SAS OUTPUT FOR A FACTORIAL ANOVA WITH EQUAL SAMPLE SIZESSAS OUTPUT COMPARING 2X2 FACTORIAL WITH CONTRASTS OF A 1-FACTOR DESIGNFactorial ANOVA for balanced designContrasts for main eff and inter in one-factor designSAS OUTPUT FOR FACTORIAL ANOVA AND 1-FACTOR ANOVA THAT IGNORES SEXFactorial ANOVA for Balanced DesignOne Factor ANOVA for TherapySPSS OUTPUT FOR FACTORIAL ANOVA AND 1-FACTOR ANOVA THAT IGNORES SEXFactorial ANOVA for Balanced DesignSPSS OUTPUT FOR FACTORIAL ANOVA AND 1-FACTOR ANOVA THAT IGNORES SEX (cont’d)One Factor ANOVA for TherapySAS Code for Welch-Satterthwaite ApproachSAS OUTPUTSource DF Type I SS Mean Square F Value Pr > FSource DF Type II SS Mean Square F Value Pr > FSource DF Type III SS Mean Square F Value Pr > FType I MeansType III MeansType II MeansEFFECTS IN A FACTORIAL DESIGNMain EffectsInteractionsANALYZING A FACTORIAL ANOVAThe Variance Partitioning Approach to Factorial ANOVAThe Model Comparison Approach to Factorial DesignsWe can test the factorial ANOVA using the GLM procedure. In the classes statement we list the categorical (or nominal) independent variables (i.e., sex and therapy). In the model statement we indicate that the dv (depress) should be predicted by sex therapy and the Sex x Therapy interaction (sex*therapy). (Alternatively, we could have used a short cut for expressing all main effects and interactions by typing a vertical bar between each independent variable (e.g., sex|therapy); The model statement also includes the Type III sums of squares option (ss3) – we’ll discuss this when we examine situations involving unequal sample sizes. Finally, we use the “Lsmeans” statement (least square means) and specify the main effect and interaction means. We’ll discuss the difference between the “means” and Lsmeans” statements when we discuss unequal sample sizes. When sample sizes are equal both statements produce the same values.The top section of the output contains an ANOVA table corresponding to the omnibus ANOVA for a one-factor design with 6 levels (the 6 levels being the six samples from the crossing of sex and therapy). The “Model” line corresponds to the between group variation and the “Error” line corresponds to within-group variation. Notice that with 6 groups there are 5 df in the numerator and SSbetween = 91.83. Also notice that Mean Square Error (0.55) is MSW.INTERESTING TANGENTS OF FACTORIAL ANOVAThe point of this laborious exercise was to demonstrate that main effects and interactions are simply a set of orthogonal contrasts. And, those orthogonal contrasts account for the total variation between groups.Factorial ANOVA Versus One-Factor ANOVAFOLLOW-UP TESTS FOR MAIN EFFECTSFollow-up Tests Assuming Homogeneity of VarianceFollow-up Tests with Heterogeneity of VarianceFOLLOW-UP TESTS FOR THE INTERACTIONSimple EffectsCourse: Analysis of Variance Topic: AxB Factorial ANOVA 1 A x B FACTORIAL ANOVAAnalysis 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 thedata 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 / Female1, 2, 3 (2X) 6, 7, 8 (7X) 1, 2, 3 (2X) 1, 2, 3 (2X)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:PillSex Placebo DrugMale 1, 2, 3 (2X) 6, 7, 8 (7X)Female 1, 2, 3 (2X) 1, 2, 3 (2X)To the doctor’s dismay, it appears as if the pill differentially affects males and females. Inparticular, it appears as if the love drug is effective only for males!EFFECTS IN A FACTORIAL DESIGNThe 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 EffectsA 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.PillSex Placebo DrugMale 1, 2, 3 (2X) 6, 7, 8 (7X)Female 1, 2, 3 (2X) 1, 2, 3 (2X)Marginal means 2 4.5The 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
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