Reweighted estimators for additive hazard model with censoring indicators missing at random

Abstract

Survival data with missing censoring indicators are frequently encountered in biomedical studies. In this paper, we consider statistical inference for this type of data under the additive hazard model. Reweighting methods based on simple and augmented inverse probability are proposed. The asymptotic properties of the proposed estimators are established. Furthermore, we provide a numerical technique for checking adequacy of the fitted model with missing censoring indicators. Our simulation results show that the proposed estimators outperform the simple and augmented inverse probability weighted estimators without reweighting. The proposed methods are illustrated by analyzing a dataset from a breast cancer study.

Appendix

1.1 Regularity conditions and sketch proofs of the main results

For proofs of the theorems, we list the following regularity conditions.

• (C1): $\Lambda _{0}(\tau )<\infty$ and $\mathrm {Pr}\{Y(\tau )=1\}>0$;

• (C2): Z is bounded with probability 1 and time-independent;

• (C3): The matrix A is positive definite.

• (C4): The observation probability $\pi (W,\alpha )$ is bounded away from 0; $\pi (W,\alpha )$ is twice continuously differentiable in $\alpha$; There exists a compact neighborhood $\mathcal {A}$ of $\alpha _{0}$ such that $E[\mathrm {sup}_{\alpha \in \mathcal {A}}\{\Vert \dot{\pi }(W,\alpha )\Vert ^{2} +\Vert \ddot{\pi }(W,\alpha )\Vert \}]<\infty$, where $\dot{\pi }(W,\alpha )=\partial \pi (W,\alpha )/\partial \alpha$ and $\ddot{\pi }(W,\alpha )=\partial ^{2} \pi (W,\alpha )/\partial \alpha \partial \alpha ^{T}$; There exists $\alpha _{*}$ satisfying the equations $E(S_{\alpha _{*}}^{*})=0$, where $S_{\alpha }^{*}=(\xi -\pi (W,\alpha ))[\pi (W,\alpha )(1-\pi (W,\alpha ))]^{-1}\dot{\pi }(W,\alpha )$.

• (C5): $\rho (W,\gamma )$ is twice continuously differentiable in $\gamma$; There exists $\gamma _{*}$ satisfying the equations $E(S_{\gamma _{*}}^{*})=0$, where $S_{\gamma }^{*}=\xi (\delta -\rho (W,\gamma ))[\rho (W,\gamma )(1-\rho (W,\gamma ))]^{-1}\dot{\rho }(W,\gamma )$.

All these conditions are standard for the derivation of asymptotic results in the survival analysis and parametric inference.

Proof of Theorem 1

By some simple algebraic calculations, it can be seen that

$$\begin{aligned} n^{\frac{1}{2}}(\hat{\beta }_{sr}-\beta _{0})= & {} \Big [\frac{1}{n}\sum \limits _{i=1}^{n}\int \limits _{0}^{\tau }\xi _{i}\frac{\hat{\pi }^{*}(t)}{\pi (W_{i},\hat{\alpha })} \{Z_{i}-\bar{Z}_{sr}(t,\hat{\alpha })\}^{\otimes 2}Y_{i}(t)dt\Big ]^{-1} \\&\times \Big [n^{-\frac{1}{2}}\sum \limits _{j=1}^{n}\int \limits _{0}^{\tau }\xi _{j} \frac{\hat{\pi }^{*}(t)}{\pi (W_{j},\hat{\alpha })} \{Z_{j}-\bar{Z}_{sr}(t,\hat{\alpha })\}dM_{j}(t)\Big ]. \end{aligned}$$

Under conditions (C1), (C2) and (C4), it can be shown that

$$\begin{aligned} \mathrm {sup}_{t\in [0,\tau ]}\Vert \hat{\pi }^{*}(t)-\pi ^{*}(t)\Vert= & {} o_{p}(1), \end{aligned}$$

(A.1)

$$\begin{aligned} \mathrm {sup}_{t\in [0,\tau ]}\Vert S_{sr}^{(k)}(t,\hat{\alpha })-s^{(k)}(t)\Vert= & {} o_{p}(1), k=0,1,2, \end{aligned}$$

