Background: For a project, I am fitting a conditional logit model where I have 5 control cases for every realized case. To do that I use the
clogit() function in the package
survival. I wanted to graph interactions with the
effects package by John Fox et al. It turns out that this package can't handle
clogit objects (output of
As I believed I remembered that conditional logit were a special case of GLM, I thought the clever/lazy way to get my interaction plots would be to refit the model using a fixed effects glm and then use
The documentation of
clogit seemed to confirm my intuition:
It turns out that the logliklihood for a conditional logistic regresson model = loglik from a Cox model with a particular data structure. [...] When a well tested Cox model routine is available many packages use this ‘trick’ rather than writing a new software routine from scratch, and this is what the clogit routine does.
In detail, a stratified Cox model with each case/control group assigned to its own stratum, time set to a constant, status of 1=case 0=control, and using the exact partial likelihood has the same likelihood formula as a conditional logistic regression. The clogit routine creates the necessary dummy variable of times (all 1) and the strata, then calls coxph.
Based on this description, it seems that I should be able to reproduce the stratification achieved through
strata() by using a random intercept for each case/control group with
lmer(). However, when I try, the results of
lmer differ. One thing is that I probably have the wrong likelihood function. I don't really know how to specify this in
lmer but more important, I am wondering what else I am missing.
I wonder whether I am completely wrong or somewhat on the right track but missing some pieces? What I would like is to understand what are the difference in terms of how the model is fitted between a conditional logit and a regular one (I understand that might be quite a long answer, so a book reference would be a great start). The my usual references for regression (Gelman and Hill, 2007; Mills 2011) are somewhat silent on the subject.