3.2 Censoring and Truncation

When dealing with survival data, some observations are usually censored meaning they are related to spells which are not completely observed. Duration data can also suffer from a selection bias which is called truncation.

3.2.1 Censoring mechanisms

Left-censoring occurs when the event of interest occurs before the beginning of the observation period. For example, an individual is included in a study of unemployment duration at $t_0$. At that time he has already been unemployed for a period but he cannot recall exactly the duration of this period. If we observe that he finds a job again at $t_1$, we can only deduce that the duration of unemployment is bigger than $t_1-t_0$, this individual is consequently left-censored. Observation 2 on figure 3.1 is associated with a left-censored spell (Liu 2019).

A spell is considered right-censored when it is observed from time $t_0$ until a censoring time $t_c$ as illustrated by observation 4 on figure 3.1. For instance, the lifetime related to a customer who has not churned at the end of the observation period is right-censored. Let us note $X_i$ the duration of a complete spell and $C_i$ the duration of a right-censored spell. We also note $T_i$ the duration actually observed and $\delta_i$ the censoring indicator such that $\delta_i = 1$ if the spell is censored. Then $(t_1, \delta_1),\dots,(t_N, \delta_N)$ are the realizations of the following random variables:

$$\begin{equation} \begin{aligned} T_i & = \min(X_i, C_i) \\ \delta_i & = \pmb{1}_{X_i > C_i} \end{aligned} \tag{3.1} \end{equation}$$

3.2.2 Selection bias

Survival data suffers from a selection bias (or truncation) when only a sub-sample of the population of interest is studied. A customer entering the firm’s portfolio after the end of the study is said to be right-truncated, whereas a client who has left the portfolio before the beginning of the study is considered left-truncated. Mathematically, a random variable $X$ is truncated by a subset $A \in \mathbb{R}^+$ if instead of $\Omega(X)$, we solely observe $\Omega(X)\bigcap A$. On figure 3.1, the first and fifth observations suffers from a selection bias.

Figure 3.1: Censored and truncated data

References

Liu, Weinan. 2019. “Inclusive Underwriting: The Case of Cardiovascular Risk Calculator.” PhD thesis, ENSAE ParisTech.