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For ${\sigma }^2>0$, a gamma distribution with unit mean and variance ${\sigma }^2$ has shape and rate $\psi =1/{\sigma }^2$. Substituting its Laplace transform into the preceding result givesIts negative logarithmic derivative is the population hazard function:Consequently, at times with $h_0(t)>0$, the gamma frailty hazard ratio isAt a time when both hazards vanish this formula is a continuous extension; the literal ratio $0/0$ is undefined.

Write $a=e^{\beta }>0$ and $x={\sigma }^2{H}_0(t)\ge 0$. Two useful identities areFor small cumulative exposure $x$, $r=a\{1+(1-a)x+O(x^2)\}$, so it starts at the conditional hazard ratio $e^{\beta }$. If $\beta >0$, it decreases towards one; if $\beta <0$, it increases towards one; and if $\beta =0$, it equals one throughout. It stays between $e^{\beta }$ and one and never crosses one.

For fixed ${\sigma }^2>0$, the late-time limit is one provided $H_0(t)\to \infty$. If instead the integrated hazard tends to a finite value $K$, the limiting hazard ratio is $a(1+{\sigma }^{2}K)/(1+a{\sigma }^{2}K)$, which need not be one. Elapsed time alone does not imply unbounded exposure.

For small ${\sigma }^2$ at a fixed time, $r\to e^{\beta }$. At ${\sigma }^2=0$ the frailty degenerates to $U=1$, giving an ordinary proportional hazards model. For large ${\sigma }^2$ at a fixed time with $H_0(t)>0$, $r\to 1$; at time zero, however, it remains $e^{\beta }$. Thus that variance limit is not uniform near zero. Larger heterogeneity accelerates the movement of the population hazard ratio towards one. The mechanism is survival selection: frailer individuals leave the population sooner, particularly in the higher-hazard group, while each individual's conditional multiplier remains fixed.