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Weighted and Unweighted MEANS ANOVA. Data Set “ Int ”. Notice that there is an interaction here. Effect of gender at School 1 is 155-110 = 45. Effect of gender at School 2 is 135-120 = 15. Weighted means School 1: [10(155) + 20(110)]/30 = 125. School 2: [20(135) + 40(120)]/60 = 125.

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## Weighted and Unweighted MEANS ANOVA

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**Data Set “Int”**Notice that there is an interaction here. Effect of gender at School 1 is 155-110 = 45. Effect of gender at School 2 is 135-120 = 15.**Weighted means**School 1: [10(155) + 20(110)]/30 = 125. School 2: [20(135) + 40(120)]/60 = 125. Unweighted means School 1: (155 + 110)/2 = 132.5 School 2: (135 + 120)/2 = 127.5**Simple Effects of School, Weighted means**125 = 125, no simple effect. Simple Effects of School, Weighted means 132.5 127.5, is a simple effect.**Weighted Means ANOVA**• See calculations on handout.**Unweighted Means ANOVA**• Compute harmonic mean sample size. • Prepare table of adjusted cell sums. See the handout.**The cell sizes here are proportional, a 2 on them would**yield a value of 0. Some say OK to do weighted ANOVA in that case, but, as you can see, the results differ depending on whether you do unweighted or weighted ANOVA.**Data Set “ ”**Notice that there is no interaction here. Effect of gender at School 1 is 155-140 = 15. Effect of gender at School 2 is 135-120 = 15. With no interaction, it does not matter how you weight the means.**Non-Proportional Sample Sizes**There is a greater proportion of boys at School 1 than at School 2. Gender and School are no longer independent of each other. The weighted means show School 1 > School 2. But for the boys, School 2 > School 1. And for the girls, School 2 > School 1. The unweighted means show School 2 > School 1.**Reversal Paradox**• This is known as a reversal paradox. • The direction of the effect in the aggregate data is in one direction. • But at each level of a third variable the direction is opposite what it was in the aggregate data.**Sex Bias in Graduate Admissions**Which sex is the victim of discrimination?**Orthogonal versus Nonorthogonal Factorial ANOVA**• When the sample sizes are equal, or proportional, the two ANOVA factors are independent of each other (aka “orthogonal.”) • If they are not independent of each other (aka “nonorthogonal”) then the sums of squares cannot be as simply partitioned. • With nonorthogonal data, the model sums of squares includes variance that is shared by the two main effects.**VarY**Error A B ? VarB VarA**Variance “?”**• What should we do with this variance ? • Usually we exclude it from error but assign it to neither the main effect of A nor the main effect of B. • In a sequential analysis we assign it to one and only one of the ANOVA effects.**Sequential Analysis**• Suppose that A was measured at Time 1, B at Time 2, and Y at Time 3. • Since most of us consider causes to precede effects, we are more comfortable thinking that A might be a cause of B than we are thinking that B might cause A. • In this case, we might decide to allocate the “?” variance to A rather than to B.

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