# Conditional logistic regression vs GLMM in R

I have paired data (GWAS case/control study) and I have heard using conditional logistic regression or generalized linear mixed models (GLMM) is appropriate. Which should I use in this case? Why would you use one over the other. More importantly can you guys point me towards resources for doing these methods in R? I'm finding a lot of material for SAS, which I do not prefer. I can provide more details if necessary.

• The conditional logistic regression applies fixed effects (in the context of econometrics),

$$logit(p_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta+u_i.$$ where each pair of subjects has an individual intercept ($u_i$). It can be implemented with clogit() of package survival or clogistic() of package Epi.

• Generalized linear mixed models (GLMM) for binary data can adopt link functions like logit, probit and cloglog. The mixed logistic regression is as,

$$logit(p_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\boldsymbol z_{ij}^{'}\boldsymbol u_i$$ where $\boldsymbol u_i$ are random variables and can have the distribution assumption (e.g. normal distribution). Of course you can use a random intercept model, i.e. $\boldsymbol z_{ij}^{'}=1$ and $\boldsymbol u_i$ is a scalar. You can estimate GLMM using glmer() of package lme4.

• As to the choice between conditional logistic regression and GLMM for binary data, some people are in favor of conditonal (fixed-effects) logistic regression and GLMM with probit link, but against fixed-effects probit or GLMM with logit link. The reason may be that some of the consistency properties break down, especially with small within-cluster sample size ($n_i=2$ for your case).

You can find the clarification of fixed effects and random effects (and marginal models) in different contexts here.

• how can case-control even be analyzed with GLMM ? isn't one assumption is that observations are independent conditioned on the strata? Case-Controls destroys this assumption... – Maverick Meerkat Sep 7 '20 at 13:59