Frailty model

A frailty model uses an unobserved positive random effect to represent individual heterogeneity in event hazards. Conditioning on survival progressively selects smaller frailties.

A shared frailty model assigns the same latent positive hazard multiplier $V_i$ to correlated event histories or group members. Gamma or lognormal frailty distributions are common examples. Modeling dependence through this latent effect yields conditional covariate effects; integrating out frailty can change the form of marginal hazards. Eligibility and event history still have to be encoded in the at-risk indicator.

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.

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 $H_0(t)$.

Let a frailty random variable have a gamma distribution with mean one and variance $v>0$. Its shape and rate are both $1/v$. For conditional hazard function $U{a}^z{h}_0(t)$, $z\in \{0,1\}$, averaging the conditional survivor function gives $(1+v{a}^z{H}_0(t){)}^{-1/v}$. Differentiating its logarithm yields the population hazard function $a^z{h}_0(t)/(1+v{a}^z{H}_0(t))$ and hence the displayed hazard ratio. Writing $x=v{H}_0(t)$ gives $r-1=(a-1)/(1+ax)$. Therefore the population hazard ratio moves monotonically from $a$ towards one as exposure $x$ increases, even though the individual conditional hazard ratio remains $a$. The limit one requires $H_0(t)\to \infty$ for fixed positive $v$, or $v\to \infty$ for fixed positive $H_0(t)$; it is not a consequence of elapsed time alone. This is survival selection through preferential removal of larger frailties.

Let the frailty random variable be uniform on $[a,b]$ with $0<a<b$, and let the conditional individual hazard function be the constant $U\theta$, $\theta >0$. Averaging conditional exponential survivor functions gives the displayed population survivor function for $t>0$, with value one at zero. Its population hazard of a survival mixture isIt starts at $\theta (a+b)/2$ and tends to $a\theta$: the frailty distribution among survivors progressively concentrates near its smallest frailty. Although each individual's hazard is constant, the population hazard decreases through survival selection.

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 $1+H_0(t)$.