Chapter Preview. Survival modeling focuses on the estimation of failure time distributions from observed data. Failure time random variables are defined on the non-negative real numbers and might represent time to death, time to policy termination, or hospital length of stay. There are two defining aspects to survival modeling. First, it is not unusual to encounter distributions incorporating both parametric and nonparametric components, as is seen with proportional hazard models. Second, the estimation techniques accommodate incomplete data (i.e., data that are only observed for a portion of the time exposed as a result of censoring or truncation). In this chapter, we apply R's survival modeling objects and methods to complete and incomplete data to estimate the distributional characteristics of the underlying failure time process. We explore parametric, nonparametric, and semi-parametric models; isolate the impact of fixed and time-varying covariates; and analyze model residuals.

Survival Distribution Notation

Frees (2010) provides an excellent summary of survival model basics. This chapter adopts the same notation.

Let y denote the failure time random variable defined on the non-negative real numbers. The distribution of y can be specified by any of the following functions:

1. • f(t) = the density of y

2. • F(t) = Pr(y ≤ t), the cumulative distribution of y

3. • S(t) = Pr(y > t) = 1 − F(t), the survival function

4. • h(t) = f(t)/S(t), the hazard function

5. • H(t) = − 1n(S(t)), the cumulative hazard function