# Reporting pseudo p-values for bootstrap-estimated linear regression coefficients

I've just received peer reviews for an applied paper that reported a table of multiple linear regression coefficients. While I reported 95% confidence intervals for these coefficients, a reviewer has requested I report exact p-values as well. The difficulty is that the confidence intervals were estimated via an ordinary non-parametric bootstrap, given the peculiarities of the data set.

Is there a way I can estimate 'pseudo p-values' from these confidence intervals? If so, under what assumptions/conditions are these p-values reasonable?

For bootstrapping, I'm assuming you resampled with replacement, computing the beta value each time to generate a distribution from which you computed the 2.5%ile and 97.5%ile. This gave you 95% confidence intervals.

For permutation, you'd resample from without replacement (shuffling up the case/control status or whatever the main factor is), get a distribution of null p values, and then compute you empirical p value based on wherever your observed falls in this null distribution. For details on the math, check out the book recommended on the link I posted above. Sorry for the piecemeal reply.

• Thanks, this is helpful. Is it reasonable to report both p-values calculated from a permutation test and confidence intervals calculated by a non-parametric bootstrap? Apr 28 '15 at 6:06
• Another obvious question (I'm new to permutation tests): Would I "get a distribution of null p values" or get a distribution of beta coefficients under the null hypothesis of no relationship? Apr 28 '15 at 6:10