F-Test for Multiple Correlation and Regression (MCR)

 

For power analyses of MCR, we use the effect size index f2 (see Cohen, 1977) which, in general, reflects the proportion of variance accounted for by some source in the population (PVs) relative to the residual variance proportion (PVe): 

 
f2 = PVs/PVe
 

The relation of f2 to the noncentrality parameter lambda is given by

 
lambda = f2 * N
 

We will consider two cases, one in which we have

Note: For F-Test (MRC) (as well as for F-Test (ANOVA)), you can choose whether you want to perform power analyses for global (i.e., omnibus) tests or for special tests. Global test is the default option. This test refers to the H0 that all means in the design are equal (ANOVA) or that all regression coefficients (next to the additive constant) are zero (MRC).

MCR, One Set of Predictors

In MCR with one set of predictors, we test the H0 that the correlation of a Set A of predictors with a dependent variable is zero in the population. More formally,

H0:

R2(Y*A) = 0

H1:

R2(Y*A) = c, c > 0.

The effect size index f2 can be derived from R2(Y*A): 

           R2(Y*A)
f2 = __________________
  
        (1 - R2(Y*A))  
 

Assume that our H1 specified R2 (Y*A) = 0.1. Thus, f2 = .1/(1-.1) = 0.1111.

You can easily calculate f2 with G*Power. Simply click onto the "Calc 'f2'" button, insert the R2(Y*A) value, and click onto "Calc & Copy."

Given alpha = .05, a set of five predictors, and a total sample size of N = 110, what is the power of this regression analysis?

 

Select:

Type of Power Analysis:

Post-hoc

Type of Test:

F-Test (MCR), Global

Accuracy mode calculation

Input:

Alpha:

.05

Effect size "f2":

0.1111

Total sample size:

110

Predictors:

5

(This is a new item that pops up only when you perform power analyses for MCRs.)

Result:

Power (1-beta):

0.7518

Critical F:

F(5,104) = 2.3017

Lambda:

12.2210

 

MCR, Two Sets of Predictors

In MCR with two sets of predictors, we test the H0 that the correlation between a dependent variable and a Set B of predictors, added to a Set A of predictors, is zero in the population. In other words, the H0 is that the added Set B of predictors explains no variance that has not already been accounted for by Set A. Let

R2(Y*(A,B)) denote the variance explained jointly by Set A and Set B,

R2(Y*A) denote the variance explained by Set A, and

R2(Y*(B*A)) = R2(Y*(A,B)) - R2(Y*A) denote the proportion of variance in the dependent variable accounted for by Set B over and above what is accounted for by Set A (see Cohen, 1977, p. 423f).

 

Thus,

H0:

R2(Y*(B*A)) = 0

H1:

R2(Y*(B*A)) = c

The effect size index f 2 can be derived as before. Remember that R2(Y*(A,B)) is the variance explained jointly by Set A and Set B. The f 2 for the additional variance explained by Set B is

 
          R2 (Y*(B*A))
f2 = ____________________
  
        (1 - R2(Y*(A,B)))  
 

Alternatively, you can compute f 2 from explained and residual variance components:

f 2 = PVs/PVe.

In order to calculate f 2 from variance components within G*Power, simply click onto the "Calc 'f2'" button, insert the variance values, and click onto "Calc & Copy." Make sure to insert the proper error term.

 

Assume now that our H1 postulates that R2(Y*(B*A)) = .04. This is what we think we gain by adding Set B to the predictor set.

Let R2(Y*(A,B)) = .1. Then f 2 = .0444 (see the formula above). Set A has 2 predictors, Set B has 3 predictors (i.e., we have 5 predictors as in the previous example).

Given alpha = .05 and a sample size of N = 95, what is the power of this test?

Select:

Type of Power Analysis:

Post-hoc

Type of Test:

F-Test (MCR), Special

Accuracy mode calculation

Input:

Alpha:

.05

Effect size "f2":

0.0444

Total sample size:

95

Predictors:

5

Numerator DF:

3

(the 3 Set B predictors)

Result:

Power (1-beta):

0.3617

Critical F:

F(3,89) = 2.7070

Lambda:

4.2180

This power is ridiculous, we better start running a lot more subjects.

Math

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Please report suggestions for improvements to
Axel Buchner, Franz Faul, or Edgar Erdfelder.