A new approach to regression analysis of censored competing-risks data

Abstract

An approximate likelihood approach is developed for regression analysis of censored competing-risks data. This approach models directly the cumulative incidence function, instead of the cause-specific hazard function, in terms of explanatory covariates under a proportional subdistribution hazards assumption. It uses a self-consistent iterative procedure to maximize an approximate semiparametric likelihood function, leading to an asymptotically normal and efficient estimator of the vector of regression parameters. Simulation studies demonstrate its advantages over previous methods.

Appendix

Proof of Lemma 1

To begin with, we have applied at the beginning of Sect. 3.2 the theory of integral equations to show the existence of $(\widetilde{A}_j,\widetilde{F}_j)$, $j=1,2$. We now use arguments involving exponential inequalities of empirical processes to relate quantities like ${\hat{A}}_j(t;B)$ and ${\hat{F}}_j(s-;Z_i,B)$ to their population versions, similar to those in Lai & Ying (1991, Appendix A; 1994, Sect. 4.2) to show that there exists $\rho >0$ such that

$$\begin{aligned}&\sup _{||b_1-\beta _1||+||b_2-\beta _2||\le \rho , 0<t < \tau } \left\{ \left| {\hat{A}}_{j}(t;B)-\widetilde{A}_{j}(t;B)\right| + ||D{\hat{A}}_j(t;B) - D\widetilde{A}_j(t;B)||\right\} \nonumber \\&\quad =O_p(n^{-1/2})\quad \text {for } j=1,2, \end{aligned}$$

(31)

where Dg denotes the Jacobian matrix of the partial derivatives of g with respect to the components of B and $||C|| = \max |c_{ij}|$ is the max-norm of a matrix $C=(c_{ij})$. A key tool in these arguments is provided by the aforementioned empirical process theory that enables us to approximate the averages of random functions by their expectations. Another important tool is to introduce the smooth weight function $p_n$ that satisfies (C1) and (C2) to circumvent difficulties when the denominator $1-{\hat{F}}_+(s-;Z_i,B)$ becomes too small. The smoothness of ${\hat{S}}_j$ and ${\hat{A}}_j$ makes the arguments considerably simpler than those in Lai and Ying (1991, (1994) since we can differentiate them with respect to the components of B and use assumptions (12) and (C2) to bound the derivatives. Before analyzing the partial derivatives in $D{\hat{A}}_j - D\widetilde{A}_j$, we first prove the result for ${\hat{A}}_j - \widetilde{A}_j$ in (31) and the corresponding result for ${\hat{F}}_j - \widetilde{F}_j$. Recalling that $\pi _n(u)=p_n(u)/u$, our basic idea is to identify ${\hat{A}}_j(t;B)$ in (8) and ${\hat{F}}_j(t-;Z, B)$ as sample versions of (16) that replaces $E\{p_n(\cdots )dN_j(s)\}$ in (16) by

$$\begin{aligned} n^{-1} \sum _{i=1}^n p_n(1 - {\hat{F}}_+(s-;Z_i, B))dN_{ij}(s), \end{aligned}$$

and $E\{I_{\{T\ge s\}} \cdots \exp (b_j^T Z(s))\}$ in (16) by $n^{-1}{\hat{S}}_j(s;B)$ defined in (9), in which ${\hat{F}}_j(t-;Z_i, B)$ corresponds to the second equation in (16) with $\widetilde{A}_j$ replaced by ${\hat{A}}_j$. Perturbation analysis (Bellman 1964; Khalil 2002) of the system of integral equations (16) together with exponential inequalities of empirical processes (Lai and Ying 1988, 1991) can then be used to show uniform convergence of these sample versions of (16) to their population counterparts $\widetilde{A}_j$ and $\widetilde{F}_j$ at rate $O_p(n^{-1/2})$. Making use of this and the exponential inequalities for empirical processes again, we can replace sample means in the integrands and integrators in the stochastic integral representation (35) of $n^{-1/2}U_j(B)$, thereby proving the uniform convergence of $n^{-1} U_j(b_1, b_2) - \widetilde{U}_j(b_1, b_2)$ in (19). Since $U_j(B)$ and $\widetilde{U}_j(B)$ are smooth functions of the components of B, $D{\hat{A}}_j - D\widetilde{A}_j$ and $D{\hat{F}}_j - D\widetilde{F}_j$ can be analyzed similarly, thereby proving (31), which can then be used to prove the corresponding result in $(\partial /\partial b_n)U_j(b_1, b_2)/n$ in (19). $\square$