(A.2)

$$\begin{aligned} \mathrm {sup}_{t\in [0,\tau ]}\Vert \bar{Z}_{sr}(t,\hat{\alpha })-\bar{z}(t)\Vert= & {} o_{p}(1). \end{aligned}$$

(A.3)

By (A.1), (A.3) and the fact that $\hat{\alpha }\xrightarrow {\mathrm {P}}\alpha _{0}$, we have

It is easy to see that

$$\begin{aligned} n^{\frac{1}{2}}U_{sr}(\beta _{0},\hat{\alpha })= n^{-\frac{1}{2}}\sum \limits _{i=1}^{n}\int \limits _{0}^{\tau }\xi _{i} \frac{\hat{\pi }^{*}(t)}{\pi (W_{i},\hat{\alpha })} \{Z_{i}-\bar{Z}_{sr}(t,\hat{\alpha })\}dM_{i}(t). \end{aligned}$$

So we can conclude that

$$\begin{aligned} n^{\frac{1}{2}}(\hat{\beta }_{sr}-\beta _{0})=(A+o_{p}(1))^{-1} n^{\frac{1}{2}}U_{sr}(\beta _{0},\hat{\alpha }). \end{aligned}$$

(A.4)

By the Taylor expansion of $n^{\frac{1}{2}}U_{sr}(\beta _{0},\hat{\alpha })$ at $\alpha _{0}$,

$$\begin{aligned} n^{\frac{1}{2}}U_{sr}(\beta _{0},\hat{\alpha })=n^{\frac{1}{2}}U_{sr}(\beta _{0},\alpha _{0}) +\frac{\partial U_{sr}(\beta _{0},\alpha )}{\partial \alpha ^{T}}\Big |_{\alpha =\alpha _{0}} n^{\frac{1}{2}}(\hat{\alpha }-\alpha _{0})+o_{p}(1), \end{aligned}$$

(A.5)

where

$$\begin{aligned}&-\frac{\partial U_{sr}(\beta _{0},\alpha )}{\partial \alpha ^{T}} \\&\quad =\frac{1}{n}\sum \limits _{i=1}^{n}\frac{\xi _{i}}{\pi ^{2}(W_{i},\alpha )} \int \limits _{0}^{\tau }\hat{\pi }^{*}(t)\{Z_{i}-\bar{Z}_{sr}(t,\alpha )\}dM_{i}(t)\dot{\pi }^{T}(W_{i},\alpha )\\&\qquad +\frac{1}{n}\sum \limits _{i=1}^{n}\frac{\xi _{i}}{\pi (W_{i},\alpha )} \int \limits _{0}^{\tau }\hat{\pi }^{*}(t) \Big \{\frac{S_{sr}^{(1)}(t,\alpha )}{(S_{sr}^{(0)}(t,\alpha ))^{2}} \Big [\frac{1}{n}\sum \limits _{j=1}^{n}\frac{\xi _{j}}{\pi ^{2}(W_{j},\alpha )}\dot{\pi }^{T}(W_{j},\alpha )Y_{j}(t)\Big ]\\&\qquad -\frac{1}{S_{sr}^{(0)}(t,\alpha )} \Big [\frac{1}{n}\sum _{j=1}^{n}\frac{\xi _{j}}{\pi ^{2}(W_{j},\alpha )}Y_{j}(t)Z_{j}\dot{\pi }^{T}(W_{j},\alpha )\Big ] \Big \} dM_{i}(t). \end{aligned}$$

