Constant hazard survival model

A constant hazard survival model takes $h(t)=\lambda$, giving survivor function $S(t)=e^{-\lambda t}$ and an exponential event-time distribution.

Let the survival time $T$ have an exponential distribution with rate $\lambda =0.2$ per year. Its hazard function is $h(t)=0.2$ per year, its cumulative hazard is $H(t)=0.2t$ for $t$ in years, and its survival function is $S(t)=e^{-0.2t}$. For a subject still event-free after five years, the conditional probability of an occurrence within the next two years isThe original population's probability of an occurrence between those same endpoints is instead $S(5)-S(7)=e^{-1}-e^{-1.4}\approx 0.121282$. Dividing by $S(5)\approx 0.367879$ recovers the conditional answer. The memoryless property makes the two-year conditional answer independent of the preceding five years.

For just $\Delta =0.01$ year, the exact conditional event probability is $1-e^{-0.002}\approx 0.001998$, close to the instantaneous approximation $h(5)\Delta =0.002$. For two years that approximation gives $0.4$, whereas the exact answer is about $0.329680$. At the single exact time five, the event probability is zero. These are three different quantities: an instantaneous hazard function with inverse-time units, a short-interval approximation, and an exact finite-interval probability.

For independent exponentially distributed failure times subject to independent right censoring, let $d$ be observed failures and $E={\sum }_i{x}_i$ total follow-up exposure. The parameter-dependent likelihood is ${\theta }^d{e}^{-\theta E}$, giving maximum-likelihood estimator $\hat{\theta }=d/E$. The fitted individual cumulative hazards sum to $d$, but this estimating identity alone does not establish that a constant hazard model fits.