Parametric estimation of association in bivariate failure-time data subject to competing risks: sensitivity to underlying assumptions

Abstract

There has arisen a considerable body of research addressing the estimation of association between paired failure times in the presence of competing risks. In a 2002 paper, Bandeen-Roche and Liang proposed the conditional cause-specific hazard ratio (CCSHR) as a measure of this association and a parametric method by which to estimate it. The method features an interpretable decomposition of the CCSHR into factors describing the association between a pair’s times to first failure among multiple failure causes and the association in pair members’ propensities to fail due to a common cause. There were indications of sensitivity to model assumptions, however, in the 2002 work. Here we report a detailed study of the method’s sensitivity to its parametric assumptions. We conclude that the method’s performance is most sensitive to mis-specification of temporality in the association between pair members’ first-failure times and of correlation between propensity to fail early or late and the propensity to fail of a specific cause. Implications for methods development are highlighted.

A Generation of correlated failure times with marginally exponential distribution

To generate ‘disease’ failure time of the first component of a pair, we generated gamma distributed random numbers as in the third set of simulation studies (see methods). Then we used the fact that $\dfrac{\log (1-\log (U)/A)}{l_1 \times (t-1)}$ is exponentially distributed where U is uniformly distributed, $l_1$ is the exponential parameter, and A is gamma distributed with a shape parameter $1/(t-1)$ and a scale parameter 1. The ‘disease’ failure time for the second component and the ‘death’ failure times for two components were generate similarly.

To see that this method yields the distributions as claimed, let us consider a univariate frailty model with a random effect denoted by $\alpha$, with distribution G and Laplace transformation $p(x) = E( e^{-x \alpha })$, where the marginal survival function for individual j in the cluster is $S_j(t) = \int \lbrace S_j^*(t) \rbrace ^a dG(a)$. Then, $-\,\log S_j^* (t) = q[S_j(t)]$, that is, $S_j(t) = p[-\,\log S_j^*(t)]$ where q is the inverse function of p [see Eq. (1) of Bandeen-Roche and Liang (1996)]. For exponential distribution, $S_j(t) = e^{-\lambda t}$ and for Clayton copula, $p(u) = (1+u)^{\frac{1}{1-\theta }}$. Thus,

$$\begin{aligned} e^{-\lambda T}= & {} \left( 1 - \log S_j^* (T)\right) ^{\frac{1}{1-\theta }} \nonumber \\ - \lambda T= & {} \dfrac{1}{1-\theta } \log \left( 1 - \log S_j^*(t)\right) \\ T= & {} \dfrac{1}{\lambda (\theta -1)} \log \left( 1 - \log S_j^*(t)\right) \nonumber \end{aligned}$$

(13)

For gamma frailty, conditionally on frailty, $S_j^*(T)^A$ is uniformly distributed, thus $\log ( S_j^* (T)) = \log (U)/A$. Then,

$$\begin{aligned} T = \dfrac{1}{\theta -1} \log \left( 1-\log (U)/A\right) . \end{aligned}$$

(14)