I have $i>100$ Cox regression models which differ only in one explanatory variable, $x_i$. They have the same outcome data and the same other explanatory (adjustment) variables.
I want to control the global Type 1 error probability/adjust for multiple testing of the significance of the coefficients of the $x_i$.
Some of the $x_i$ show high pairwise correlations. I inferred that therefore their coefficients and p-values are dependent. Neither Bonferroni nor Holm correction take into account these dependencies. The most attractive/exact option in my eyes is a permutation test that does consider them.
I thought about scrambling the rows of the outcome survival matrix, thereby conserving the correlations between the $x_i$ while destroying their relationship with the outcome. Then I compute the coefficients of all the $x_i$. In doing this for a few thousand different resamples I get the multivariate null distribution of the coefficients.
And this is where I'm stuck. I wanted to take the proportion of the null/resampled distribution for each coefficient estimated in the original data that is higher than it (or lower in the case of negative coefficients) as its p-value. But that would, again, destroy the dependency structure, as it treats the multivariate distribution as $i$ separate univariate ones.
How would one go about doing a permutation test for this problem? Is it even possible?