Bayesian dynamic models for survival data with a cure fraction

Abstract

In this paper, we propose a new class of semi-parametric cure rate models. Specifically, we construct dynamic models for piecewise hazard functions over a finite partition of the time axis. Allowing the size of partition and the levels of baseline hazard to be random, our proposed models provide a great flexibility in controlling the degree of parametricity in the right tail of the survival distribution and the amount of correlations among the log-baseline hazard levels. Several properties of the proposed models are derived, and propriety of the implied posteriors with improper noninformative priors for regression coefficients based on the proposed models is established for the fixed partition of the time axis. In addition, an efficient reversible jump computational algorithm is developed for carrying out posterior computation. A real data set from a melanoma clinical trial is analyzed in detail to further demonstrate the proposed methodology.

Appendix: Proofs of Properties and Theorem

Proof of Property 2.1

From the definition of E(ξ _j | λ₀) given in (2.8), we have

$$ \begin{aligned} E(\xi_j \vert \lambda_0) &= \log\left[\frac{H_0(s_j) - H_0(s_{j-1})}{s_j - s_{j-1}} \right ] \\ &= \log\left[\frac{H_0\left(\frac{s_j + s_{j-1}}{2} + \frac{s_j - s_{j-1}}{2}\right) - H_0\left(\frac{s_j + s_{j-1}}{2} - \frac{s_j - s_{j-1}}{2}\right)}{s_j - s_{j-1}}\right]. \end{aligned} $$

Now using the definition of a derivative of the function ${H_0(\cdot)}$ and the assumption that ${\frac{s_j + s_{j-1}}{2} \rightarrow t}$, we have ${\lim_{s_j - s_{j-1} \rightarrow 0} E(\xi_j \vert \lambda_0) = \log ( \frac{\hbox{d}}{\hbox{d}t} H_0(t)) = \log h_0(t)}$. □

Proof of Property 2.2

Note that for ${t\in (s_{j-1}, s_j)}$,

$$ E[H^*(t)\vert \lambda_0] = E[\hbox{e}^{\xi_j}\vert \lambda_0](t-s_{j-1}) + \sum_{g=1}^{j-1} E[\hbox{e}^{\xi_g}\vert \lambda_0] (s_g-s_{g-1}), $$

where

$$ \begin{aligned} E[\hbox{e}^{\xi_j}\vert \lambda_0] &= E\left[\hbox{e}^{\mu_j + \sum_{l=1}^{j} \left\{\prod_{k = l+1}^j c_k \right\} w_l}\vert \lambda_0 \right] \nonumber \\ &= \left[\frac{H_0(s_j) - H_0(s_{j-1})}{s_j - s_{j-1}} \right] \times E\Bigg[\exp \Bigg(\frac{ \sum_{l=1}^j \Big\{\prod_{k=l+1}^{j} c_k^2 \Big\} b_l W}{2} \Bigg) \Bigg]. \end{aligned} $$

As α → ∞, b _j → 0 and c _j → r for ${j = 1,2, \ldots, J}$. Thus, we have

$$ \lim_{\alpha \rightarrow \infty} E[\hbox{e}^{\xi_j}\vert \lambda_0] = \frac{H_0(s_j) - H_0(s_{j-1})}{s_j - s_{j-1}}, $$

and

$$ \lim_{\alpha \rightarrow \infty} E[H^*(t)\vert \lambda_0] = \left( \frac{t-s_{j-1} }{ s_j-s_{j-1}} \right) \Big[H_0(s_j)-H_0(s_{j-1}) \Big] + H_0(s_{j-1}). $$

(A.1)

Note that the right hand side of (A.1) equals H ₀(t) when ${t \in \{s_1, s_2, \ldots, s_J\}}$ and for other values of t is only approximately equal in the sense that as ${s_j-s_{j-1}\rightarrow 0}$, the right hand side of (A.1) approaches H ₀(t). Again,

$$ \hbox{Var}[H^*(t)\vert \lambda_0] = \hbox{Var}[\hbox{e}^{\xi_j}\vert \lambda_0](t-s_{j-1})^2 + \sum_{g=1}^{j-1} \hbox{Var}[\hbox{e}^{\xi_g}\vert \lambda_0] (s_g-s_{g-1})^2, $$

where

$$ \hbox{Var}[\hbox{e}^{\xi_j}\vert \lambda_0] = E[\hbox{e}^{2 \xi_j}\vert \lambda_0] - E[\hbox{e}^{ \xi_j}\vert \lambda_0]^2, $$

