Unweighted MEANS ANOVA Data Set Int Notice that there is an interaction here Effect of gender at School 1 is 155110 45 Effect of gender at School 2 is 135120 15 Weighted means ID: 435625
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Slide1
Weighted and Unweighted MEANS ANOVASlide2
Data Set “Int”
Notice that there is an interaction
here and that the cell sizes are proportional.
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 = 127.5Slide4
Main Effect
of School, Weighted means
125 = 125, no simple effect.
Main Effect
of School,
Unweighted
means
132.5
127.5, is a simple effect.Slide5
Weighted Means ANOVASee calculations on handout.Slide6
Unweighted Means ANOVACompute 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.Slide8
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
.Slide9
Data Set “ ”
Effect
of
school
w weighted means = 20.
Effect of
school
w unweighted means = 20.
With no interaction, it does not matter how you weight the means.Slide10
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.Slide11
Reversal ParadoxThis 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.Slide12
Sex Bias in Graduate Admissions
Which sex is the victim of discrimination?Slide13
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.Slide14
Error
A
B
?
Var
Y
Var
A
Var
BSlide15
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.Slide16
Sequential AnalysisSuppose 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.