If we are trying to find the power of a paired t-test, SAS offers this template (among others) in PROC POWER:

   proc power;
      pairedmeans test=diff
         meandiff = 7
         corr = 0.4
         stddev = 12
         npairs = 50
         power = .;

R, on the other hand, has this:

pwr.t.test(n = NULL, d = NULL, sig.level = 0.05, power = NULL, 
    type = c("two.sample", "one.sample", "paired"),
    alternative = c("two.sided", "less", "greater"))

where there is no way to specify the correlation. But the correlation makes a huge difference to the results. d (the effect size) is Cohen's d (difference in the means divided by pooled standard deviation).

Is this an error in the R program (that seems very unlikely) or have I missed something (far more likely) and, if I have missed something, what is it I have missed?

  • 1
    $\begingroup$ I presume you're using the function from the R package pwr. See the second example in the help of pwr.t.test (that sqrt(1-0.6) in the calculation of d is where the correlation comes in; the correlation in that example was 0.6). Compare against Cohen (1988) Statistical power analysis for the behavioral sciences (2ed.) as indicated in the help. $\endgroup$
    – Glen_b
    Mar 4 '19 at 4:53
  • $\begingroup$ Ah. Now I see. Thanks. I still think the SAS method is easier to use, but at least I realize that R is also correct (which I was pretty sure of, anyway). $\endgroup$
    – Peter Flom
    Mar 4 '19 at 11:15

The formula for the pooled standard deviation accounts for the covariance between time point 1 and time point 2. R requires Cohen's d, which makes use of the difference between the two means and the pooled standard deviation. R, perhaps, directly makes use of Cohen's d to calculate power. In SAS you can specify the correlation separately, but if you change your correlation, your SD will also change.

  • $\begingroup$ I am sure you are right, but I don't quite see it. The pooled sd includes the sample sizes of each sample and the variance of each sample. Where does the correlation come in? $\endgroup$
    – Peter Flom
    Mar 3 '19 at 20:16
  • 1
    $\begingroup$ If X1 and X2 are not independent, then Var(X1-X2) = Var(X1) + Var(X2) - 2 COV(X1, X2). COV(X1, X2) = SD(X1) SD(X2) COR(X1, X2). That's my understanding of it. I could be wrong. $\endgroup$
    – NB21
    Mar 7 '19 at 6:39

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