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Power in bivariate and multiple linear regression analyses in G*Power
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In bivariate regression, there is no difference between the power needed to reject H0: R = 0 and H0: b1 = 0. Despite this, there are
two ways to do a power analysis here. Because researchers are usually focused on analysis of individual predictors, I’ve presented
below how to do a power analysis for the latter null hypothesis in G*Power.
1.
2.
Test family: t tests > Statistical test: Linear bivariate regression: One group, size of slope
Choose either an a priori (planning) or post-hoc (observed) power analysis
Enter the following input parameters:
3.
4.
5.
6.
7.
8.
9.
10.
Tail(s): usually two
Slope H1: the predicted (a priori) or observed (post hoc) slope; may be standardized or not, pay attention to SDs in 9 & 10
α err prob: usually .05
Power (1 – β): usually .8, .9, .95 (for a priori analyses)
Total sample size: how many subjects (for post-hoc analyses)
Slope H0: usually 0
Std dev σ_x: depends
Std dev σ_y: depends
6 is needed for a priori analyses and 7 for post-hoc analyses, but not vice versa. I strongly urge you to do your analysis on
standardized slope rather than unstandardized slope if you can. If you can do this, then the values to enter for 10 and 11 above are 1
(because z-scores always have SD = 1). Otherwise you’ll have to enter actual (or estimated for a priori analyses) SDs for x and y.
In the case of multiple regression, there are at least three different null hypotheses that one might be interested in testing, and
there is a separate procedure for doing a power analysis for each. The three null hypotheses are as follows:
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A. H0: R = 0 (that is, does the whole model – with all predictors – explain a significantly-greater-than-0 proportion of variance)
B. H0: bi = 0 (that is, does one predictor make a significant unique contribution to explained variance)
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C. H0: ΔR = 0 (that is, does a set of predictors explain a significant proportion of variance above and beyond some other set’s
prediction)
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A. H0: R = 0
B. H0: bi = 0
1.
1.
2.
Test family: F tests > Statistical
test: Linear multiple regression:
2
Fixed model, R deviation from
zero
Choose either an a priori
(planning) or post-hoc (observed)
power analysis
Enter the following input parameters:
3.
4.
5.
6.
7.
2
2
2
Effect size: f = R / (1 – R ) [note
2
this is R for the whole model]
α err prob: usually .05
Power (1 – β): usually .8, .9, .95
(for a priori analyses)
Total sample size: how many
subjects (for post-hoc analyses)
number of predictors (in the full
model)
5 is for a priori analyses, 6 for post-hoc,
but not vice versa
2.
Test family: t tests > Statistical
test: Linear multiple regression:
Fixed model, single regression
coefficient
Choose either an a priori
(planning) or post-hoc (observed)
power analysis
Enter the following input parameters:
3.
4.
5.
6.
7.
8.
Tail(s): usually two
2
2
2
Effect size: f = sr / (1 – R ) [note
2
2
this is R for the whole model; sr
is the variance explained by only
the predictor of interest]
α err prob: usually .05
Power (1 – β): usually .8, .9, .95
(for a priori analyses)
Total sample size: how many
subjects (for post-hoc analyses)
number of predictors (in the full
model)
6 is for a priori analyses, 7 for post-hoc,
but not vice versa
2
C. H0: ΔR = 0
1.
2.
Test family: F tests > Statistical
test: Linear multiple regression:
2
Fixed model, R increase
Choose either an a priori
(planning) or post-hoc (observed)
power analysis
Enter the following input parameters:
3.
4.
5.
6.
7.
8.
2
2
2
Effect size: f = R / (1 – R ) [note
2
this is R for the model including
all predictors up to the step of
interest]
α err prob: usually .05
Power (1 – β): usually .8, .9, .95
(for a priori analyses)
Total sample size: how many
subjects (for post-hoc analyses)
Number of tested predictors
(entered on the step of interest)
Total number of predictors (all
the predictors in the model on
the step of interest)
5 is for a priori analyses, 6 for post-hoc,
but not vice versa