 4 5 for the drug condition Notice that these are the means Dr Infomercial initially used to demonstrate the effectiveness of the love drug when he averaged across the scores of males and females Consequently Dr Infomercial was reporting the main effect of pill At the risk of being redundant the following table displays the marginal means of sex Pill Sex Placebo Drug Marginal means Male 1 2 3 X 2 6 7 8 X 7 4 5 Female 1 2 3 X 2 1 2 3 X 2 2 Notice that the marginal mean for males 4 5 is obtained by averaging the score for males who take the placebo with the score for males who take the drug Likewise the marginal mean for females 2 is obtained by averaging across the placebo and drug conditions The main effect of sex involves the difference between males and females averaging across the levels of pill We will return to the issue of averaging when we discuss factorial designs in which sample size varies across conditions For the time being notice that with equal samples sizes averaging the mean of each sample produces the same marginal mean as does averaging the scores of each sample Interactions There are multiple ways of describing an interaction The easiest way to conceptualize an interaction is when the effect of one variable changes across levels of the other variable Or as we will see when we discuss designs with more than 2 factors an interaction occurs when one effect changes across levels of another variable An interaction can involve a change in the magnitude or direction of an effect For example we saw previously that the effect of pill appeared to change as a function of sex For males the drug produced a larger love score M 7 than did the placebo M 2 For females however there was no difference between the drug M 2 and the placebo M 2 That is the magnitude of the pill effect is 5 for males i e 7 2 and 0 for females i e 2 2 Consequently there appears to be an interaction in which the magnitude of the effect of pill changes across levels of sex Course Analysis of Variance Topic AxB Factorial ANOVA 3 The following tables display hypothetical patterns of interactions A Change in Direction B Change in Magnitude Pill Pill Sex Placebo Drug Sex Placebo Drug Male 2 7 Male 2 7 Female 7 2 Female 3 6 C No Interaction Pill Sex Placebo Drug Male 2 7 Female 2 7 Table A depicts an interaction that involves a change in direction of the effect of pill For males the drug produces higher love scores M 7 than does the placebo M 2 For females the drug produces lower love scores M 2 than does the placebo M 7 That is the effect of pill changes direction across levels of sex in that it is 5 for males and 5 for females Table B depicts an interaction that involves a change in the magnitude of the effect of pill Notice that the drug increases love scores relative to the placebo for both males and females However the magnitude of the increase is stronger form males 7 2 5 than for …
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