By (A.1) to (A.3) and the law of large numbers, it can be proven that

(A.6)

where

$$\begin{aligned} V_{\alpha _{_0}}= & {} E\Big [\int \limits _{0}^{\tau }\{Z-\bar{z}(t)\} \frac{1}{\pi (W,\alpha _{_0})}\dot{\pi }^{T}(W,\alpha _{_0})\pi ^{*}(t)dM(t)\Big ]\\&-E\Big [\int \limits _{0}^{\tau }\Big \{\frac{h^{(2)}(t)}{s^{(0)}(t)} -\frac{s^{(1)}(t)h^{(1)}(t)}{(s^{(0)}(t))^{2}}\Big \}\pi ^{*}(t)dM(t)\Big ],\\ h^{(1)}(t)= & {} \mathrm {E}\Big [\frac{1}{\pi (W,\alpha _{0})}\dot{\pi }^{ T}(W,\alpha _{0})Y(t)\Big ] \end{aligned}$$

and

$$\begin{aligned} h^{(2)}(t)=\mathrm {E}\Big [\frac{1}{\pi (W,\alpha _{0})}Y(t)Z\dot{\pi }^{ T}(W,\alpha _{0})\Big ]. \end{aligned}$$

By (A.1), (A.3) and Lemma A.1 of Qi et al. (2005), we can obtain that

$$\begin{aligned} n^{\frac{1}{2}}U_{sr}(\beta _{0},\alpha _{0})= n^{-\frac{1}{2}}\sum \limits _{i=1}^{n} \frac{\xi _{i}}{\pi (W_{i},\alpha _{0})} \int \limits _{0}^{\tau }\pi ^{*}(t)\{Z_{i}-\bar{z}(t)\}dM_{i}(t)+o_{p}(1).\qquad \quad \end{aligned}$$

(A.7)

Define

$$\begin{aligned} S_{\alpha _{_0}}=\frac{\xi -\pi (W,\alpha _{_0})}{\pi (W,\alpha _{_0})\{1-\pi (W,\alpha _{_0})\}}\dot{\pi }(W,\alpha _{_0}) \end{aligned}$$

and

$$\begin{aligned} I_{\alpha _{_0}}=E\Big [S_{\alpha _{_0}}S_{\alpha _{_0}}^{T} -\frac{\xi -\pi (W,\alpha _{_0})}{\pi (W,\alpha _{_0})\{1-\pi (W,\alpha _{_0})\}}\ddot{\pi }(W,\alpha _{_0})\Big ], \end{aligned}$$

which are score and information matrices of $\pi (W,\alpha )$ respectively. Then under condition (C4), it can be shown that

$$\begin{aligned} n^{\frac{1}{2}}(\hat{\alpha }-\alpha _{0})= n^{-\frac{1}{2}}\sum \limits _{i=1}^{n}I_{\alpha _{_0}}^{-1}S_{\alpha _{_0},i}+o_{p}(1), \end{aligned}$$

(A.8)

where $S_{\alpha ,i}$ is obtained through replacing $\xi$ and W by $\xi _{i}$ and $W_{i}$ in $S_{\alpha }$ respectively. By (A.4) to (A.8), we finally arrive at

$$\begin{aligned}&n^{\frac{1}{2}}(\hat{\beta }_{sr}-\beta _{0}) \\&\quad = A^{-1}n^{-\frac{1}{2}}\sum \limits _{i=1}^{n} \Big [\frac{\xi _{i}}{\pi (W_{i},\alpha _{0})} \int \limits _{0}^{\tau }\pi ^{*}(t)\{Z_{i}-\bar{z}(t)\}dM_{i}(t) -V_{\alpha _{_0}}I_{\alpha _{_0}}^{-1}S_{\alpha _{_0},i}\Big ]+o_{p}(1). \end{aligned}$$

