How to test assumptions for a large number of statistical tests? I am running a logistic regression. The outcome is a clinical variable, and there are two predictors: gene expression (continuous), hormone levels (continuous), and the interaction term between them.
There is only one hormone, but there are ~10K genes.
I am running independently 10K regressions, one for each gene.
The hypothesis is that interaction between some of the genes and the hormone leads to the clinical outcome. I would like to test if it is so.
The most important statistical assumption one should check in logistic regression (afaik) is linearity. The logit of the probability log(p/1-p) should be in a linear relationship with each of the predictors. This can be checked using a plot.
But, for a large number of genes this is not feasible. Even if I take only the ones that are found to be interesting (as a result of the regression) there are about 100 such genes.
What would be an efficient way to check the assumption?
The same goes for checking for outliers. How does one do it in an efficient way on ~100 regressions?

Often the data do not possess the information needed to make judgments about nonlinearity and additivity.  It would be better to fit logistic models making no such assumptions, e.g., use a regression spline in log gene expression and in log hormone level (logs will make them require slightly fewer knots) and cross-product terms.  Then you can use Wald "chunk" tests to test general hypotheses.
The clinical outcome variable you are using is the lowest information variable you can have (a binary Y).   What is that variable?  Is it truly dichotomous in its rawest form?  You need all the effective sample size you can get for your difficult problem.
You might also use a penalized regression joint analysis of all gene expressions.
