A tutorial on frailty models

Abstract

The hazard function plays a central role in survival analysis. In a homogeneous population, the distribution of the time to event, described by the hazard, is the same for each individual. Heterogeneity in the distributions can be accounted for by including covariates in a model for the hazard, for instance a proportional hazards model. In this model, individuals with the same value of the covariates will have the same distribution. It is natural to think that not all covariates that are thought to influence the distribution of the survival outcome are included in the model. This implies that there is unobserved heterogeneity; individuals with the same value of the covariates may have different distributions. One way of accounting for this unobserved heterogeneity is to include random effects in the model. In the context of hazard models for time to event outcomes, such random effects are called frailties, and the resulting models are called frailty models. In this tutorial, we study frailty models for survival outcomes. We illustrate how frailties induce selection of healthier individuals among survivors, and show how shared frailties can be used to model positively dependent survival outcomes in clustered data. The Laplace transform of the frailty distribution plays a central role in relating the hazards, conditional on the frailty, to hazards and survival functions observed in a population. Available software, mainly in R, will be discussed, and the use of frailty models is illustrated in two different applications, one on center effects and the other on recurrent events.

Keywords: Correlated failure times; frailty models; random effects models; survival analysis; unobserved heterogeneity.

Figure 1.

Changes in the mean and variance of a covariate x over time among survivors in a proportional hazards model.

Figure 2.

Plot of scaled Schoenfeld residuals-based $\beta (t)$ induced by omitting a covariate from a proportional hazards model.

Figure 3.

Correlation between x₁ and x₂ among survivors over time.

Figure 4.

Frailty distribution of survivors, gamma frailty, $\lambda (t)=t^2/20$.

Figure 5.

Marginal hazard ratio between two groups of individuals: a low risk one with ${\lambda }_0(t)=t^2/20$, and a high risk one with ${\lambda }_1(t)=3{\lambda }_0(t)$. For comparability, the distributions are matched by the squared coefficient of variation of the distribution of survivors at time t = 1, with $C{V}^2(1)=\text{var}[Z|T\ge 1]/\text{E}{[Z|T\ge 1]}^2$.

Figure 6.

Conditional survival function of T₂, given $t_1=0.1$ and given $t_1=2$; the conditional distribution of T₁ and T₂ given X = x is exponential with rate $\lambda e^{\beta x}$, λ = 1 and β = 1, and X has a normal distribution with mean 0 and standard deviation ${\sigma }^2$, with different values of σ.

Figure 7.

Correlation between T₁ and T₂ as a function of ${\sigma }^2$; the conditional distribution of T₁ and T₂ given X = x is exponential with rate $\lambda e^{\beta x}$ and β = 1, and X has a normal distribution with mean 0 and variance ${\sigma }^2$.

Figure 8.

Cross-ratio (top row) and adjusted cross-ratio (bottom row, at $t_1=1$) for the gamma, inverse Gaussian and positive stable distributions, for different values of Kendall’s tau. The individual hazard, conditional on Z = 1, is taken as $\lambda (t)=t^2/20$.

Figure 9.

Histogram of center sizes.

Figure 10.

Kaplan–Meier survival estimates, overall and by center.

Figure 11.

Center effects from the fixed effects and frailty models, expressed in hazard ratios.

Figure 12.

Event history of the CGD data. The length of the line indicates the length of follow-up, and the dots indicate the infections.

Figure 13.

Histogram of number of events per individual.

Figure 14.

Histogram of the estimated frailties.