While doing a project involving survival analysis a while back, I learned a useful technique for reducing survival analysis to binary regression. The following discussion is adapted from Chapter 7 of Survival Analysis by David G. Kleinbaum.
To illustrate how the technique works, consider a small data set with three subjects. Subjects 1 and 3 are administered a certain treatment for an “event” while Subject 2 is the control. From subsequent follow-ups, we learned that Subject 1 got the event in the first interval, Subject 2 got the event in the third interval, and Subject 3 got the event in the second interval.
To turn the data into a form suitable for binary regression, we introduce three dummy variables D1, D2, D3 which are coded 1 if we are in the corresponding interval and 0 otherwise. This yields the following table:
A logistic regression model can then be formulated as
where Pr(Event = 1) is the probability of the event under study for a given interval conditioned on survival of previous intervals.
The dummy variables play a similar role to that of the intercept in a standard regression model: it provides a baseline outcome, one for each interval, in the case where all other variables are zero.
This is how we interpret the coefficients: is the log odds of the event occurring in the first interval among the control group; is the log odds of the event occurring in the second interval conditioned on survival of the first interval among the control group; is the log odds of the event occurring in the third interval conditioned on survival of the first two intervals among the control group; and is the log odds ratio for the Treatment variable.
Why is this technique useful?
- While logistic regression is now a standard feature on big data platforms, some scalable machine learning libraries do not yet support survival analysis models like hazards models. The technique described here is a way around such constraints.
- Technical folks who are not otherwise familiar with survival analysis seems to have an easier time understanding this alternative formulation of survival analysis compared to the more standard hazards models.
- The alternative formulation easily supports time-dependent variables which are not easy to handle in standard hazards models.
- The alternative formulation also allows us to move beyond linear models for survival analysis. Any machine-learning algorithm that deals with binary regression can now be considered.