By the central limit theorem, the desired result is proved. $\square$

Proof of Theorem 2

It is easily verified that

$$\begin{aligned}&n^{\frac{1}{2}}(\hat{\beta }_{ar}-\beta _{0}) \nonumber \\&\quad = \left[ \frac{1}{n}\sum _{i=1}^{n}\int \limits _{0}^{\tau }\hat{\pi }^{*}(t) \left\{ Z_{i}-\bar{Z}_{ar}(t)\right\} ^{\otimes 2}Y_{i}(t)dt\right] ^{-1} \nonumber \\&\qquad \times \left[ n^{-\frac{1}{2}}\sum \limits _{j=1}^{n}\int \limits _{0}^{\tau } \left\{ Z_{j}-\bar{Z}_{ar}(t)\right\} \hat{\pi }^{*}(t) \left\{ \frac{\xi _{j}}{\pi (W_{j},\hat{\alpha })}dN_{j}(t)\right. \right. \nonumber \\&\left. \left. \qquad +\left( 1-\frac{\xi _{j}}{\pi (W_{j},\hat{\alpha })}\right) \rho (W_{j},\hat{\gamma })dN_{j}^{*}(t) -Y_{j}(t)\beta _{0}^{T}Z_{j}dt-Y_{j}(t)d\Lambda _{0}(t)\right\} \right] \nonumber \\&\quad =\left[ \frac{1}{n}\sum \limits _{i=1}^{n}\int \limits _{0}^{\tau }\hat{\pi }^{*}(t) \left\{ Z_{i}-\bar{Z}_{ar}(t)\right\} ^{\otimes 2}Y_{i}(t)dt\right] ^{-1}\nonumber \\&\,\,\,\,\qquad \times \left[ n^{-\frac{1}{2}}\sum _{j=1}^{n}\int \limits _{0}^{\tau } \left\{ Z_{j}-\bar{Z}_{ar}(t)\right\} \hat{\pi }^{*}(t)d\hat{M}_{j}^{*}(t)\right] , \end{aligned}$$

(A.9)

where

$$\begin{aligned}&d\hat{M}_{j}^{*}(t)\\&\quad =\frac{\xi _{j}}{\pi (W_{j},\hat{\alpha })}dN_{j}(t) +\left( 1-\frac{\xi _{j}}{\pi (W_{j},\hat{\alpha })}\right) \rho (W_{j},\hat{\gamma })dN_{j}^{*}(t)\\&\qquad -Y_{j}(t)\beta _{0}^{T}Z_{j}dt-Y_{j}(t)d\Lambda _{0}(t). \end{aligned}$$

By the fact that

$$\begin{aligned} \mathrm {sup}_{t\in [0,\tau ]}\Vert \bar{Z}_{ar}(t)-\bar{z}(t)\Vert =o_{p}(1) \end{aligned}$$

(A.10)

and (A.1), we can conclude that

(A.11)

By (A.11), we have

$$\begin{aligned} n^{\frac{1}{2}}(\hat{\beta }_{ar}-\beta _{0})=(A+o_{p}(1))^{-1} n^{\frac{1}{2}}U_{ar}(\beta _{0},\hat{\alpha },\hat{\gamma }). \end{aligned}$$

(A.12)

By the Taylor expansion of $n^{\frac{1}{2}}U_{ar}(\beta _{0},\hat{\alpha },\hat{\gamma })$ at $\alpha _{*}$ and $\gamma _{*}$,

$$\begin{aligned}&n^{\frac{1}{2}}U_{ar}(\beta _{0},\hat{\alpha },\hat{\gamma }) \nonumber \\&\quad =n^{\frac{1}{2}}U_{ar}(\beta _{0},\alpha _{*},\gamma _{*}) +\frac{\partial U_{ar}(\beta _{0},\alpha ,\gamma )}{\partial \alpha ^{T}}\Big |_{\alpha =\alpha _{*},\gamma =\gamma _{*}} n^{\frac{1}{2}}(\hat{\alpha }-\alpha _{*}) \nonumber \\&\qquad +\,\frac{\partial U_{ar}(\beta _{0},\alpha ,\gamma )}{\partial \gamma ^{T}}\Big |_{\alpha =\alpha _{*},\gamma =\gamma _{*}} n^{\frac{1}{2}}(\hat{\gamma }-\gamma _{*}) +o_{p}(1), \end{aligned}$$

