When predicting case status, what is the appropriate way to examine differences in proteomics by gender (/the interaction between protein and gender)? I have a dataset I'm working with containing measurements of several thousand proteins per person, with about 150 people split pretty evenly between two groups (cases and controls). Case status is split evenly within men and women, but gender is split unevenly overall, with about 50 men and 100 women. Everyone has a measurement for every protein.
I'm ultimately interested in what proteins are associated with/predictive of case status. I've evaluated this in the entire group using a lasso regression model, which zeroed out all but about 50 proteins (which is fine, just providing more information).
I want to know if there is a difference in the proteins that predict case status in men versus women. Specifically, I want to know:

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*Are there different proteins predicting case status in men versus women?


*Are there differences in the strength or direction of association of matching predictive proteins between men and women?


*Are there proteins that predict case status well in both men and women? If so, do they do so in the same way and to the same degree in men and women?


*Do proteomic models (either the overall model or in models specifically fitted by gender) fit better among men versus among women?

Edited for clarity:
I'm currently stuck on how to answer these questions statistically. I'm deliberating between two approaches and would like to get input on the advantages and disadvantages of each method (assuming both are valid) and any suggestions of alternative method if you have any. The approaches and some of my thoughts about them are:

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*Split the data into two sets; one with all men, one with all women. Repeat the lasso regression model in each of the two subsets, then directly compare the proteins and coefficients that come from each gendered model and the overall model. I'm uncertain whether the information gleaned from this approach would accurately be considered the interaction between protein and gender on case status in the strict statistical sense. I think I would be learning about model-level associations/interactions, not protein-level associations/interactions (but I am very unsure on this, so please correct me if I'm wrong).


*Run a new model on the full dataset, adding the interaction between gender and each protein to the formula from my overall model, and evaluate the interaction with gender by which gender x protein coefficients remain in the resulting model. But I don't know if that answers the question I'm asking; or how I would interpret the interaction (the non-simple/complex effect) in the resulting model. Could I expect it to predict case status equally well among men as among women?
Edited to add:
I gather what an interaction would mean for any single protein x gender coefficient. However, I don't understand what it would mean for the entire proteomic model, or combination of proteins, and gender. It's possible that what I'm referring to and confused about can only be illuminated by comparing the overall and the interaction model, or perhaps not, but that's a large part of what I'm trying to ask.
Further points of consideration:

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*On multiple testing and power: Approach 1. uses fewer tests per model, but creates two models. Approach 2. uses doubly many tests in one model, but that is the only model. The total number of participants included is the same in the end, but in Approach 1., they are included across two models, one with ~50 people, and one with ~100 people, and in Approach 2., the only model has ~150 people. Is there a definite answer to which approach minimizes multiple testing concerns and maximizes power? If so, are they both the same approach, or does one minimize multiple testing concerns where the other maximizes power?


*If I were to use Approach 1., fitting to men and women separately, would that result in differing fits by gender? I would imagine so, as one model has only one fit (perhaps at best, the fit of the model in Approach 2. would reflect an average of the fit for men and the fit for women, if not a different fit altogether). If this is the case, would I validly be able to compare the gendered models to each other, or even to the overall model, considering differences in fit and sample size?
End Edit

There also remains the possibility that neither approach is suitable, or appropriate, to answer the questions I am asking. If that is the case, please do let me know what you would suggest as an alternative.
With that, what approach would you use to examine the difference in the relationship between protein and case status by gender, and why?
Any thoughts or suggestions are greatly appreciated!
Edited to add:
In reference to this answer as a suggested answer to this question, I appreciate the suggestion and have previously reviewed this answer, but it does not answer my question. It partially addresses what I've written with respect to interactions for what in this case would be single specific protein x gender coefficients, but not what the interaction would be between the proteomic model and gender. Also, as it only really reflects Approach 2., it does not address why one would use Approach 1. versus Approah 2., or essentially, what are the benefits and drawbacks of each approach?

Approach 3: instead of modeling case/control against proteins and gender, model the set of protein expression levels against case/control and gender. That's been a standard approach dating back to the days of spotted microarrays for measuring mRNA expression. Standard software tools for that are provided for example by the Bioconductor limma package, which can be adapted to several different types of expression data. It uses appropriately pooled error estimates and takes multiple comparisons into account.
With predictors in the model of gender, case/control, and their interaction, that will tell you specifically which proteins are associated with case/control regardless of gender, which are associated with gender regardless of case/control status, and which differ depending on the combination of case/control and gender.
