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I wish to conduct a power analysis using the bootstrap method to find the difference in means of two groups. I wonder how we can calculate the "power" if we do not know in our data, the null is indeed false. In the book, An introduction to the bootstrap Book by Bradley Efron, the probability of rejecting the null hypothesis at different sample sizes is computed, but this is different from power (which calculates the same quantity conditioned on the Null hypothesis being false). In some papers, I have seen that a constant is added to one of the groups but I am not clear how to choose which group (in a two tailed setting).

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Your concerns are justified - the two samples might come from populations with different mean. To be on the safe side, you can generate two artificial samples by resampling just one of the groups, and add the constant to one of the artificial samples. Do this separately for each of the original groups, and average the power. This should give reasonable results provided your sample sizes are not small.

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