(A.13)

where

$$\begin{aligned}&-\frac{\partial U_{ar}(\beta _{0},\alpha ,\gamma )}{\partial \alpha ^{T}} \\&\quad {=}\frac{1}{n}\sum \limits _{i=1}^{n}\int \limits _{0}^{\tau }\left\{ Z_{i}{-}\bar{Z}_{ar}(t)\right\} \hat{\pi }^{*}(t) \frac{\xi _{i}}{\pi ^{2}(W_{i},\alpha )}\dot{\pi }^{T}(W_{i},\alpha ) \left\{ dN_{i}(t){-}\rho (W_{i},\gamma )dN_{i}^{*}(t)\right\} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial U_{ar}(\beta _{0},\alpha ,\gamma )}{\partial \gamma ^{T}} =\frac{1}{n}\sum \limits _{i=1}^{n}\int \limits _{0}^{\tau }\left\{ Z_{i}{-}\bar{Z}_{ar}(t)\right\} \hat{\pi }^{*}(t) \left( 1-\frac{\xi _{i}}{\pi (W_{i},\alpha )}\right) \dot{\rho }^{T}(W_{i},\gamma )dN_{i}^{*}(t). \end{aligned}$$

By (A.1), $\hat{\alpha } \xrightarrow {\mathrm {P}}\alpha _{*}$ and $\hat{\gamma } \xrightarrow {\mathrm {P}}\gamma _{*}$, we have

(A.14)

and

(A.15)

where

$$\begin{aligned} V_{\alpha _{*}}^{*}=E\left[ \int \limits _{0}^{\tau }\left\{ Z-\bar{z}(t)\right\} \pi ^{*}(t) \frac{\dot{\pi }^{T}(W,\alpha _{*})}{\pi (W,\alpha _{*})} \left\{ dN(t)-\rho (W,\gamma _{*})dN^{*}(t)\right\} \right] \end{aligned}$$

and

$$\begin{aligned} V_{\gamma _{*}}^{*}=E\left[ \int _{0}^{\tau }\{Z-\bar{z}(t)\}\pi ^{*}(t) \frac{\xi -\pi (W,\alpha _{*})}{\pi (W,\alpha _{*})} \dot{\rho }^{T}(W,\gamma _{*})dN^{*}(t)\right] . \end{aligned}$$

Similar to (A.7), we have

$$\begin{aligned} n^{\frac{1}{2}}U_{ar}(\beta _{0},\alpha _{*},\gamma _{*})= n^{-\frac{1}{2}}\sum \limits _{i=1}^{n} \int \limits _{0}^{\tau }\pi ^{*}(t)\left\{ Z_{i}-\bar{z}(t)\right\} dM_{i}^{*}(t)+o_{p}(1),\qquad \end{aligned}$$

(A.16)

where

$$\begin{aligned}&dM_{i}^{*}(t)\\&\quad =\frac{\xi _{i}}{\pi (W_{i},\alpha _{*})}dN_{i}(t) +\left( 1-\frac{\xi _{i}}{\pi (W_{i},\alpha _{*})}\right) \rho (W_{i},\gamma _{*})dN_{i}^{*}(t)\\&\qquad -\,Y_{i}(t)\beta _{0}^{T}Z_{i}dt-Y_{i}(t)d\Lambda _{0}(t). \end{aligned}$$

Define

$$\begin{aligned} S_{\alpha _{*}}^{*}= & {} \frac{\xi -\pi (W,\alpha _{*})}{\pi (W,\alpha _{*})(1-\pi (W,\alpha _{*}))}\dot{\pi }(W,\alpha _{*}),\\ S_{\gamma _{*}}^{*}= & {} \frac{\xi (\delta -\rho (W,\gamma _{*}))}{\rho (W,\gamma _{*})(1-\rho (W,\gamma _{*}))} \dot{\rho }(W,\gamma _{*}),\\ I_{\alpha _{*}}^{*}= & {} E\Big [S_{\alpha _{*}}^{*}S_{\alpha _{*}}^{*T} -\frac{\xi -\pi (W,\alpha _{*})}{\pi (W,\alpha _{*})(1-\pi (W,\alpha _{*}))} \ddot{\pi }(W,\alpha _{*})\Big ], \end{aligned}$$

and

$$\begin{aligned} I_{\gamma _{*}}^{*}=E\Big [S_{\gamma _{*}}^{*}S_{\gamma _{*}}^{*T} -\frac{\xi (\delta -\rho (W,\gamma _{*}))}{\rho (W,\gamma _{*})(1-\rho (W,\gamma _{*}))} \ddot{\rho }(W,\gamma _{*})\Big ]. \end{aligned}$$

