thanks for the link. This is a new usage for me, so I'll be interested to see if anyone else has any thoughts. My feeling is that subtracting the mean when evaluating is still the correct thing to do, but I'm not sure.

Okay, and your 'age' variable in the model is actually 'age minus mean age', correct? In that case, you absolutely need to subtract the mean age when calculating the HR because the term 'age' in the model is actually 'years from the mean age' and not the subject's age.

Are you doing two models, one for the linear predictor and then a Cox, or am I just misunderstanding your simplification? It would help if you wrote out the Cox regression as modeled.

There are only 11 dummy variables - there is no $\beta_{12}$, and I've never seen a requirement to omit the intercept for the global null. The global null is just for the month variables and doesn't depend on other covariates, etc (in so far as those variables have independent effects).

Yeah, I doubt the 1,2 coding is a problem. I would have expected more power with the clogit as well. With appropriately matched case-control data I would trust the conditional logistic results more than the regular logistic results. There could be a problem with insufficient control of confounding. Alternately, the matched variable may be inappropriate - is it definitely associated with both the outcome and exposure? If not, you can have decreased precision in the clogit and/or bias in the regular regression.

Other than the fact that I would code control = 0 and case = 1, the clogit looks fine, so the problem is likely with your initial regression. Can you post that? It's possible that using 1,2 for the outcome is causing some weird behavior, I suppose, but that seems less likely. I don't think the FDR adjustment would affect things. You could be right that the matching hasn't worked very well - are you only matching on one variable?

+1 This is a really clear, comprehensive, and well referenced answer. I wish I could upvote you more than once.

It might help if you posted your code for each, as well as a description of the matching variables. If your initial regression did not account fully for matching, then it may be biased upwards.

Also, I am surprised that I couldn't come up with good example papers. I'm going to keep looking and will update my answer when I find some, but it seemed like some background would be useful to the oort.

It sounded like you have data on the clusters at multiple time points, which is why I suggested clogit. You could also try generalized additive models (GAM).