= Gamma frailty hazard ratio
{title2=$r(t)=a\frac{1+vH_0(t)}{1+vaH_0(t)}$}
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> $Ua^zh_0(t)$, $z\in\{0,1\}$, averaging the conditional <survivor function> gives $(1+va^zH_0(t))^{-1/v}$. Differentiating its logarithm yields the population <hazard function> $a^zh_0(t)/(1+va^zH_0(t))$ and hence the displayed <hazard ratio>. Writing $x=vH_0(t)$ 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.
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