# Bootstrapping coefficient p-values under null hypothesis in case-resampling regression

I am fitting a multiple linear mixed-effects regression model in lme4. The residual fit is near-linear, enough to warrant not assuming residual homoscedasticity. One way to model regression without explicitly making this assumption is to use case-resampling regression (Davison & Hinkley 1997), an application of the bootstrap (Efron & Tibshirani 1993).

In case-resampling regression, rather than assuming a normal distribution for the $T$-statistic, we estimate the distribution of $T$ empirically. We mimic sampling from the original population by treating the original sample as if it were the population: for each bootstrap sample of size $n$ we randomly select $n$ values with replacement from the original sample and then fit regression giving estimates, repeating this procedure $R$ times.

Having applied this procedure, I am trying to calculate empirical $p$-values for my regression coefficients. As in parametric regression, I want to conduct the two-tailed hypothesis test of significance for slope with test statistic $T$ under the null hypothesis $H_0 : \hat\beta_1 = 0$. Since we are treating the original sample as the population, our $T=t$ is the observed value from the original sample. For $\hat\beta_{0, 1, \dots, p}$ We calculate the $p$-value as follows:

(1) $min(p = \frac{1}{R} 1\{T \geq t\}, p = \frac{1}{R} 1\{T\leq t\})$

Davison and Hinkley take $t = \hat\beta_1$ so that, in practice

(2) $min(p = \frac{1\{\hat{\beta^*}_1 \geq \hat{\beta}_1\} + 1}{R + 1}, p = \frac{1\{\hat{\beta^*}_1 \leq \hat{\beta}_1\}+1}{R+1})$

The major problem here is that the bootstrap samples were not sampled under the null hypothesis, so in (1) and (2) we are evaluating the alternative hypothesis rather than the null. Efron & Tibshirani (1993) indeed caution that all hypothesis testing must be performed by sampling under the null. This is relatively simple for, say, testing the difference between two means, where the null $H_0 : \sigma_1 = \sigma_2$, and which requires a simple transformation of the data prior to sampling.

So my question here is: how do I perform significance testing under the null hypothesis in case-resampling regression? As far as I could see, neither Davison & Hinkley (1997) nor Efron & Tibshirani (1993) seem to mention how to sample under the null. Is there some adjustment that I can introduce before (to the data) or after case-resampling (to the least-squares formula) in a way that is easily implementable in lme4? Any ideas and or algorithms would be greatly appreciated.