Analysis of the time-varying Cox model for the cause-specific hazard functions with missing causes

Abstract

This paper studies the Cox model with time-varying coefficients for cause-specific hazard functions when the causes of failure are subject to missingness. Inverse probability weighted and augmented inverse probability weighted estimators are investigated. The latter is considered as a two-stage estimator by directly utilizing the inverse probability weighted estimator and through modeling available auxiliary variables to improve efficiency. The asymptotic properties of the two estimators are investigated. Hypothesis testing procedures are developed to test the null hypotheses that the covariate effects are zero and that the covariate effects are constant. We conduct simulation studies to examine the finite sample properties of the proposed estimation and hypothesis testing procedures under various settings of the auxiliary variables and the percentages of the failure causes that are missing. These simulation results demonstrate that the augmented inverse probability weighted estimators are more efficient than the inverse probability weighted estimators and that the proposed testing procedures have the expected satisfactory results in sizes and powers. The proposed methods are illustrated using the Mashi clinical trial data for investigating the effect of randomization to formula-feeding versus breastfeeding plus extended infant zidovudine prophylaxis on death due to mother-to-child HIV transmission in Botswana.

Appendices

Appendix

This Appendix introduces the notations and presents the conditions for the asymptotic results presented in Theorems 1–5.

Let $\mathcal {F}_t$ be the right continuous filtration generated by the data processes $\{N_{ik}(s), Y_i(s), Z_i(s); i=1,\dots ,n, k=1,2,\dots ,L, 0 \le s \le t\}$. Assume $E(dN_{ik}(t)=1|\mathcal {F}_{t^-})=E(dN_{ik}(t)=1|Y_i(t), Z_i(t))=Y_i(t)\lambda _{ik}(t|Z_i(t))dt$. It follows that $M_{ik}(t) = N_{ik}(t) -\int _0^t Y_i(u)\lambda _k(u|Z_i(u))du$, $i=1,\dots ,n, k=1,2,\dots ,L$, are multivariate orthogonal martingales with respect to $\mathcal {F}_t$ (Aalen and Johansen 1978). To accommodate additional information introduced due to missing data, we define the augmented filtration $\mathcal {F}_t^*$ generated by the data processes $\{N_{ik}(s), Y_i(s), Z_i(s), R_i, \delta _i A_i; i=1,\dots ,n, k=1,2,\dots ,L, 0 \le s \le t\}$. Let $\lambda _{ik}^*(t)\,dt=P\{T_i\in [t,t+dt),V_i=k|X_i \ge t, Z_i(t), R_i, \delta _i A_i\}$. Then $Y_i(t)\lambda _{ik}^*(t)$ is the intensity of $N_{ik}(t)$ with respect to $\mathcal {F}_t^*$, and $M_{ik}^*(t) = N_{ik}(t) -\int _0^t Y_i(u)\lambda _{ik}^*(u)du$, $i=1,\dots ,n, k=1,2,\dots ,L$, are multivariate orthgonal martingales with respect to $\mathcal {F}_t^*$.

Let $S^{(j)}(t,\beta _k)=n^{-1}\sum _{i=1}^{n} Y_i(t)\exp \left( \beta _k(t)^{{\textsf {T}}}Z_i(t)\right) Z_i(t)^{\otimes j}$, and $S_I^{*(j)}(t,\beta _k,\psi )=n^{-1}\sum _{i=1}^n q_i$$Y_i(t)\exp \left( \beta _k(t)^{{\textsf {T}}}Z_i(t)\right) Z_i(t)^{\otimes j}$, for $k=1,\dots ,L$ and $j=0,1,2$. Let $s^{(j)}(t,\beta _k)=ES^{(j)}(t,\beta _k)$ and $s_I^{*(j)}(t,\beta _k,\psi )=ES_I^{*(j)}(t,\beta _k,\psi )$. If the model $r(\zeta _i, A_i, \psi )$ is correctly specified, then $s^{(j)}(t,\beta _k)$$= s_I^{*(j)}(t,\beta _k,\psi )$. Define $\Sigma _k(t) = \big [s^{(2)}(t,\beta _k) - {\big (s^{(1)}(t,\beta _k)\big )^{\otimes 2}}/$${s^{(0)}(t,\beta _k)}\big ] \lambda _{k0}(t)$ and $\Sigma _k^*(t)=E\big [ \big (Z_i(t)-$${s^{(1)}(t,\beta _k)}/{s^{(0)}(t,\beta _k)} \big )^{\otimes 2}$${R_i}{\pi ^{-2}(Q_i)} Y_{i}(t) \lambda _{ik}^*(t) \big ]$.