with

$$ \begin{aligned} E[\hbox{e}^{2 \xi_j}\vert \lambda_0] &= E\left[\hbox{e}^{2 \mu_j + 2 \sum_{l=1}^{j} \left\{\prod_{k = l+1}^j c_k \right\} w_l}\vert \lambda_0\right] \nonumber \\ &= \left[\frac{H_0(s_j) - H_0(s_{j-1})}{s_j - s_{j-1}} \right]^2 \times \; E \left[\exp \left(2 \sum_{l=1}^j \left\{\prod_{k=l+1}^{j} c_k^2 \right\} b_l W \right) \right]. \end{aligned} $$

Now letting α → ∞, we obtain

$$ \lim_{\alpha \rightarrow \infty} E[\hbox{e}^{2 \xi_j}\vert \lambda_0] = \left[\frac{H_0(s_j) - H_0(s_{j-1})}{s_j - s_{j-1}}\right]^2. $$

Thus,

$$ \hbox{Var}[H^*(t)\vert \lambda_0] \rightarrow 0, \quad \hbox{as $\alpha \rightarrow \infty$}. $$

This shows that H ^*(t| λ)→ H ₀(t) in probability for ${t\in \{ s_1, \ldots, s_J\}}$ and that F ^*(t| λ) = 1−exp(−H ^*(t| λ))→ F ₀(t| λ₀) in probability. Thus, given θ, the random survival function S ^* _p (t| λ) = exp[−θ F ^*(t| λ)] converges to exp[−θ F ₀(t| λ₀)] in probability as α → ∞. This convergence is exact for ${t\in \{s_1, \ldots, s_J\}}$ and approximate for other values of t. □

Proof of Property 2.3

Using (2.6), we have

$$ \hbox{Cov}(\xi_j, \xi_{j-1}) = E[\xi_j - \mu_j][\xi_{j-1} - \mu_{j-1}] = E[c_j(\xi_{j-1} - \mu_{j-1}) + w_j][\xi_{j-1} - \mu_{j-1}] = c_j \hbox{Var}(\xi_{j-1}), $$

and

$$ \hbox{Var}(\xi_j) = c_j^2 \hbox{Var}(\xi_{j-1}) + b_j E[W]. $$

Then the correlation of ξ _j and ${\xi_{j-1}}$ is given by

$$ \begin{aligned} \hbox{corr}(\xi_j, \xi_{j-1}) &= \sqrt{\frac{ c_j^2 \hbox{Var}(\xi_{j-1})}{\hbox{Var}(\xi_{j})}} = \sqrt{1 - \frac{ b_j E[W]}{\hbox{Var}(\xi_{j})}} \\ &= \sqrt{1 - \frac{ b_j}{\sum_{l=1}^j \left\{\prod_{k=l+1}^{j}c_k^2 \right\} b_l}}\\ & = \sqrt{1 - \frac{ 1}{\sum_{l=1}^j\left\{\prod_{k=l+1}^{j} \frac{1}{(1 + b_k)^2} \right\} r^{2(j-l)}\frac{b_l}{b_j}}}.\end{aligned}$$

Now letting α → 0, we obtain

$$ \hbox{corr} (\xi_j, \xi_{j-1}) = \sqrt{1 - \frac{1}{\sum_{l=1}^j \left(\frac{r}{2}\right)^{2(j-l)}}} = \sqrt{1 - \frac{1 - (\frac{r}{2})^2}{1 - (\frac{r}{2})^{2j}}}. $$

Therefore,

$$ \hbox{corr}(\xi_j, \xi_{j-1}) \rightarrow \frac{r}{2}, \quad \hbox{as $\alpha \rightarrow 0$ and $j \rightarrow \infty$}. $$

Also letting α → ∞ gives

$$ \hbox{corr}(\xi_j, \xi_{j-1}) \rightarrow 1. $$

□

Proof of Theorem 3.1

In (2.1), summing out the unobserved latent variable N _i yields

$$ L(\beta,\lambda\vert D)= \sum_{N} L(\beta,\lambda\vert D_{\rm comp}) = \prod^n_{i=1} \Big(\theta_i f(y_i\vert \lambda)\Big)^{\nu_i} \exp\{-\theta_i(1-S(y_i\vert \lambda))\}, $$

where θ _i  = exp(x _i ′β). If ν _i =  0, for ${i = 1, 2, \ldots, n}$, then

$$ L(\beta,\lambda\vert D)= \prod^n_{i=1} \exp\{-\theta_i(1-S(y_i\vert \lambda))\} \; < \; \infty. $$

