The distribution of unobserved heterogeneity in competing risks models

Abstract

We show that in a large class of proportional hazard competing risks models, the distribution of the bivariate frailty among survivors converges to a limiting distribution. This generalizes the result of Abbring and van den Berg (Biometrika 94(1):87–99, 2007), who show that in a single spell duration model, the frailty distribution converges to the gamma distribution. The resulting limiting distribution has an interpretation in terms of partition of a gamma distribution, and allows for both positive and negative correlation between the survival variables. This result provides a natural and flexible specification for the frailty in competing risks models.

Appendices

Appendix 1

Proofs

Proof of Property 2

$$\begin{aligned} \frac{\mathbb {P}[T_1<T_2|\min (T_1, T_2)=t]}{\mathbb {P}[T_1>T_2|\min (T_1, T_2)=t]}= & {} \frac{h_1(t)}{h_2(t)}=\frac{\lambda _1(t)\mathbb {E}[U|T_1>t, T_2>t] }{\lambda _2(t)\mathbb {E}[V|T_1>t, T_2>t]}\\\sim & {} \ell \frac{\mathbb {E}[\Lambda (t)U|T_1>t, T_2>t] }{\mathbb {E}[\Lambda (t)V|T_1>t, T_2>t]}, \end{aligned}$$

which converges to a positive number $\ell '$ if the distribution $\Lambda (t)(U, V)| T_1>t, T_2>t$. Thus $\mathbb {P}[T_1<T_2|\min (T_1, T_2)=t]=\frac{h_1(t)}{h_1(t)+h_2(t)}$ converges to $\frac{\ell '}{1+\ell '}$.

Proof of Theorem 1

The convergence of the distribution of $(\Lambda (t)U,\Lambda (t)V)|T_1>t, T_2>t$, or equivalently that of $(\Lambda _1(t)U,\Lambda _2(t)V)| T_1>t, T_2>t$, is equivalent to the convergence of the Laplace-Stieltjes transform of the cdf of the latter:

$$\begin{aligned} \mathcal {L}_2(x,y)&= \displaystyle \frac{\mathbb {E}\big [e^{-\Lambda _1(t)Ux-\Lambda _2(t)Vy}e^{-\Lambda _1(t)U-\Lambda _2(t)V}\big ]}{\mathbb {E}\big [e^{-\Lambda _1(t)U-\Lambda _2(t)V}\big ]} = \displaystyle \frac{\mathbb {E}\big [e^{-(x+1)\Lambda _1(t)U-(y+1)\Lambda _2(t)V}\big ]}{\mathbb {E}\big [e^{-\Lambda _1(t)U-\Lambda _2(t)V}\big ]}\\&=\displaystyle \frac{\iint e^{-(1+x)u-(1+y)v}dF(\frac{u}{\Lambda _1(t)}, \frac{v}{\Lambda _2(t)})}{\mathbb {E}\big [e^{-\Lambda _1(t)U-\Lambda _2(t)V}\big ]} \end{aligned}$$

Thus its convergence implies the pointwise convergence of: $\frac{F(\frac{u}{\Lambda _1(t)}, \frac{v}{\Lambda _2(t)})}{\mathbb {E}\big [e^{-\Lambda _1(t)U-\Lambda _2(t)V}\big ]}$ to a measure k(u, v), say. In particular, by taking $u=v=1$ we also get: $\displaystyle \frac{F(\frac{u}{\Lambda _1(t)}, \frac{v}{\Lambda _2(t)})}{F(\frac{1}{\Lambda _1(t)}, \frac{1}{\Lambda _2(t)})} \rightarrow \frac{k(u, v)}{k(1, 1)}$, or equivalently (by the monotonic property of F): $\displaystyle \frac{F(\frac{u}{\Lambda (t)}, \frac{v}{\Lambda (t)})}{F(\frac{1}{\Lambda (t)}, \frac{1}{\Lambda (t)})} \rightarrow \frac{k(u/a_1, v/a_2)}{k(1/a_1, 1/a_2)}$. Thus F is regularly varying at zero.

Conversely, if F is regularly varying at zero, then, under some regularity conditions, the extended continuity theorem (Feller 2008, Theorem 2, Chapter XIII.1) applies and the previous steps can be reversed. $\square$

Proof of Property 9

(1) By Feller (2008), Chapter XIII.5, the regular variation at zero of $a_1 U+a_2 V$ is equivalent to the regular variation of its Laplace transform at infinity. When t goes to infinity, we have:

$$\begin{aligned} \displaystyle \frac{\mathbb {E}[e^{-(a_1U+a_2V)ct}]}{\mathbb {E}[e^{-(a_1U+a_2V)t}]} = \frac{\mathbb {E}[e^{-(a_1U+a_2V)ct}]/\mathbb {E}[e^{-(U+V)t}]}{\mathbb {E}[e^{-(a_1U+a_2V)t}]/\mathbb {E}[e^{-(U+V)t}]} \sim \frac{\mathcal {L}_{\nu }(a_1c, a_2c)}{\mathcal {L}_{\nu }(a_1, a_2)} =c^{-\alpha } \end{aligned}$$

by the homogeneity of $\nu$. Thus $a_1 U+a_2 V$ is $RV_0(\alpha )$.

Proof of Property 5

If (U, V) follows (6), then the pdf of $\Lambda (t)(U, V)$ given $T_1>t, T_2>t$ is

$$\begin{aligned} f_t(u, v) \propto e^{-\frac{\Lambda _1(t)}{\Lambda (t)}u-\frac{\Lambda _2(t)}{\Lambda (t)}v}l\left( \frac{u}{\Lambda (t)}, \frac{v}{\Lambda (t)}\right) \mu \left( \frac{u}{\Lambda (t)},\frac{v}{\Lambda (t)}\right) . \end{aligned}$$

(19)

Since $\Lambda _1(t) \sim a_1 \Lambda (t)$, $\Lambda _2(t) \sim a_2 \Lambda (t)$, $l(\frac{u}{\Lambda (t)}, \frac{v}{\Lambda (t)}) \sim l(\frac{1}{\Lambda (t)}, \frac{1}{\Lambda (t)})$, and $\mu$ is homogeneous, the distribution (19) converges to: $f_{\infty }(u, v) \propto e^{-a_1u-a_2v}\mu (u, v)$. Therefore, (U, V) is $BRV_0(\nu )$ by Theorem 1.

Conversely, if the cdf F is regularly varying at zero with a limit measure $\nu$, then, under some regularity conditions (see e.g., Omey and Willekens 1989), the pdf f is also regularly varying, with:

$$\begin{aligned} \displaystyle \lim _{a \rightarrow 0} \frac{f(ax, ay)}{f(a, a)}= \frac{\mu (x, y)}{\mu (1, 1)}. \end{aligned}$$

(20)

Let us define $l(x, y)=\frac{f(x, y)}{\mu (x, y)}$; then by Eq. (20), we can check that l is slowly varying at zero. $\square$