A <frailty random variable> is an unobserved positive multiplicative risk factor. A proportional frailty model has
$$
h(t\mid U)=Uh_0(t),
\qquad
S(t\mid U)=e^{-UH_0(t)}.
$$
The scale of $U$ is not separately identifiable from $h_0$: multiplying $U$ by a constant and dividing $h_0$ by it leaves the model unchanged. We may therefore normalize $\mathbb EU=1$, which lets $h_0$ represent the mean initial hazard multiplier and makes relative frailty interpretable.

If $U\sim\operatorname{Exponential}(1)$ and $h_0(t)=\lambda$, then the <Laplace transform> of $U$ gives
$$
S(t)=\mathbb E[e^{-U\lambda t}]
=\frac1{1+\lambda t},
\qquad
h(t)=-\frac d{dt}\log S(t)
=\boxed{\frac\lambda{1+\lambda t}}.
$$
Thus $h(0)=\lambda$, while $h(t)\to0$ as $t\to\infty$. Survivors become increasingly enriched for low-frailty individuals.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 6 / a / Solution

{title2=$U$}

A frailty random variable is the positive latent multiplier applied to an individual's hazard. Its scale is conventionally normalized, often by setting $\mathbb E U=1$, because it is confounded with the scale of the baseline hazard.


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 Incoming links: Frailty random variable