2
$\begingroup$

Can anybody please explicate the following statement by Andrew Gelman?

If you have 80% power, then the underlying effect size for the main effect is 2.8 standard errors from zero. That is, the z-score has a mean of 2.8 and standard deviation of 1, and there’s an 80% chance that the z-score exceeds 1.96 (in R, pnorm(2.8, 1.96, 1) = 0.8).

$\endgroup$
  • 2
    $\begingroup$ What exactly do you find unclear about it? $\endgroup$ – Tim Feb 15 at 21:46
  • $\begingroup$ Please see my answer below. I think I got confused by the code given. It is a nice trick if one wants to avoid typing lower.tail = FALSE. $\endgroup$ – Ivan Feb 16 at 7:22
1
$\begingroup$

A two-tail hypothesis with a significance level of 0.05 are assumed. The right-tail critical value is 1.96. The power is the mass of the sampling distribution under the alternative to the right of this decision boundary. Then we want to find a Gaussian with a standard deviation of 1 so that 80% of its mass is to the right of 1.96. Then a mean of 2.8 gives the desired outcome.

In R, it could be verified by pnorm(1.96, 2.8, 1, lower.tail = FALSE), which is a more direct and verbose translation of the reasoning compared to the original code.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Ivan, @Tim - Given that we are thinking of a two-tailed test, doesn't the code need to be something like pnorm(1.96, 2.8, 1, lower.tail = FALSE) + pnorm(-1.96, 2.8, 1, lower.tail = TRUE)? Granted, one expression or the other is likely to be negligible...or am I not thinking about this in the right way? $\endgroup$ – user697473 Mar 9 at 23:56
  • $\begingroup$ The statement was about the right tail, and hence the calculation is limited to that tail. $\endgroup$ – Ivan Mar 11 at 9:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.