Survival analysis with cures when it is known that for some subjects the event (death) will never occur

Question

Say we have the following set up. At time t=0 there are N infected patients. There is a treatment which, if taken until t=T, cures 100%. However, some patients will be cured before t=T while others will die before t=T. I need to analyse times to death. How do I treat those patients that have cured? They are not really censored, in the way I understand it, since we in fact know that they are cured. I am looking for (references to) exact instructions on how to construct distribution function for death times. Many thanks.

Added after an answer from DWin I realise that hazards, which I understand to mean (in discrete time framework) the probability of dying at time $t=n$ conditional on having survived and not cured by time $t=n-1$. But what about unconditional probabilities?

If I didn't have "cures" and a conditional probability of dying at time $t=n$ is $d_n$, then the unconditional probability of dying, say, at time $t=2$ is $P(D=2) = (1-d_1) \cdot d_2$.

But with "cures", do I not need to take into account the fact that at each time $t$ I now have three outcomes: (1) dead, (2) not dead but not cured, (3) cured? And is the probability of dying at $t=2$ now given by $P(D=2)=(1-d_1)\cdot(1-c_1)\cdot d_2$ where $c_n$ is the conditional probability of curing at time $t=n$?

From playing around with numbers, this seems to be the right way to go in the sense that the cumulative probabilities constructed in this way recover the correct number of deaths that accrued by time $t$. But how to recover hazard rates from a cumulative distribution function constructed in this way is not clear to me.

Answer 1

If you can establish that they are "cured", then they are no longer in the risk set for dying of the condition under treatment. Estimating hazards requires that you divide the events by the number at risk. So they really should be censored at the first time that you know they have been cured. This is a bit likethe data presentations done by oncologists and transplant surgeons where Kaplan-Meier plots of "progression-free survival" or "rejection-free survival" are presented, but when you read the methods it's apparent that death from other causes is considered to be a censoring time. It rather stands the meaning of "survival" on its head. The specialists are throwing up their hands and saying "not my fault!"

Generally the notation is to use either m or q for the rate or probability of dying and to use d for the number of deaths. Terms like your (1-q) would be useful in calculating the probability of surviving to the next interval (as you were doing).

I still haven't figured out your goal and the possibility of having other not yet specified processes (such a death rate for the cured group). If you are worried about keeping track of all non-censored individuals, then you can use Markov models. If the "cured" individuals have different rate of dying you would have three (or four) states; not-cured-alive, cured-alive, (possibly a censored group and a separate censoring process for going from cured to censored)) and dead. The dead and censored states are called "absorbing", since once in that state, there is no exit (or transition") to another state. If the rates are constant, it is a simple matter to model with matrix methods and there are some lovely theoretical results you could draw on. If the rates vary over time it is called a time-inhomogeneous Markov model.

Answer 2

OK, I have figured this out (I think). First, we need to construct two empirical distribution functions: one for deaths and one for cures. If $D$ is time of death and $C$ is time of cure, then the two distributions are

$F_D(t) = P(D \le t) = \frac{1}{N} \sum_{i=1}^t d_i$

and

$F_C(t) = P(C \le t) = \frac{1}{N} \sum_{i=1}^t c_i$

where $N$ is the total number of observations (patients) at the beginning and $d_i (c_i)$ is the number of deaths (cures) at time $t=i$ (note that the notation here is different from the one I sued in the OP).

Also note that if either (or both) of the two events (i.e. death or cure) are not guaranteed to happen for all subjects (like in the OP, where those subject that do not die before the end of the treatment at $t=T$ will be cured and hence for them the event of death does not happen, and, obviously, those that die do not cure), then the corresponding distribution function (functions) will not add up to 1. To make them into proper distribution functions, they would need to be conditioned on the appropriate event taking place.

The hazard and cure rates are recovered in the following way:

$\lambda_D(t) = \frac{P(D=t)}{1-P(D \le t-1)-P(C \le t-1)} = \frac{F_D(t)-F_D(t-1)}{1-F_D(t-1)-F_C(t-1)} \rightarrow \frac{f_D(t)}{1-F_D(t)-F_C(t)}$

where the last expression is (I believe should be true) for a continuous case with $f_D(t) = F'_D(t)$ and similarly for cures

$\lambda_C(t) = \frac{P(C=t)}{1-P(D \le t-1)-P(C \le t-1)} = \frac{F_C(t)-F_C(t-1)}{1-F_D(t-1)-F_C(t-1)} \rightarrow \frac{f_C(t)}{1-F_D(t)-F_C(t)}.$

How does it work on here - do I now accept my own answer or do I delete the question?