High Censoring Rate in Survival Analysis; Much higher survival time among censored patients

Question

I am trying to understand censoring in survival analysis and wondering about how to tell when standard use of censoring breaks down. In one case, the number of censored patients is fairly high (low death rate), yet the median or mean "survival" time (times of last confirmed observation of the censored patient) among these censored patients with death unconfirmed is nearly twice the equivalent survival times found among the uncensored patients with confirmed dates of death.

In the limiting case of a "wonder-drug" where nearly all of the patients survived beyond the trial close-out date, nearly all patients would be censored and using only the remaining uncensored patients (who may have died in the short time before the drug "kicked-in") might lead to a very poor looking survival curve that belies the activity of the drug.

I understand that censoring is usually a useful method, but it seems a bit counter-intuitive to assess the effectiveness of a drug based on patients for whom the drug is least effective.

Answer 1

I also found the word "censoring" to be confusing when I first started survival analysis.

"Censored" individuals aren't removed from analysis; they're just treated differently from those with events/deaths. If censoring is non-informative, as @swmo discussed, then a censored individual provides information that the event did not occur up to the censoring time. Just doesn't provide the exact time.

A standard survival curve includes the censored patients, noting the censoring times with a mark on the curve at the censoring time. The survival curve only drops at times of (noncensored) events, with a drop given by the ratio of events at that time to the total at risk at that time, including those with later censoring times. So the survival curves for the wonder drug in your example would in fact look quite good, with the few early events leading to small drops in the curve (as the fraction of individuals dying early was small) and then a high survival fraction thereafter.

Also, you're not usually comparing censored to uncensored patients within a single treatment group, as the question seems to suggest. Rather, you're comparing the timing of events in treatment group A to those in treatment group B. So in a test of a poor drug A versus wonder drug B, there would be many events/deaths in group A and few in group B, or at least events would tend to happen earlier in group A.

If most patients in group B are "cured" and they are not otherwise at high risk of death, then the "survival times" of the censored individuals in that group would mostly be determined by the duration of the study. A longer survival time for censored versus non-censored individuals may just mean that the study went on long enough to pick up most of those who were not "cured" by drug B.

Answer 2

Survival analysis is often done under the assumption of non-informative censoring, e.g. censoring is independent of failure time. To give an example of when this breaks down is not too difficult: think of the situation where censoring is clearly informative. An extreme example of this would be to censor every patient right before death, leaving us with only survivors.

The above example is of course silly, but a more realistic one is the case where patients undergoing different treatments have to report to some hospital every month. If not, they are censored from the study. Then censoring might become more likely when a patient's situation is getting worse as the patient doesn't have the energy to go to the hospital for some purely scientific purpose.

In your example of a wonder-drug, we can (under appropriate assumptions) still estimate survival using standard methods. If the drug takes some time to kick-in there'll be an increased hazard (compared to later) right after taking the drug, if we start follow-up right when the drug is taken. This seems only fair. But the drug will be assessed using all the patients in the study, also the censored ones, as they contribute survival time to the calculations.

Otherwise, you could include in your study only patients that had survived e.g. the first week after having received the drug. But then you'd be conditioning on surviving some period of time, of course.

Another quite common problem is the introduction of survival bias in terms of "immortal time". For instance, in some observational studies, the subjects may be known from some date, but only because they survived. Thus, the first immortal time (where had the subject died it would not have been included in the study) should be discarded when analysing data. A different example of immortal time bias can be studied by reading more on the Stanford heart transplant data. Originally, researchers working on this study calculated survival time for the patient given a heart transplant as the time from being accepted for transplantation until death or censoring. However, in the time from being accepted until the actual operation the patients were not exposed to the treatment (transplantation). This time should have counted as survival of unexposed patients, not exposed ones, as the researchers did.

In conclusion, bias is all around, but I don't see it in your example.