Let $S_i^{\psi }$ and $I^{\psi }$ be the score vector and information matrix for ${{\widehat{\psi }}}$ under (4). Then,

$$\begin{aligned}&S_i^{\psi } = \frac{\delta _i(R_i-r(\zeta _i, A_i,\psi _0))}{r(\zeta _i, A_i,\psi _0)(1-r(\zeta _i, A_i,\psi _0))} \frac{\partial r(\zeta _i, A_i,\psi _0)}{\partial \psi },\\&I^{\psi } = E\Bigg \{ \frac{\delta _i}{r(\zeta _i, A_i,\psi _0)(1-r(\zeta _i, A_i,\psi _0))} \frac{\partial r(\zeta _i, A_i,\psi _0)}{\partial \psi } \Bigg (\frac{\partial r(\zeta _i, A_i,\psi _0)}{\partial \psi } \Bigg )^{{\textsf {T}}} \Bigg \}, \end{aligned}$$

and ${{\widehat{\psi }}} - \psi = n^{-1}\sum _{i=1}^n(I^{\psi })^{-1}S_i^{\psi }+o_p(n^{-1/2})$, where $\psi _0$ is the true value of $\psi$. We also define the following notations:

$$\begin{aligned}&{\mathcal {A}}_{i}(t,\beta _k)=\int _0^{\tau } K_h(u-t) H^{-1}\Bigg (Z_i(u)-\frac{s^{(1)}(u,\beta _k)}{s^{(0)}(u,\beta _k)} \Bigg ) q_{i0}\ dM_{ik}(u),\\&{\mathcal {B}}_{i}(t,\beta _k)=\int _0^{\tau } K_h(u-t) H^{-1}\Bigg (Z_i(u)-\frac{s^{(1)}(u,\beta _k)}{s^{(0)}(u,\beta _k)} \Bigg )(1-q_{i0})\ E(dM_{ik}(u)|Q_{i}),\\&\mathcal {D}^n(t,\beta _k) = n^{-1}\sum _{i=1}^n \int _0^{\tau }K_h(u-t)\Bigg (Z_i(u)-\frac{s^{(1)}(u,\beta _k)}{s^{(0)}(u,\beta _k)}\Bigg ) \dfrac{-R_i}{(\pi (Q_i,\psi _0))^2}\\&\quad \Bigg (\dfrac{\partial \pi (Q_i,\psi _0)}{\partial \psi }\Bigg )^{\textsf {T}}dM_{ik}(u),\\&\mathcal {O}_i(t,\beta _k) = \mathcal {D}^n(t,\beta _k)(I^{\psi })^{-1}S_i^{\psi }. \end{aligned}$$

The following conditions are assumptions we use to prove the theorems:

• (C.1) For $k=1,\dots ,L$, $\beta _k(t)$ has componentwise second derivatives on $[0,\tau ]$. The sample path of the covariate process $Z_i(t)$ is left continuous and of bounded variation, and satisfies the moment condition $E[||Z_i(t)||^4\exp (2M||Z_i(t)||)]<\infty$, where M is a constant such that $(t,\beta _k(t))\in [0,\tau ]\times [-M,M]^p$ for all t and $||A||=\max _{k,l}|a_{kl}|$ for a matrix $A=(a_{kl})$.

• (C.2) The kernel function $K(\cdot )$ is bounded and symmetric with bounded support $[-1,1]$. The bandwidth h satisfies $nh^2\rightarrow \infty$ and $nh^5$ is bounded as $n \rightarrow \infty$.

• (C.3) The matrix $\Sigma _k(t)$ is positive definite for all $t \in [0, \tau ]$.

• (C.4) For $k=1,\dots ,L$ and for $j=0,1,2$, the functions $s^{(j)}(t,\beta _k)$ and $s_I^{*(j)}(t,\beta _k,\psi )$ are componentwise continuous on $t\in [0,\tau ],\beta _k\in [-M,M]^p,\psi \in \varTheta _{\psi }$, where $\varTheta _{\psi }$ is a compact set. $\sup _{t\in [0,\tau ],\beta _k\in [-M,M]^p}||S^{(j)}(t,\beta _k)-s^{(j)}(t,\beta _k)||=O_p(n^{-1/2})$, and $\sup _{t\in [0,\tau ],\beta _k\in [-M,M]^p,\psi \in \varTheta _{\psi }}$$||S_I^{*(j)}(t,\beta _k,\psi )-s_I^{*(j)}(t,\beta _k,\psi )||=O_p(n^{-1/2}).$

• (C.5) The function $r(\zeta _i,A_i,\psi )$ is twice differentiable with respect to $\psi$ on a compact set $\varTheta _{\psi }$, $r^\prime (\zeta _i,A_i,\psi )=\partial r(\zeta _i,A_i,\psi )/\partial \psi$ is uniformly bounded, and there is an $\varepsilon > 0$ such that $r(\zeta _i,A_i,\psi ) \ge \varepsilon$ for all i. The function $f(A_i |k,T_i,Z_i,\varphi _k)$ is also twice differentiable with respect to $\varphi _k$ on a compact set $\varTheta _{\varphi _k}$ for $k=1,\dots ,L$.

Supplementary materials

The Web-based Supplementary Materials referenced in the manuscript are available with this paper at the journal’s online website.