I was wondering what is the difference between these three models (discrete time cox proportional hazard, conditional logistic regression and logistic regression). I would appreciate it, if you could explain it simply and without using too many technical terms.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
These are all ways of working with an outcome that is binary: no/yes, 0/1, an event happened/didn't, etc. With a discrete-time survival model, the outcome isn't just whether an event happened but when.
Logistic regression is a particular type of binomial regression for modeling/predicting a binary outcome as a function of a linear combination of predictor variables. Logistic regression uses a "logit link" to model the log-odds of the outcome ("logit link").
Conditional logistic regression is a way to extend logistic regression by allowing for grouping of individuals ("strata") who share sets of characteristics that are related to outcome. For example, you might have groups defined by the hospital that treated them. Or you might have groups defined by matching on variables that you want to control for without modeling them directly. Conditional logistic regression uses a different type of calculation ("conditional likelihood") from the "maximum likelihood" used in standard logistic regression, to get around problems that arise from that grouping.
A discrete-time survival model of the time it takes for an event to occur can be set up as a binomial regression. You can have multiple data rows for each individual, one for each time interval during which the individual was at risk. Each row includes the time, the predictor values in place during that time interval, and a binary indicator of whether the event occurred during that interval. You stack up all the data rows for all the individuals. You then model the probability of an event as a function of time and other predictors. If the binomial regression uses the logit link of logistic regression, it models the log-odds of the outcome.
Strictly speaking, a Cox proportional hazards model works in continuous instead of discrete time. It turns out, however, that if you use a "complementary log-log link" instead of a logit link in the binomial regression, you get what's called a "grouped proportional hazards" survival model. That is, you model the log-hazard of the event (as in a Cox model) instead of the log-odds. See this page and its links.
One possible source of confusion is that the code used for a continuous-time Cox proportional hazards model can be re-purposed to do the calculations for conditional logistic regression. In that application you don't model time (all individuals are treated as having the same value of time); you take advantage of how the Cox software handles strata. The R clogit() function for conditional logistic regression is thus distributed as part of the survival package that includes functions for Cox models.
