Conditional logistic regression

Conditional logistic regression is an extension of logistic regression that allows one to take into account stratification and matching. Its main field of application is observational studies and in particular epidemiology. It was designed in 1978 by Norman Breslow, Nicholas Day, K. T. Halvorsen, Ross L. Prentice and C. Sabai. It is the most flexible and general procedure for matched data.

Observational studies use stratification or matching as a way to control for confounding. Several tests existed before conditional logistic regression for matched data as shown in related tests. However, they did not allow for the analysis of continuous predictors with arbitrary strata size. All of those procedures also lack the flexibility of conditional logistic regression and in particular the possibility to control for covariates.

Logistic regression can take into account stratification by having a different constant term for each strata. Let us denote ${\displaystyle Y_{i\ell }\in \{0,1\}}$ the label (e.g. case status) of the ${\displaystyle \ell }$th observation of the ${\displaystyle i}$th strata and ${\displaystyle X_{i\ell }\in \mathbb {R} ^{p}}$ the values of the corresponding predictors. Then, the likelihood of one observation is

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