Thus, it suffices to prove the propriety of the posterior distribution for the case where ν _i  = 1. Let δ _ij =  1 if j = j _i and 0 otherwise. Then ${f(y_i\vert \lambda) = \lambda_{j_i} \exp \{ - [ \lambda_{j_i} (y_i - s_{j_i-1}) +}\sum_{g=1}^{j_i-1} \lambda_g (s_g-s_{g-1}) ] \}$ and ${ S(y_i\vert \lambda) = \exp\left\{ - \left[ \lambda_{j_i} (y_i - s_{j_i-1}) + \sum_{g=1}^{j_i-1} \lambda_g (s_g - s_{g-1}) \right] \right\}}$.

Since X ^* is of full rank, there exists ${i^\ast_1, \ldots, i^\ast_p}$ such that ${\nu_{i^{*}_1} = \nu_{i^{*}_2} = \ldots= \nu_{i^{*}_p} = 1}$, and ${X_p^{*}=(x_{i_1^*}, x_{i_2^*},\ldots, x_{i_p^*})'}$ is of full rank. We make the transformation ${u = (u_1, \ldots, u_p)' = X_p^{*} \beta}$. This linear transformation is a one-to-one. Thus, we have

$$ \begin{aligned} \; & \int_{R^p} \prod_{l=1}^{p} f(y_{i^{*}_l}\vert \lambda) \exp(x_{i^{*}_l}^\prime \beta) \exp\left\{ - \exp(x_{i^{*}_l}^\prime \beta) (1- S(y_{i^{*}_l}\vert \lambda))\right\} \ \hbox{d} \beta \\ = &\vert X_p^{*}\vert ^{-1} \prod_{l=1}^{p} \int_{-\infty}^{\infty} f(y_{i^{*}_l}\vert \lambda) \exp(u_l) \exp\left\{ - \exp(u_l) (1-S(y_{i^{*}_l}\vert \lambda)) \right\} \ \hbox{d} u_l \nonumber \\ = &\vert X_p^{*}\vert ^{-1} \prod_{l=1}^{p} \frac{f(y_{i^{*}_l}\vert \lambda)}{1-S(y_{i^{*}_l}\vert \lambda)}. \end{aligned} $$

Since s _j-1 <  y _{i_j} ≤  s _j , we have

$$ \begin{aligned} \frac{f(y_{i_j}\vert \lambda)}{1-S(y_{i_j}\vert \lambda)} & = \frac{\lambda_j \exp\left\{ - \left[ \lambda_j (y_{i_j} - s_{j-1}) + \sum_{g=1}^{j-1} \lambda_g (s_g - s_{g-1}) \right] \right\}} {1 - \exp\left\{ - \left[ \lambda_j (y_{i_j} - s_{j-1}) + \sum_{g=1}^{j-1} \lambda_g (s_g - s_{g-1}) \right] \right\}} \\ & \leq \frac{\lambda_j \exp\left\{ - \lambda_j (y_{i_j} - s_{j-1})\right\}} {1 - \exp(- \lambda_j (y_{i_j} - s_{j-1}))} \leq K_1 \lambda_j, \end{aligned} $$

where K ₁ >  0.

Note that ξ _j =  log(λ _j ), $\xi_1 \vert W \sim N(\mu_1, b_1 W)$ and $\xi_j \vert \xi_{j-1}, W \sim N(\mu_j + c_j(\xi_{j-1} - \mu_j), b_j W)$, for $j = 2, 3, \ldots, J$. Let $p^{*}(\beta, \xi, W, \lambda_0\vert D)= L(\beta, \xi\vert D) \pi^\ast(\xi, W, \lambda_0)$. Then, we have

$$ \int p^{*}(\beta, \xi, W, \lambda_0\vert D) \hbox{d} \beta \hbox{d} \xi \hbox{d} W \hbox{d} \lambda_0 \leq K_1 \int \hbox{e}^{\xi^\prime 1} \pi^\ast(\xi, W, \lambda_0) \hbox{d} \xi \hbox{d} W \hbox{d} \lambda_0, $$

(A.2)

where the vector 1 has all elements equal to one. It is easy to see that given that the prior distributions of W and λ₀ are proper, the integral of the right hand side of (A.2) is finite. This completes the proof. □