What exactly is meant by bias in this context?

Question

I'm working through an example of survival-time analysis with censored and un-censored data. We're given the survival times of 94 patients. Some of these survival times are censored i.e.in this context a censored survival time represents the "last time point when the patient was known to be alive." We don't know whether and if so when a patient actually died. (could be that patient dropped out of the study, death by other causes etc.)

The textbook then goes on to say the following: "we now now consider the $n=47$ non-censored survival times and assume they are exponentially distributed. We emphasize that this approach is not generally acceptable as ignoring the censored observations will introduce bias if the distributions of censored and uncensored events differ."

My questions are:

1. What exactly do we mean by bias in this context? (do we simply mean that we might be missing out on getting a better idea of what the true distribution of the survival times might be?)

Answer 1

As you mentioned we do not know what happens with the censored individuals. Let's say the event in question is death. In this case, censorship means we have not observed the patient die, therefore that information is unavailable. Censored patients may die in the near or far future, but the important thing is that we do not know when they will die (or whether they already did). If we drop censored individuals from our study, we remove some information from our model. The only situation in which it would be ok to do this is if the distribution of time-of-death for censored and uncensored individuals is the same, otherwise, our analysis is biased.

Example: If we drop all censored individuals and they all actually died much later on after the last time they were observed our model estimates will be higher (higher rate ratio or hazard ratio, everyone appears to die faster than in reality) than it would actually be had we observed these individuals we have dropped.

As it may say in your book this is not good, assuming an exponential distribution corresponds to a Poisson regression approach, which in its standard format cannot deal with censoring. To deal with censoring it is better to use Cox regression. If you use R, have a look at the survival package.

Please let me know if anything is not clear.

Answer 2

It just means that as you increase your sample size (i.e. number of non-censored survival times available to you gets very large), the distribution you observe will not get closer and closer to the "true" distribution of survival times. That is, it will be biased, even as the variance is reduced by increasing your sample size.