# Permutation strategy for testing coefficients of Generalized Linear Models

What is a good strategy for performing permutation test for coefficients of Generalized Linear Models (GLM) such as logistic regression? Specially when the response is binomial.

Following the questions here and here, and @Geln_b's informative answer (the former link), I was wondering if one could apply the same concept to logistic regression? If not, is there any other way to perform permutation test?

I have case-control cohorts of healthy/diseased individuals with respect to different diseases. The response variable $Y$ is binomial represting whether individual is healthy, and the predictors are genetic mutations.

For example, here is a model one would try to fit that includes pairwise interactions:

$Y = \beta_0 + \beta_1 \; mutation_1 + \beta_2 \; mutation_2 \; + \beta_3 \; mutation_1 * mutation_2 + \beta_4 \; age \; + \beta_5 \; gender \; + \beta_6 \; PC_1 + ... + \beta_9 \; PC_4 \;$

$PC_n$ are first four/five principal components to correct for population sub-structure e.g. ancestry (there are better ways to handle that, but doing this for now).

I'm interested in the interaction coefficient, and find it hard to set up a permutation strategy as I will be testing thousands of pairs of mutations separately (thousands of separate models - the matrix is fat $M << N$). And, I often hear that there is no solid/valid permutation strategy for interactions.

I was hoping to do something like what @GregSnow suggested here for a set of pairwise interactions (permutation distribution of multiple interactions in separate models as a multivariate gaussian distribution):

One possibility is to compute the mean vector from your permutation distribution (should be close to a vector of 0's) and the variance/covariance matrix. Then use that information to compute the Mahalanobis distance for all the points, the p-value would then be the proportion of points with a Mahalanobis distance greater than the original coefficients.

Since I'm dealing with diseases, I could use disease prevalence in population, and form the response as a probability instead of a binary variable $Y$. For example, disease A occurs in 0.1% of population, can we somehow use this as the response (like a probability of getting disease) instead of binomial variable, and would it make sense?