I'm working with some permutation tests where my main aim is to evaluate a treatment effect, and I have a question about choice of test statistic.
I've seen that some use β for the variable of interest as the test statistic, while others use the t-value (β/SE β) as test statistic.
I've not seen good explanations of advantages of choosing one over the other, and wonder if someone can tip me on relevant references or provide a quick explanation.
The best I've seen so far is here (p. 6):
Suppose that we really are interested in the standardized difference between means, but we are reluctant to use the parametric t-test because it assumes that the sampling distribution for t values is based on an underlying normal distribution. With resampling we can use a calculated t-value as a measure of the group difference, but we can test it against an empirical sampling distribution for the t-value ... We can randomly reshuffle the data into two groups of N=20 each and recompute the t-value. If we do this for many reshuffles of the data (e.g., 10,000) we can generate an empirical distribution of the t-value. This distribution is NOT necessarily distributed according to the parametric t distribution
From what I gather, the main point is that by using the t-value as a test statistic the assumption of an underlying normal distribution can be relaxed. If so, are there any other arguments for or against using the t-value?
I see that the t-value is also used in permutation tests in Efron & Hastie (2017) Computer Age Statistical Inference, p. 49-50.