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).

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    $\begingroup$ 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

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.

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  • $\begingroup$ 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

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