Survival Analysis in the Absence of Censoring

Question

I am interested in better understanding the effects of "censoring" in Survival Analysis.

I have heard that two of the main motivations which started the field of Survival Analysis were:

1) Survival Regression Models allow you to estimate hazard and survival probabilities over a period of time (e.g. 1021 days from now, what is the survival probability of someone in a specific cohort?) - whereas Classical Regression Models (e.g. time_of_event = b_0 + b1_age + b2_weight) would only allow you to measure the average expected survival time (i.e. a point estimate, e.g. someone in a specific cohort will experience the "event" 200 days from now, plus-minus 15 days).

2) Survival Regression Models allow for Censored Data (e.g. a patient in a medical study has to move to a new country before the study is finished - we have some information about this patient, but are still missing the "response" for this patient) . For example, in the case of the Cox Proportional Hazards, a special likelihood function was developed to estimate model parameters in the presence of censored data. In short, Survival Models allow you to use the "complete" part of "incomplete data", whereas standard regression models would require you to discard these observations all together.

My Question: I have consulted other questions (e.g. Time to event with no censoring - use survival or normal regression?) which suggest that "censoring" is not required for Survival Analysis Models, and that Survival Analysis Models (e.g. Kaplan-Meier, Cox PH) can full work in the absence of censoring. I have also consulted other questions (e.g. How do you compare two "survival times" when there is no censoring per se?) which state that not only Survival Models can work without "censoring", but the absence of censoring can be considered as a "blessing".

Are there any theoretical results that show for Survival Models (e.g. Kaplan-Meier, Cox PH) that show "(for a given dataset) no censoring is better than censoring?" Or that "less censoring is better than more censoring"? Perhaps the confidence intervals become tighter when censoring is less?

Do any such results exist within the domain of Survival Analysis?

Answer 1

Survival regression fits a probability distribution to time-to-event data. If you have complete, uncensored data then that's reasonably straightforward. For a survival model with a probability distribution $f(t)$ of parameterized event times fit by maximum likelihood, an event at time $t$ contributes a factor proportional to $f(t)$ to the likelihood whose maximization provides parameter estimates.

If you have censored survival data and censoring is uninformative, then you can use the information those cases provide to help fit the model. This page shows how censored and truncated data contribute factors to the likelihood. You will note that those contributions are related to survival probabilities at censoring or truncation times rather than the time-specific values of the distribution, $f(t)$, provided by known event times. Censored or truncated event times thus provide less information about the detailed shape of the distribution of survival times than do known event times.

There's an additional problem if censoring is informative, with the fact of censoring related to survival-associated variables. Then the assumptions underlying model fitting don't necessarily hold.

So there are two major reasons why censored event times are less than ideal: they provide less information about the shape of the survival curve, and they introduce the risk of informative censoring. In practice, however, censored survival times are inevitable, and ignoring censored data will generally lead to bias.