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Survival analysis

Survival analysis concerns statistical models of the time until the first occurance of some event. In a biological setting, the event in question might be the death of an organism. In mechanics, it could be the failure of a machine. In economics, it could be entering the workforce.

Model

The survivor function is the fundamental part of such models, denoting the probability that the time of death is later than some time $t$:

where $T$ is a random variable denoting the time of death. Let $f(t)$ denotes the pdf of $T$ and $F(t)$ denote the cdf of $T$. We can see that $S$ must be non-increasing since

and $f$ is nonnegative.

Another fundamental object is the hazard function of $T$, $\lambda (t)$, which gives the rate at which the probability of death is changing at some time $t$:

Using the definition of conditional probability and the cdf, we can write

The cumulative hazard function of $T$, $\Lambda (t)$, is defined as

$\Lambda (t)$ is also called the integrated hazard. It follows that $\Lambda (t)=-\ln S(t)$.

Common Distributions

Some commonly used survivor functions (survival time distributions) include:

• The exponential distribution?:

• The Weibull distribution?:

  $$F(t)=1-e^{-\gamma t^{\alpha }}\text{or}\lambda (t)=\gamma \alpha t^{\alpha -1}.$$

  F(t) = 1 - e^{-\gamma t^\alpha} \quad \text{or} \quad \lambda(t) = \gamma \alpha t^{\alpha - 1}.

  There is positive or negative duration dependence as $\alpha \gtrless 1$.