G*Power: 3-Way Factorial Independent Samples ANOVA
The analysis is done pretty much the same as it is with a two-way ANOVA.
Suppose we are planning research for which an A x B x C, 2 x 2 x 3 ANOVA would be
appropriate. We want to have enough data to have 80% power for a medium sized
effect. The omnibus analysis will include seven F tests – three with one df each (A, B,
and A x B) and four with two df each (C, A x C, B x C, and A x B x C). We plan on
having sample size constant across cells.
For the tests of A, B, and A x B:
F tests - ANOVA: Fixed effects, special, main effects and interactions
Analysis:
A priori: Compute required sample size
Input:
Effect size f
= 0.25
α err prob
= 0.05
Power (1-β err prob)
= .80
Numerator df
= 1
Number of groups
= 12
Noncentrality parameter λ
= 8.0000000
Critical F
= 3.9228794
Denominator df
= 116
Total sample size
= 128
Output:
Actual power
=
0.8009381
Remember that Cohen suggested .25 as the value of f for a medium-sized effect.
1. The number of groups here is the number of cells in the factorial design, 2 x 2 x 3 =
12. When you click “Calculate” you see that you need a total N of 128. That works out
to 10.67 cases per cell, so bump the N up to 11(12) = 132.
For the effects with 2 df:
F tests - ANOVA: Fixed effects, special, main effects and interactions
Analysis:
A priori: Compute required sample size
Input:
Effect size f
= 0.25
α err prob
= 0.05
Power (1-β err prob)
= .80
Numerator df
= 2
Number of groups
= 12
Noncentrality parameter λ
= 9.8750000
Critical F
= 3.0580504
Denominator df
= 146
Total sample size
= 158
Actual power
= 0.8016972
Output:
That works out to 13.2 cases per cell, so bump the N up to 14(12) = 168.
GPower3-3WayFactorial.doc
Suppose that you anticipate obtaining a significant triple interaction and following
that with analysis of the A x B simple interactions at each level of C. Playing it
conservative by using individual error terms, you will then need at each level of C
F tests - ANOVA: Fixed effects, special, main effects and interactions
Analysis:
A priori: Compute required sample size
Input:
Effect size f
= 0.25
α err prob
= 0.05
Power (1-β err prob)
= .80
Numerator df
= 1
Number of groups
= 4
Noncentrality parameter λ
= 8.0000000
Critical F
= 3.9175498
Denominator df
= 124
Total sample size
= 128
Actual power
= 0.8013621
Output:
That is 128/4 = 32 cases for each A x B cell. Since there are three levels of C,
the total sample size needed is now 3 x 128 = 384.
Suppose the A x B interaction were to be significant one or more of the levels of
C. You likely would then test the simple, simple, main effects of A at each level of B (or
vice versa). For each such comparison (which would involved only two cells):
F tests - ANOVA: Fixed effects, special, main effects and interactions
Analysis:
A priori: Compute required sample size
Input:
Effect size f
= 0.25
α err prob
= 0.05
Power (1-β err prob)
= .80
Numerator df
= 1
Number of groups
= 2
Noncentrality parameter λ
= 8.0000000
Critical F
= 3.9163246
Denominator df
= 126
Total sample size
= 128
Actual power
= 0.8014596
Output:
You need 128 scores, 64 per cell. Since we have a total of 12 cells, that works
out to 768 cases. You might end up deciding that you can get by with having less
power for detecting simple effects than for detecting effects in the omnibus analysis.
Suppose you ended up with 20 scores per cell, total N = 20(12) = 240. How
much power would you have for detecting medium-sized effects in the omnibus
analysis?
For the one df effects:
F tests - ANOVA: Fixed effects, special, main effects and interactions
Analysis:
Post hoc: Compute achieved power
Input:
Effect size f
= 0.25
α err prob
= 0.05
Total sample size
= 240
Output:
Numerator df
= 1
Number of groups
= 12
Noncentrality parameter λ
= 15.0000000
Critical F
= 3.8825676
Denominator df
= 228
Power (1-β err prob)
= 0.9710633
For the two df effects:
F tests - ANOVA: Fixed effects, special, main effects and interactions
Analysis:
Post hoc: Compute achieved power
Input:
Effect size f
= 0.25
α err prob
= 0.05
Total sample size
= 240
Numerator df
= 2
Number of groups
= 12
Noncentrality parameter λ
= 15.0000000
Critical F
= 3.0354408
Denominator df
= 228
Power (1-β err prob)
= 0.9411531
Output:
How much power would you have if you got down to the level of comparing one
cell with one other cell:
F tests - ANOVA: Fixed effects, special, main effects and interactions
Analysis:
Post hoc: Compute achieved power
Input:
Effect size f
= 0.25
α err prob
= 0.05
Total sample size
= 40
Numerator df
= 1
Number of groups
= 2
Noncentrality parameter λ
= 2.5000000
Critical F
= 4.0981717
Denominator df
= 38
Power (1-β err prob)
= 0.3379390
Output:
Links
 Karl Wuensch’s Statistics Lessons
 Internet Resources for Power Analysis
Karl L. Wuensch
Dept. of Psychology
East Carolina University
Greenville, NC USA