11.3 Frailty models
Some subjects are more prone to the event than their covariates say, because of genes, environment or unmeasured severity. A frailty model puts this into the hazard as an unobserved multiplier \(Z\), \[ h(t \mid x, Z) = Z\,h_0(t)e^{\beta ^{\mathsf T}x} , \] usually with \(Z\) gamma distributed with mean \(1\) and variance \(\theta \). In a shared frailty model, all subjects of one cluster (twins, a litter, the repeated events of one patient) share the same \(Z\), which makes their times correlated, with larger \(\theta \) meaning stronger dependence; \(\theta = 0\) gives the ordinary Cox model.
Proposition 11.3.1. With gamma frailty of variance \(\theta \), the population survivor function and hazard are \[ S(t \mid x) = \big \{1 + \theta H_0(t)e^{\beta ^{\mathsf T}x}\big \}^{-1/\theta } , \qquad h(t \mid x) = \frac {h_0(t)e^{\beta ^{\mathsf T}x}}{1 + \theta H_0(t)e^{\beta ^{\mathsf T}x}} . \]
Proof. Given \(Z\), \(S(t \mid x, Z) = \exp \{-ZH_0(t)e^{\beta ^{\mathsf T}x}\}\). Averaging over \(Z\) with density \(z^{1/\theta - 1}e^{-z/\theta }/\{\Gamma (1/\theta )\theta ^{1/\theta }\}\) gives the Laplace transform of the gamma distribution, \(E[e^{-sZ}] = (1 + \theta s)^{-1/\theta }\), at \(s = H_0(t)e^{\beta ^{\mathsf T}x}\). The hazard is \(-\frac {\mathrm d}{\mathrm dt}\log S\). □
The population hazard ratio between \(x = 1\) and \(x = 0\) is therefore \(e^{\beta }\{1 + \theta H_0(t)\}/\{1 + \theta H_0(t)e^{\beta }\}\): it equals \(e^\beta \) at the start and moves towards \(1\) as \(H_0(t)\) grows, because the frail die early in the high-risk group and the survivors become a selected, robust population. Unmeasured heterogeneity alone can thus produce the fading hazard ratios of the section on checking proportional hazards, and a model that ignores it estimates a marginal effect closer to \(1\) than the conditional effect \(e^\beta \) for a given subject. An alternative that estimates the marginal effect directly is an ordinary Cox model with robust (sandwich) standard errors clustered on the subject, cluster(id) in R, which corrects the inference for the correlation without modelling it.
Example 11.3.2. The kidney data record, for \(38\) patients on portable dialysis, the times to infection at the catheter insertion point, twice per patient. A shared gamma frailty model with age and sex gives \(\hat \theta = 0.41\) (\(p = 0.04\) for \(\theta = 0\)): patients differ in their susceptibility. The conditional hazard ratio for women against men is \(0.20\) (\(p = 0.0006\)), while the marginal model with robust errors gives \(0.44\) (\(p = 0.09\)) (Software 11.5.3). Both say women have fewer infections; the conditional ratio, comparing a woman with a man of the same frailty, is further from \(1\), as the discussion above predicts.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.