Its main field of application is observational studies and in particular epidemiology.
It was devised in 1978 by Norman Breslow, Nicholas Day, Katherine Halvorsen, Ross L. Prentice and C.
[1] It is the most flexible and general procedure for matched data.
Observational studies use stratification or matching as a way to control for confounding.
Logistic regression can account for stratification by having a different constant term for each stratum.
For example, consider estimating the impact of exercise on the risk of cardiovascular disease.
If people who exercise more are younger, have better access to healthcare, or have other differences that improve their health, then a logistic regression of cardiovascular disease incidence on minutes spent exercising may overestimate the impact of exercise on health.
To address this, we can group people based on demographic characteristics like age and zip code of their home residence.
contains information about the variable of interest (in this case, minutes spent exercising) for individual
(which, in this example, is just a scalar) is the quantity of interest --- the impact of exercise on cardiovascular disease.
Logistic regression as described above works satisfactorily when the number of strata is small relative to the amount of data.
If we hold the number of strata fixed and increase the amount of data, estimates of the model parameters (
Pathological behavior, however, occurs when we have many small strata because the number of parameters grow with the amount of data.
For example, if each stratum contains two datapoints, then the number of parameters in a model with
In these settings, as we increase the amount of data, the asymptotic results on which maximum likelihood estimation is based on are not valid and the resulting estimates are biased.
Conditional logistic regression fixes this issue.
In fact, it can be shown that the unconditional analysis of matched pair data results in an estimate of the odds ratio which is the square of the correct, conditional one.
However, they did not allow for the analysis of continuous predictors with arbitrary stratum size.
All of those procedures also lack the flexibility of conditional logistic regression and in particular the possibility to control for covariates.
When the strata are pairs, where the first observation is a case and the second is a control, this can be seen as follows With similar computations, the conditional likelihood of a stratum of size
Conditional logistic regression is available in R as the function clogit in the survival package.
It is in the survival package because the log likelihood of a conditional logistic model is the same as the log likelihood of a Cox model with a particular data structure.
[3] It is also available in python through the statsmodels package starting with version 0.14.