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.

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.

7.17

401

201

201

101

4~

N

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.

Error

AB?

VarY

VarAVarB

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.