Proportional frailty model

A proportional frailty model has conditional hazard $h(t\mid U=u)=u{h}_0(t)$. Its conditional survival function is $e^{-u{H}_0(t)}$, so its population survival function is the Laplace transform of the frailty distribution evaluated at $H_0(t)$.

In a proportional frailty model, Bayes theorem tilts the original density $g(u)$ among subjects surviving to $t$ toThis survival selection progressively favors smaller frailties.

For unit-rate exponential distribution frailty, the population survival and hazard in a proportional frailty model areThe surviving frailty distribution is exponential with rate $1+H_0(t)$.