Other t-Tests

With this option, we can perform power analyses for any test that depends on the t-distribution. All parameters of the noncentral t-distribution can be manipulated independently. Note that with "Other t-Tests" you cannot do a priori power analyses, the reason being that there is no definite association between N and df (the degrees of freedom). You need to tell G*Power the values of both N and df explicitly.

We consider 3 examples here:

In addition, we give hints on how to do power analyses for nonparametric tests such as the

Matched-Pairs t-Tests

In t-tests for matched pairs, we have differences of the values from N matched pairs,

y1 = xA1 - xB1
:     :     :
yN = xA1 - xB1
 

The H0 we test is that the pairs do not differ, that is, the population mean muY of the differences is zero. More formally,

H0:

muY = 0

H1:

muY = c, c <> 0.

When computing the standard deviation sigmaY of the distribution of differences, we need to take into account the correlation r between A and B in the population:

 

              __________________________________________
             /                                          \
            /        2        2
sigmaY =   /   sigmaA + sigmaB - 2 r sigmaA sigmaB
          /
        \/
 

where sigmaA and sigmaB are the standard deviations of x in the populations A and B, respectively, and r is the population correlation between A and B as paired.

In matched-pairs t-tests, N is the total sample size (i.e., total number of pairs), df = N-1, and the effect size is: 

       muY
f = ________
 
     sigmaY
 

where muY is the difference between the means as specified by H1.

 

Example

Assume that we are faced with a repeated measures design in which the same subject is observed under each of two treatments. We have data from 40 subjects. Previous research has shown that the standard deviation of the differences is approximately 20. We consider mean differences of 8 or larger as important. Thus, the effect size we need to enter is f = 8/20 = .4. We fix alpha at .05.

Select:

Type of Power Analysis:

Post-hoc

Type of Test:

Other t-Tests, two-tailed.

Accuracy mode calculation

Input:

Alpha:

.05

Effect size "f":

0.4

N:

40

df:

39

(Df = N-1 in matched pairs t-tests.)

Result:

Power (1-beta):

0.6940

Critical t:

t(39) = 2.0227

Delta:

2.5298

As we said before, you cannot perform a priori power analyses directly, but you can, of course, perform repeated post-hoc power analyses, adjusting N and df until you arrive at the power value you desire. For instance, if you want, in the above example, the power to be .95, you simply increase N and df (= N-1) until the power is as close as possible to .95 (which will be the case with N = 84 and df = 83 for the present example).

 

One-Sample t-Tests

We want to compare the mean of a population from which we sample to a constant c. The effect size index d is computed according to

	
       | mu - c |
 f = ______________
         sigma
 
 

where mu and sigma are the mean and the standard deviation in the population, respectively. As Cohen (1977, p. 46) writes, the interpretations of f (Cohen's d3') as well as the effect size conventions are identical to those for d.

N is the total sample size, and df = N-1. Thus, we're all set to do this power analysis analogously to the one for matched pairs t-tests (above).

z-Tests

We can easily do power analyses for z-tests with G*Power because, as df approaches infinity, the t-distribution asymptotically converges with the normal distribution with mean

 
           __
          /  \
d = f *  / n 
       \/
      
 
 

and standard deviation 1. In other words, the critical t(32000) is virtually identical to the critical z value. As the effect size index, we use

 
 
     |mu1 - mu2| 
f = _____________
        sigma
 
 

N is again the total sample size.

mu1 and mu2 are the the means in populations 1 and 2, respectively.

However, for df we specify df = 32000. That's it.

In this way, you can perform power analyses for all sorts of z-tests (e.g., approximate z-test for hypotheses about binomial probabilities, comparisons of correlations between two different samples etc.). Note, however, that N and f have to be specified such that they make sense for the test you want to consider.

Nonparametric Tests

Some variants of power analyses for nonparametric tests can be conducted by adjusting the result obtained for the corresponding parametric test (cf. Bredenkamp, 1980; Singer, Lovie & Lovie, 1986).

For example, an a priori power analysis for the Wilcoxon-Mann-Whitney U test can be conducted by first performing an a priori power analysis for the t-test for means. If the t-test model is valid, and Nt designates the sample size necessary for the t-test to achieve some given power (1-beta), then the sample size Nu = Nt/A.R.E. yields approximately the same power for the U test.

A.R.E. denotes the asymptotic relative efficiency (or Pitman efficiency) of the U test relative to the t-test which is 3/pi = .955 (see Lehmann, 1975).

The same procedure may often be used to approximate the power of randomization tests (Onghena, 1994, pp. 144-176). In this case, the A.R.E. of the randomization test relative to the corresponding parametric test is 1. For power analyses in randomization tests which do not have a corresponding parametric test, special computer software is in preparation (Onghena, 1994; Onghena & Van Damme, 1994).

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Axel Buchner, Franz Faul, or Edgar Erdfelder.