Survival Analysis Theory

Property 1:

S(t) = e–H(t)

or equivalently

H(t) = −ln S(t)

Proof: Since S(t) = 1 − F(t), the derivative S′(t) = − F′(t) = − f(t).  Thus

Taking the integral of both sides yields

From which we get H(t) = -ln S(t). Taking the exponential of both sides yields the first result.

Observation: Some properties of the hazard function are

Observation: If the hazard function is constant then t has an exponential distribution; i.e. a constant hazard functions corresponds to an exponentially distributed lifetime. To see this, let

Then c(1−F(t)) = F′(t), which has a solution when F(t) = 1 − ce-ct. Alternatively