A Bayesian proportional hazards mixture cure model for interval-censored data

Abstract

The proportional hazards mixture cure model is a popular analysis method for survival data where a subgroup of patients are cured. When the data are interval-censored, the estimation of this model is challenging due to its complex data structure. In this article, we propose a computationally efficient semiparametric Bayesian approach, facilitated by spline approximation and Poisson data augmentation, for model estimation and inference with interval-censored data and a cure rate. The spline approximation and Poisson data augmentation greatly simplify the MCMC algorithm and enhance the convergence of the MCMC chains. The empirical properties of the proposed method are examined through extensive simulation studies and also compared with the R package “GORCure”. The use of the proposed method is illustrated through analyzing a data set from the Aerobics Center Longitudinal Study.

Appendix

After initializing values for the parameters, the proposed MCMC algorithm proceeds in the following steps.

• 1. Let $Z_{i} = 0$ and $W_{i} = 0$ for all i, $Z_{il} = 0$ and $W_{il} = 0$ for all i and l. If $\delta _{1i}=1$, then sample

  $$\begin{aligned}{} & {} Z_{i} \sim \text {Poi}(\varLambda _0(R_{i})e^{{\varvec{\beta }}'{} \textbf{x}_{i}})1{(Z_{i}>0)}, \\{} & {} (Z_{i1},\ldots ,Z_{iK}) \sim \text {Multinomial}(Z_{i}; \gamma _{1}I_{1}(R_{i}),\ldots ,\gamma _{K}I_{K}(R_{i})). \end{aligned}$$

  If $\delta _{2i}=1$, then sample

  $$\begin{aligned}{} & {} W_{i} \sim \text {Poi}(\{\varLambda _0(R_{i})-\varLambda _0(L_{i})\}e^{{\varvec{\beta }}'{} \textbf{x}_{i}})1{(W_{i}>0)}, \\{} & {} (W_{i1},\ldots ,W_{iK}) \sim \text {Multinomial}(W_{i}; \gamma _{1}\{I_{1}(R_{i})-I_{1}(L_{i})\},\ldots ,\gamma _{K}\{I_{K}(R_{i})-I_{K}(L_{i})\}). \end{aligned}$$

• 2. For $\beta _p$ corresponding to a numeric covariate, use the ARMS algorithm to sample from its full conditional distribution

  $$\begin{aligned} p(\beta _p|\cdot ) \propto [\sum _{i=1}^{n}\{x_{ip}\beta _{p}(Z_{i}\delta _{1i}+W_{i}\delta _{2i})-e^{{\varvec{\beta }}'{} \textbf{x}_{i}}(\varLambda _{0}(R_{i})(\delta _{1i}+\delta _{2i})+\varLambda _{0}(L_{i})\delta _{3i}u_{i})\}]e^{-\frac{\beta _{p}^{2}}{2\sigma _{01}^2}}. \end{aligned}$$

• 3. For $\beta _p$ corresponding to a dummy variable, let $\zeta _p=\exp (\beta _p)$, sample $\zeta _p$ from

  $$\begin{aligned}{} & {} \text {Ga}(a_\zeta +\sum _{i=1}^{n}{x_{ip}(Z_{i}\delta _{1i}+W_{i}\delta _{2i})}, \\{} & {} b_\zeta +\sum _{i=1}^{n}e^{{\varvec{\beta }}_{-p}'\textbf{x}_{i,-p}}\{\varLambda _0(R_{i})(\delta _{1i}+\delta _{2i})+\varLambda _0(L_{i})\delta _{3i}u_{i}\}x_{ip}), \end{aligned}$$

  where ${\varvec{\beta }}_{-p}=\{\beta _{k}: k\ne p\}$ and $\textbf{x}_{i,-p}=\{x_{ik}: k\ne p\}$.

• 4. If $\delta _{1i}=1$ or $\delta _{2i}=1$ then $u_{i}=1$ for $i=1,\ldots ,n$. If $\delta _{3i}=1$ then sample $u_{i} \sim \text {Bernoulli}(p_i)$, where

  $$\begin{aligned} p_{i}=E(u_{i}|\cdot )=\frac{\pi _{i}e^{-\varLambda _{u0}(L_i)\exp ({\varvec{\beta }}'{} \textbf{x}_{i})}}{(1-\pi _i)+\pi _{i}e^{-\varLambda _{u0}(L_i)\exp ({\varvec{\beta }}'{} \textbf{x}_{i})}}. \end{aligned}$$

• 5. Sample $\gamma _l$, $l=1,\ldots ,K$, from

  $$\begin{aligned}{} & {} \text {Ga}\left( 1+\sum _{i=1}^{n}{(Z_{il}\delta _{1i}+W_{il}\delta _{2i}}\right) , \\{} & {} \eta +\sum _{i=1}^{n}e^{{\varvec{\beta }}'{} \textbf{x}_{i}}\{I_{l}(R_{i})(\delta _{1i}+\delta _{2i})+I_{l}(L_{i})\delta _{3i}u_{i}\}). \end{aligned}$$

• 6. Sample $\eta$ from $\text {Ga}(a_{\eta }+K,b_{\eta }+\sum _{l=1}^{K}\gamma _l)$.

• 7. Sample $\alpha _{q}$, $q=0,\ldots ,Q$, using the ARMS algorithm from its full conditional distribution

  $$\begin{aligned} p(\alpha _{q}|\cdot ) \propto \prod _{i=1}^{n}{\frac{\exp \{(\alpha _{0}+\alpha _{1}x_{i1}+\alpha _{2}x_{i2})u_{i}\}}{1+\exp (\alpha _{0}+\alpha _{1}x_{i1}+\alpha _{2}x_{i2})}}e^{-\frac{\alpha _{q}^{2}}{2\sigma _{02}^2}}. \end{aligned}$$