In survival analysis there is a relationship between the survival function, $S(t)$ and the hazard function, $h(t)$, in that $$ h(t) = -\frac{d}{dt} \log S(t)~~~~~~~~~(1)$$

from which we can form the differential equation $$\frac{dS(t)}{dt} = -h(t)S(t)~,~ S(0) = 1 ~~~~~~~~~~~~(2)$$ with solution given by $$ S(t) = \exp\left(-\int_0^t h(u)\,du\right)~.$$

Now when the assumption of a parametric distribution is made, the usual likelihood function is modified to incorporate censoring. My question is this: Does the differential equation (2) change at all if we incorporate censoring?

  • $\begingroup$ Let $T$ be a random variable with survival function $S(t)$, i.e. $S(t) = \Pr(T>t)$. If by "modified to incorporate censoring" you mean that $S(t)$ has jump discontinuities, then the hazard function, as the rate of change of $\log S(t)$ is not defined, since $\log S(t)$ is not differentiable at points of discontinuity. $\endgroup$ – Sasha Apr 7 '15 at 13:48

This is just off the top of my head, but fundamentally censoring does not change the relationship between the hazard function and the survival function if censoring is uninformative (which it is usually assumed to be). Unlike in competing risks, after an item is censored it is still expected to fail at some point afterwards (unless the hazard rate converges to zero sufficiently quickly that $\lim_{t\rightarrow\infty}{S (t)} $ converges to nonzero). Censoring just means we make a limited observation of the failure time, not that it is changed.

Here is an attempt to coerce into differential equation form:

\begin{align} \frac{dX (t)}{dt}&=-(h (t) + h_c (t)) X (t)\\ \frac{dC (t)}{dt}&=h_c (t) X (t) - h (t) C (t)\\ \frac{dF (t)}{dt}&=h (t)(X (t)+C (t))\\ S (t)&=1-F (t)\\ X (0)&=1\\ C (0)&=0\\ F (0)&=0 \end{align}

In this system $ X (t) $ is the population which has neither failed nor been censored, $ C (t) $ is the population which has been censored but has not yet failed and $ F (t) $ is the population which has failed (with or without prior censoring).

This is equivalent to your system since $ S (t)=X (t) +C (t) $.

Censoring is about observations which are assumed to be generated from this system, and likelihood refers to the probability of making such observations. We change the likelihood function in the case of censoring because we have different data, but the underlying data generating process is unchanged.

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  • $\begingroup$ Thanks for your answer, Tristan. How is $S(t) = X(t) + C(t)$. $S(t)$ is a probability, hence $0<S(t)<1$. Am I missing something? $\endgroup$ – Gorg May 17 '15 at 6:36
  • $\begingroup$ Also, how did you come up with the those differential equations. $\endgroup$ – Gorg May 17 '15 at 6:50
  • $\begingroup$ @Gorg S(t) is the total proportion of those surviving. X(t)/S(t) are still under observation while C(t)/S(t) are alive but censored. I assumed a rate function for censoring process h_c (t) and a failure rate function h (t) which is assumed to be independent of censoring. People leave X(t) through failure or censoring, they enter C(t) through censoring and leave through failure. $\endgroup$ – tristan May 17 '15 at 7:05

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