Proof of Lemma 2

Let $\hat{p}_n(t-;Z_i)=p_n(1-{\hat{F}}_+(t-;Z_i,B_0))$ and define $p_n(t-;Z_i)$ similarly with ${\hat{F}}_+$ replaced by $F_+$. Note that

$$\begin{aligned}&\sqrt{n}(d{\hat{A}}_{j}(t)-dA_j(t))= \frac{1}{n^{-1}{\hat{S}}_{j}(t)}n^{-1/2}\sum _{i=1}^n\Bigg [\hat{p}_n(t-;Z_i)(dN_{ij}(t)-d\varLambda _{ij}(t))\nonumber \\&\quad -I_{\{T_i\ge t\}}e^{\beta _j^\top Z_i(t)}\left( \frac{\hat{p}_n(t-;Z_i)(1-{\hat{F}}_{j}(t-;Z_i))}{1-{\hat{F}}_{1}(t-;Z_i)-{\hat{F}}_{2}(t-;Z_i)}-\frac{\hat{p}_n(t-;Z_i)(1-F_j(t;Z_i))}{1-F_1(t;Z_i)-F_2(t;Z_i)}\right) dA_j(t)\Bigg ].\nonumber \\ \end{aligned}$$

(32)

To simplify the arguments and highlights the main ideas, we focus on the case where $Z_i$ does not vary with time, for which $1-F_j(t;Z_i)=\exp (-e^{\beta _j^\top Z_i}A_j(t))$, thus expressing $F_j(\cdot ;Z_i)$ as a smooth increasing function of $A_j(\cdot )$. In this case, the right-hand side of (32) has two terms, the first of which is

$$\begin{aligned} n^{-1/2}\sum _{i=1}^n\frac{p_n(t-;Z_i)}{S_j(t)}dM_{ij}(t)+ n^{-1/2}\sum _{i=1}^n\left\{ \frac{\hat{p}_n(t-;Z_i)}{n^{-1}{\hat{S}}_j(t)}-\frac{p_n(t-;Z_i)}{S_j(t)}\right\} dM_{ij}(t).\nonumber \\ \end{aligned}$$

(33)

The second summand in (33) can be written as $n^{-1/2}\sum _{i=1}^n(\epsilon _{n,ij}(t)/S_j(t))dM_{ij}(t)$, in which $\epsilon _{n,ij}(t)$ is a predictable process such that $\max _{t<\tau , 1 \le i \le n}|\epsilon _{n,ij}(t)|/p_n(t-;Z_i)\mathop {\longrightarrow }\limits ^{P}0$. Similarly, the second term on the right-hand side of (32) can be expressed as

$$\begin{aligned} \left\{ n^{1/2}\left( {\hat{A}}_j(t)-A_j(t)\right) \psi _{jj}(t)+n^{1/2}\left( {\hat{A}}_{\bar{j}}(t)-A_{\bar{j}}(t)\right) \psi _{\bar{j}j}(t)+\delta _{n,j}(t)\right\} dA_j(t)\Big /S_j(t), \end{aligned}$$

in which $\bar{j}\ne j$, $\delta _{n,j}(t)$ is a predictable process such that $\int ^{\tau }_0(|\delta _{n,j}(t)|/S_j(t))dA_j(t)\mathop {\longrightarrow }\limits ^{P}0$, and $\psi _{jj}(t)dA_j(t)/S_j(t)$ and $\psi _{\bar{j}j}(t)dA_j(t)/S_j(t)$ are the (j, j)th and $(\bar{j},j)$th elements of $d\varPsi _n(t)$, respectively. This follows from applying the first-order Taylor expansion to the summands and then the law of large numbers to the sum in the second term on the right-hand side of (32).

Expressing $1-F_j(\cdot ;Z_i)$ as a function of $A_j(\cdot )$ as noted at the beginning of last paragraph, we can take the derivative of the quotient in (32) with respect to $A_j$, and also with respect to $A_{\bar{j}}$ because the denominator involves both $F_j$ and $F_{\bar{j}}$, explaining the formula in (23) in which the expectation is a consequence of the law of large numbers. Replacing the $o_p(1)$ term in (24) by

$$\begin{aligned}&\int _0^t\frac{1}{\sqrt{n}}\sum _{i=1}^n\left( \epsilon _{n,i1}(s)\frac{dM_{i1}(s)}{S_1(s)},\epsilon _{n,i2}(s)\frac{dM_{i2}(s)}{S_2(s)}\right) ^\top \nonumber \\&\quad +\int _0^t\left( \delta _{n,1}(s)\frac{dA_1(s)}{S_1(s)},\delta _{n,2}(s)\frac{dA_2(s)}{S_2(s)}\right) ^\top , \end{aligned}$$

the solution (25) of (24) follows from Theorem II.6.3 in Andersen et al. (1993).