Then under Condition (C4) and (C5), we have

$$\begin{aligned} n^{\frac{1}{2}}(\hat{\alpha }-\alpha _{*})= n^{-\frac{1}{2}}\sum _{i=1}^{n}(I_{\alpha _{*}}^{*})^{-1}S_{\alpha _{_*},i}^{*}+o_{p}(1) \end{aligned}$$

(A.17)

and

$$\begin{aligned} n^{\frac{1}{2}}(\hat{\gamma }-\gamma _{*})= n^{-\frac{1}{2}}\sum _{i=1}^{n}(I_{\gamma _{*}}^{*})^{-1}S_{\gamma _{_*},i}^{*}+o_{p}(1), \end{aligned}$$

(A.18)

where $S_{\alpha _{*},i}^{*}$ and $S_{\gamma _{*},i}^{*}$ are obtained through replacing $\xi , \delta$ and W by $\xi _{i}, \delta _{i}$ and $W_{i}$ in $S_{\alpha _{*}}^{*}$ and $S_{\gamma _{*}}^{*}$. By (A.9) to (A.18), we can finally conclude that

By the central limit theorem, the desired result is proved. $\square$

Proof of (7)

It is easy to see that

(A.19)

By (A.1) and Taylor expansion of $I$ at $\alpha _{0}$, we obtain

(A.20)

where $f_{4}(t,z,\alpha _{0})$ is the limit of $\hat{f}_{4}(t,z,\hat{\alpha })$. $\square$

Similar to the proof of Theorem 2.4 in Lin (2011), by (4), it can be proven that

(A.21)

By (A.21) and Taylor expansion, we have

(A.22)

where $f_{1}(t,z,\alpha _{0})$ and $f_{3}(t,z,\alpha _{0})$ are the limits of $\hat{f}_{1}(t,z,\hat{\alpha })$ and $\hat{f}_{3}(t,z,\hat{\alpha })$ respectively. It is easy to see that

(A.23)

From (A.19) to (A.23), we finally arrive at

$$\begin{aligned} \mathcal {F}(t,z)= & {} n^{-1/2}\sum \limits _{i=1}^{n}\frac{\xi _{i}\pi ^{*}(t)}{\pi (W_{i},\alpha _{0})} \int \limits _{0}^{t}\left[ I(Z_{i}\le z)-\frac{f_{1}(s,z,\alpha _{0})}{S_{sr}^{(0)}(s,\alpha _{0})} \right] dM_{i}(s)\nonumber \\&-\,n^{-1/2}\sum \limits _{i=1}^{n}\frac{\xi _{i}}{\pi (W_{i},\alpha _{0})} f_{2}^{T}(t,z,\alpha _{0})A^{-1} \int \limits _{0}^{\tau }\pi ^{*}(s)\{Z_{i}-\bar{z}_{sr}(s)\}dM_{i}(s)\nonumber \\&+\,n^{-1/2}\sum \limits _{i=1}^{n}\left[ f_{3}^{T}(t,z,\alpha _{0})+f_{2}^{T}(t,z,\alpha _{0})A^{-1}V_{\alpha } -f_{4}^{T}(t,z,\alpha _{0})\right] I_{\alpha _{_0}}^{-1}S_{\alpha _{_0},i}\nonumber \\&+ \,o_{p}(1), \end{aligned}$$

(A.24)

where $f_{2}(t,z,\alpha _{0})$ is the limit of $\hat{f}_{2}(t,z,\hat{\alpha })$. The finite dimensional convergence of $\mathcal {F}(t,z)$ can be proven by the multivariate cental limit theorem. By the techniques in Lin (2011), it can be proven that $\mathcal {F}(t,z)$ is tight. So $\mathcal {F}(t,z)$ converges weakly to a zero-mean Gaussian process which can be approximately by (7).