A calibrated Bayesian method for the stratified proportional hazards model with missing covariates

Abstract

Missing covariates are commonly encountered when evaluating covariate effects on survival outcomes. Excluding missing data from the analysis may lead to biased parameter estimation and a misleading conclusion. The inverse probability weighting method is widely used to handle missing covariates. However, obtaining asymptotic variance in frequentist inference is complicated because it involves estimating parameters for propensity scores. In this paper, we propose a new approach based on an approximate Bayesian method without using Taylor expansion to handle missing covariates for survival data. We consider a stratified proportional hazards model so that it can be used for the non-proportional hazards structure. Two cases for missing pattern are studied: a single missing pattern and multiple missing patterns. The proposed estimators are shown to be consistent and asymptotically normal, which matches the frequentist asymptotic properties. Simulation studies show that our proposed estimators are asymptotically unbiased and the credible region obtained from posterior distribution is close to the frequentist confidence interval. The algorithm is straightforward and computationally efficient. We apply the proposed method to a stem cell transplantation data set.

Appendix

We derive $\varvec{\varSigma }$ and its estimator in the Appendix. Let $dM_{li}(t) = dN_{li}(t) - Y_{li}(t) \exp \{\varvec{\beta }^T \varvec{Z}_{li}\} d\varLambda _{l}(t)$. The posterior distribution is

$$\begin{aligned} p(\varvec{\eta }|\varvec{U}_n)\sim N \left[ \left( \begin{array}{c}{} \mathbf{0} \\ \mathbf{0} \end{array}\right) , \frac{\varvec{\varSigma }}{n}=\left( \begin{array}{cc} Var(\varvec{U}_{1}) &{} Cov(\varvec{U}_{1},\varvec{U}_{2})\\ Cov(\varvec{U}_{1},\varvec{U}_{2}) &{} Var(\varvec{U}_{2}) \end{array}\right) \right] , \end{aligned}$$

where

$$\begin{aligned}&Var\{\varvec{U}_{1}(\varvec{\beta },\varvec{\phi }) \}\\&\quad = Var\left[ n^{-1} \sum _{l=1}^{L} \sum _{i=1}^{n_l} \frac{\xi _{li}}{\pi _{li}} \int _{0}^{\tau } \Big \{\varvec{Z}_{li} - \frac{\varvec{S}_{l}^{(1)}(\varvec{\beta },t)}{S_{l}^{(0)}(\varvec{\beta },t) }\Big \} dM_{li}(t) \right] \\&\quad = Var\left[ n^{-1} \sum _{l=1}^{L} \sum _{i=1}^{n_l} \frac{\xi _{li}}{\pi _{li}} \int _{0}^{\tau } \Big \{\varvec{Z}_{li} - \varvec{e}_{l}(\varvec{\beta },t)\Big \} dM_{li}(t)\right] \\&\quad = E\Big [n^{-2} \sum _{l=1}^{L} \sum _{i=1}^{n_l} \frac{\xi _{li}}{\pi _{li}} \int _{0}^{\tau } \Big \{\varvec{Z}_{li} - \varvec{e}_{l}(\varvec{\beta },t)\Big \} dM_{li}(t) \Big ]^{\otimes 2}. \end{aligned}$$

We can estimate $Var\{\varvec{U}_{1}(\varvec{\beta },\varvec{\phi }) \}$ given $\varvec{\beta }$ and $\varvec{\phi }$ as follows:

$$\begin{aligned} \widehat{ Var}\{\varvec{U}_{1}(\varvec{\beta },\varvec{\phi }) \} = \frac{1}{n^2} \sum _{l=1}^{L} \sum _{i=1}^{n_l} \frac{\xi _{li}}{\pi ^2_{li}} \int _{0}^{\tau } \left[ \left\{ \varvec{Z}_{li} - \frac{\varvec{S}_{l}^{(1)}(\varvec{\beta },t)}{S_{l}^{(0)}(\varvec{\beta },t) }\right\} \{dN_{li}(t) - d\widehat{\varLambda }_{0l}(t)\}\right] ^{\otimes 2}, \end{aligned}$$

where $d\widehat{\varLambda }_{0l}(t) = \sum _{i=1}^{n_l} dN_{li}(t)/ n_l S_{l}^{(0)}(\varvec{\beta },t)$.

We can obtain $\widehat{Var}(\varvec{U}_2)$ given $\varvec{\beta }$ and $\varvec{\phi }$ as follows:

$$\begin{aligned} \widehat{Var}(\varvec{U}_{2}) = \frac{1}{n^2} \sum _{l=1}^{L} \sum _{i=1}^{n_l} \pi _{li}(1-\pi _{li}) \varvec{\omega }_{li} \varvec{\omega }_{li}^T. \end{aligned}$$

Next, $Cov(\varvec{U}_{1},\varvec{U}_{2})$ given $\varvec{\beta }$ and $\varvec{\phi }$ can be estimated by

$$\begin{aligned} \widehat{Cov}(\varvec{U}_{1},\varvec{U}_{2})= & {} \widehat{Cov}\Big [\frac{1}{n}\sum _{l=1}^{L} \sum _{i=1}^{n_l} \frac{\xi _{li}}{\pi _{li}}\int _{0}^{\tau } \left\{ \varvec{Z}_{li} - \frac{\varvec{S}_{l}^{(1)}(\varvec{\beta },t)}{S_{l}^{(0)}(\varvec{\beta },t) }\right\} dM_{li}(t), \\&\frac{1}{n} \sum _{l=1}^{L} \sum _{i=1}^{n_l} \{\xi _{li} - \pi _{li}\} \varvec{\omega }_{li}^T\Big ]\\= & {} \frac{1}{n^2} \sum _{l=1}^{L} \sum _{i=1}^{n_l} \frac{\xi _{li} (1-\pi _{li})}{\pi _{li}} \int _{0}^{\tau } \left\{ \varvec{Z}_{li} -\frac{\varvec{S}_{l}^{(1)}(\varvec{\beta },t)}{S_{l}^{(0)}(\varvec{\beta },t)}\right\} dM_{li}(t) \times \varvec{\omega }_{li}^T. \end{aligned}$$

The estimator $\widehat{\varvec{\varSigma }}$ is

$$\begin{aligned} \frac{\widehat{\varvec{\varSigma }}}{n}=\left( \begin{array}{cc} \widehat{Var}(\varvec{U}_{1}) &{} \widehat{Cov}(\varvec{U}_{1},\varvec{U}_{2})\\ \widehat{Cov}(\varvec{U}_{1},\varvec{U}_{2}) &{} \widehat{Var}(\varvec{U}_{2}) \end{array}\right) . \end{aligned}$$