# What is independent censoring and what are its assumptions?

I'm reading the Introduction to Statistical Learning book. In chapter 11 (Survival Analysis, page 463), the authors state:

In general, we need to assume that the censoring mechanism is independent: conditional on the features, the event time T is independent of the censoring time C.

I also read the given two examples:

In order to analyze survival data, we need to make some assumptions about why censoring has occurred. For instance, suppose that a number of patients drop out of a cancer study early because they are very sick. An analysis that does not take into consideration the reason why the patients dropped out will likely overestimate the true average survival time. Similarly, suppose that males who are very sick are more likely to drop out of the study than females who are very sick. Then a comparison of male and female survival times may wrongly suggest that males survive longer than females.

But I don't think I fully comprehend it (because I'm not confident while doing the exercises). Could you explain to me what independent censoring is in another way and give me some examples?

$$\rm [I]$$ provides a nice intuitive discussion on independent censoring; its assumption and necessity.

Suppose there are two groups: $$\rm A, ~B,$$ each group consisting of $$100$$ subjects initially disese-free.

Below is the schematics of $$\rm A:$$ $$\underline{\text{Group A}}\\\begin{array}{|c|c|c|c|} \hline \text{Time} & \text{At risk} & \text{Events} & \text{Survived} \\ \hline {0-3} & {100} & {20} & 80 \\ \hline\end{array}\\\text{40 censored}\\ \begin{array}{|c|c|c|c|} \hline \text{Time} & \text{At risk} & \text{Events} & \text{Survived} \\ \hline {3-5} & {40} & {5} & 35 \\ \hline\end{array}$$

For $$\rm B:$$

$$\underline{\text{Group B}}\\\begin{array}{|c|c|c|c|} \hline \text{Time} & \text{At risk} & \text{Events} & \text{Survived} \\ \hline {0-3} & {100} & {40} & 60 \\ \hline\end{array}\\\text{10 censored}\\ \begin{array}{|c|c|c|c|} \hline \text{Time} & \text{At risk} & \text{Events} & \text{Survived} \\ \hline {3-5} & {50} & {10} & 40 \\ \hline\end{array}$$

How would one estimate $$5$$-years survival for each group?

Assume censoring is independent that is, it is random within any groups $$\rm A, ~B.$$

So under the assumption of independent and random censoring, in $$\rm A,$$ it can be expected that the $$40$$ subjects who were censored were similar to the $$40$$ who remained at risk with respect to their survival experience. Since $$5$$ out of those $$40$$ were remained under observation experienced the event, then $$5$$ would be the estimated number out of those censored $$40$$ over that same time span. So, estimated $$5$$-years survival for $$\rm A$$ would be $$\frac{20 +5 + \underbrace{5}_\text{from censored cases}}{100} = 0.70.$$

In the same vein, one can calculate for $$\rm B,$$ the estimated $$5$$-years survival which would be $$0.48.$$

The reason for stating two groups is to discern between independent and random censoring: observe $$\rm A$$ has higher proportion of censoring ($$0.50$$) than that of $$\rm B$$ ($$0.17$$). The censoring is not random. However, conditional on group status, the censoring is random. This is the basis of independent censoring.

Formally, independent censoring means

within any subgroup of interest, the subjects who are censored at time $$t$$ should be representative of all the subjects in that subgroup who remained at risk at time $$t$$ with respect to their survival experience.

The authors aptly summarized the assumption:

Assume survival experience of subjects censored at $$t$$ is as expected if randomly selected from subjects who are at risk at $$t$$.

That is,

Even though the censored subjects were not randomly selected, their survival experience would be expected to be the same as if they had been randomly selected from the risk set at time $$t$$.

## Reference:

$$\rm [I]$$ Survival Analysis: A Self‐Learning Text, David G. Kleinbaum, Mitchel Klein, Springer Science$$+$$Business Media, $$2012,$$ presentation $$\rm XI.,$$ pp. $$37-41.$$

I'm learning about survival analysis and I thought the existing answer needed more formulas. Here's what I found in "On the assumption of independent right censoring" [arXiv] by Morten Overgaard and Stefan Nygaard Hansen.

Following their notation, let $$T$$ be the latent time and $$\tilde{T}$$ be its censored form.

The pair $$(T, \tilde{T})$$ has the status-independent observation property iff $$P(\tilde{T} = t|T = t) = P(\tilde{T} \ge t|T \ge t).$$ It has the non-prognostic censoring property iff $$P(T \le t | \tilde{T} = s, T > s) = P(T \le t | T > s).$$

I haven't been able to come up with an intuitive English language description for the former. For the latter, it's saying: subjects who got censored at time $$s$$ have the same survival characteristics as subjects who survived past $$s$$ (observed or not).

The authors show the following (among many other results):

• Status-independent observation holds if and only if the Kaplan-Meier and Nelson-Aalen estimators are consistent.
• Non-prognostic censoring implies status-independent observation.
• Non-prognostic censoring is equivalent to having an independent censoring time $$C$$ with $$\tilde{T} = \min(C, T)$$.

I would translate the existing answer to $$P(T \le t | \tilde{T} = s, T > s) = P(T \le t | \tilde{T} > s).$$ which can be shown to be equivalent to non-prognostic censoring.