For time-varying $Z_i$, we can modify the preceding argument. Although the basic idea is the same, the technical details are more complicated as the Taylor expansion has to be carried out directly instead of by differentiation with respect to $A_j$; see (34) below for example. Since $dN_{i1}(t)dN_{i2}(t)=0$, $\langle M_{i1}, M_{i2} \rangle =0$, i.e., $M_{i1}$ and $M_{i2}$ are orthogonal martingales, and therefore we can apply the martingale central limit theorem (Andersen et al. 1993) and (25) to conclude that (26) converges weakly to a Gaussian martingale in $D^k[0, \tau ]$. $\square$

Proof of Lemma 3

Since ${\hat{F}}_j$ is defined by replacing $A_j$ in (5) by ${\hat{A}}_j$, it follows that

$$\begin{aligned} {\hat{F}}_j(t;Z)-F_j(t;Z)=(1-F_j(t;Z))\left[ 1-\exp \left\{ -\int _0^te^{\beta _j^TZ(s)}d({\hat{A}}_j(s)-A_j(s))\right\} \right] .\nonumber \\ \end{aligned}$$

(34)

We next combine this with (24) and apply Taylor’s theorem and the law of large numbers for the i.i.d. $(Z_i, C_i)$ to decompose $n^{1/2}U_j(B_0)$ into six martingales summands plus a $o_p(1)$ remainder.

From (13), (24) and (5), it follows that $n^{-1/2}U_j(B_0)$ can be written as

$$\begin{aligned}&\frac{1}{\sqrt{n}}\sum _{i=1}^n \int _0^{T_i}p_n\left( 1-{\hat{F}}_{{\mathrm {\scriptscriptstyle +}}}(t-;Z_i)\right) \eta _{ij}(t)dM_{ij}(t)\nonumber \\&\quad +\frac{1}{\sqrt{n}}\sum _{i=1}^n \int _0^{T_i}p_n(1-{\hat{F}}_{{\mathrm {\scriptscriptstyle +}}}\left( t-;Z_i\right) ) \phi _{ij}(t)dM_{i\bar{j}}(t)\nonumber \\&\quad -\frac{1}{\sqrt{n}} \sum _{i=1}^n\int _0^{T_i}p_n(1-{\hat{F}}_{{\mathrm {\scriptscriptstyle +}}}\left( t-;Z_i\right) )\nonumber \\&\quad \times \left[ \frac{1-{\hat{F}}_j(t-;Z_i)}{1-{\hat{F}}_1(t-;Z_i)-{\hat{F}}_2(t-;Z_i)}- \frac{1-F_j(t;Z_i)}{1-F_1(t;Z_i)-F_2(t;Z_i)}\right] \nonumber \\&\quad \times {\hat{\eta }}_{ij}(t)e^{\beta _j^\top Z_i(t)}d{\hat{A}}_j(t) \nonumber \\&\quad -\frac{1}{\sqrt{n}} \sum _{i=1}^n\int _0^{T_i}p_n(1-{\hat{F}}_{{\mathrm {\scriptscriptstyle +}}}\left( t-;Z_i\right) )\nonumber \\&\quad \times \left[ \frac{1-{\hat{F}}_{\bar{j}}(t-;Z_i)}{1-{\hat{F}}_1(t-;Z_i)-{\hat{F}}_2(t-;Z_i)}- \frac{1-F_{\bar{j}}(t;Z_i)}{1-F_1(t;Z_i)-F_2(t;Z_i)}\right] \nonumber \\&\quad \times {\hat{\phi }}_{ij}(t)e^{\beta _{\bar{j}}^\top Z_i(t)}d{\hat{A}}_{\bar{j}}(t)\nonumber \\&- \frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^{T_i}p_n(1-{\hat{F}}_{{\mathrm {\scriptscriptstyle +}}}\left( t-;Z_i\right) )\nonumber \\&\quad \times \left[ \frac{\eta _{ij}(t)\left\{ 1-F_j(t;Z_i)\right\} }{1-F_1(t;Z_i)-F_2(t;Z_i)}e^{\beta _j^\top Z_i(t)}\right] d\left( {\hat{A}}_j(t)-A_j(t)\right) \nonumber \\&-\frac{1}{\sqrt{n}}\sum _{i=1}^n\int _0^{T_i}p_n(1-{\hat{F}}_{{\mathrm {\scriptscriptstyle +}}}\left( t-;Z_i\right) )\nonumber \\&\quad \times \left[ \frac{\phi _{ij}(t)\left\{ 1-F_{\bar{j}}(t;Z_i)\right\} }{1-F_1(t;Z_i)-F_2(t;Z_i)}e^{\beta _{\bar{j}}^\top Z_i(t)}\right] d\left( {\hat{A}}_{\bar{j}}(t)-A_{\bar{j}}(t)\right) + o_p(1). \end{aligned}$$

