I am simulating data in which I ultimately test a difference in means (i.e difference between drug and placebo). I test this using a standard T-Test, and then also with a bootstrapped T-Test. I am doing this using 3 differently distributed outcomes - Normal outcome, Exponential outcome, and Gamma Outcome. And I am also interested in adjusting for a covariate.
I fit a linear regression to the data and then select the estimates for treatment effect and it's standard error to construct the test statistic. I do this same thing when I adjust for a covariate.
I am finding that the power from the bootstrapped procedure is the same as if I just computed the T-Test (without bootstrapping) when I do not adjust for a covariate and for all outcome types, but when I adjust for a covariate, the bootstrapping procedure has much less power than the T-Test (without bootstrapping). Is this to be expected? Number of simulations and sample size remain the same, the only thing that changes is the addition of a covariate.
Some background: According to the bootstrap procedure for hypothesis testing a difference in means (Efron and Tibshirani), we must "re-center" the dataset from which we bootstrap. So we compute the test statistic, and then re-center that data to create a more appropriate null distribution when we select the bootstrapped samples. I don't know if this is relevant to answering my questions, but I thought that I should give some background
Thanks for any suggestions you may have!