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Weighted and Unweighted MEANS ANOVA. & a Reversal Paradox. 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.

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data set int
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.

slide3

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

slide4

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
Weighted Means ANOVA
  • See calculations on handout.
unweighted means anova
Unweighted Means ANOVA
  • Compute harmonic mean sample size.
  • Prepare table of adjusted cell sums. See the handout.
slide7

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
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.

slide9

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
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
Sex Bias in Graduate Admissions

Which sex is the victim of discrimination?