(35)

Making use of (24), it can be shown similarly that the third summand in (35) can be expressed as

$$\begin{aligned}&\sqrt{n}\int _0^{\tau } \int _0^t E\{ \eta _{ij}(t)I_{\{T_i \ge t\}}e^{\beta _j^\top (Z_i(t)+Z_i(s))}\left[ \frac{F_{\bar{j}}(t;Z_i)(1-F_j(t;Z_i))}{(1-F_1(t;Z_i)-F_2(t;Z_i))^2}\right] \nonumber \\&\quad \times p_n(1-F_{{\mathrm {\scriptscriptstyle +}}}(t; Z_i))\}d({\hat{A}}_j(s)-A_j(s))dA_j(t) \nonumber \\&\quad +\sqrt{n}\int _0^{\tau } \int _0^t E\Big \{ \eta _{ij}(t)I_{\{T_i \ge t\}}e^{\beta _j^\top Z_i(t) + \beta _{\bar{j}}^\top Z_i(s)} \left[ \frac{(1-F_1(t;Z_i))(1-F_2(t;Z_i))}{(1-F_1(t;Z_i)-F_2(t;Z_i))^2}\right] \nonumber \\&\quad \times p_n(1-F_{{\mathrm {\scriptscriptstyle +}}}(t; Z_i))\Big \}d({\hat{A}}_{\bar{j}}(s)-A_{\bar{j}}(s))dA_j(t)+o_p(1). \end{aligned}$$

(36)

Replacing $(j,\bar{j})$ by $(\bar{j},j)$ and $\eta _{ij}$ by $\phi _{ij}$ in (36), the fourth summand in (35) can be expressed similarly. The terms inside the fifth and the sixth summands in (35) can also be replaced by the corresponding expected values, again using the law of large numbers. Hence Lemma 3 follows from (35) and the weak convergence of (26) to a Gaussian martingale. $\square$

Proof of Theorem 2

Although the nuisance parameters $F_1$ and $F_2$ in the semiparametric likelihood (7) are infinite-dimensional, we can use parametric submodels such that the information bound for estimating B in the parametric submodel is attained by ${\hat{B}}$, proving its asymptotic efficiency which is formulated in Theorem 2 in terms of the Hájek convolution theorem; see Andersen et al. (1993, Sections VIII.1.3 and VIII.2.4) and Vaart (1998, Chap. 25). Moreover, standard likelihood theory for the least favorable parametric submodel shows that $V=-\varSigma$. In fact, if we impose the stronger assumptions that (a) the support ${\mathcal {Z}}$ of $Z_i$ is bounded, (b) ess $\sup C_i=\tau <\infty$, and (c) $\inf _{z\in {\mathcal {Z}}}\{1-F_1(\tau ,z)-F_2(\tau ,z)\}>0$, then we do not need to introduce the weight function $p_n(\cdot )$ in (10), which then corresponds to the semiparametric log-likelihood function considered by Zeng and Lin (2007, (2010), whose asymptotic efficiency theory for semiparametric maximum likelihood estimators can be applied to this case. Removing these assumptions requires modification of the likelihood function to address the difficulties caused by small values of $1-{\hat{F}}_+(T_i-;Z_i,B)$. This is the idea, introduced by Lai and Ying (1991, (1992, (1994), behind the smooth weight function $p_n(1-{\hat{F}}_+(t-;Z_i,B))$ that is asymptotically negligible and can yield the same information bound as that under the additional restrictive assumptions (a), (b), and (c).