Bayesian analysis of paired survival data using a bivariate exponential distribution

Abstract

We consider a Bayesian analysis method of paired survival data using a bivariate exponential model proposed by Moran (1967, Biometrika 54:385–394). Important features of Moran’s model include that the marginal distributions are exponential and the range of the correlation coefficient is between 0 and 1. These contrast with the popular exponential model with gamma frailty. Despite these nice properties, statistical analysis with Moran’s model has been hampered by lack of a closed form likelihood function. In this paper, we introduce a latent variable to circumvent the difficulty in the Bayesian computation. We also consider a model checking procedure using the predictive Bayesian P-value.

Appendix

Derivation of (3)

Note $(Z_1,Z_3)$ and $(Z_2,Z_4)$ are IID bivariate normal random vectors with means 0 and covariance matrix (1). From the density of $(Z_1,Z_3)$ and $(Z_2,Z_4)$, change of variable technique using (1) yields the density of $(X_1,X_2,\eta_1,\eta_2)$,

$$ \frac{\lambda_1\lambda_2}{4\pi^2 (1-\rho)} \exp\left\{-\frac {\lambda_1 x_1 + \lambda_2x_2 - 2 \sqrt{\rho\lambda_1\lambda_2x_1x_2} \cos(\eta_1-\eta_2)}{1-\rho}\right\}, $$

where $x_1, x_2 \geq 0, 0 \leq \eta_1, \eta_2 \leq 2\pi$. Let $V_1 = \eta_1 -\eta_2$ and $V_2 = \eta_1.$ Then, the density of $(X_1,X_2,V_1,V_2)$ is

$$ \frac{\lambda_1\lambda_2}{4\pi^2 (1-\rho)} \exp \left\{ - \frac{1}{(1-\rho)} (\lambda_1 x_1 + \lambda_2x_2 - 2 \sqrt{\rho\lambda_1\lambda_2}\cos(v_1) \sqrt{x_1 x_2})\right\}, $$

where $-2\pi \leq v_1 \leq 2\pi,$ and $0 \leq v_2 \leq 2\pi+v_1$ for $-2\pi \leq v_1 < 0$ and $v_1 \leq v_2 \leq 2\pi$ for $0 \leq v_1 \leq 2\pi.$ Integrating out v ₂, we get the density of $(X_1,X_2,V_1)$

$$ (2\pi-|v_1|) \frac{\lambda_1\lambda_2}{4\pi^2 (1-\rho)} \exp \left\{-\frac{1}{(1-\rho)} (\lambda_1 x_1 + \lambda_2x_2 - 2 \sqrt{\rho\lambda_1\lambda_2}\cos(v_1) \sqrt{x_1 x_2})\right\}. $$

Note the density of $(X_1,X_2,V_1)$ is symmetric in v ₁ about 0. Letting $U = |V_1|$, we get the density of $(X_1,X_2,U)$, (3).

Proof of Proposition 1

It suffices to show that the posterior is proper when the data set consists of two pairs of uncensored survival times $D = \{(x_{11},x_{21},\delta_{11},\delta_{21}), (x_{12},x_{22},\delta_{12},\delta_{22})\}$ with $\delta_{ji} = 1$ for i,j = 1,2. The posterior with prior (7) is proportional to

$$ \begin{array}{lll} &&\pi(\lambda_1,\lambda_2,\rho,u_1,u_2|D) \\ &\propto&\frac{1}{\lambda_1\lambda_2} \prod\limits_{i=1}^2 \frac{\lambda_1\lambda_2 (2\pi-u_i)}{(1-\rho)} \exp\left[ -\frac{1}{(1-\rho)} \left( \lambda_1x_{1i} + \lambda_2 x_{2i} - 2\sqrt{\rho\lambda_1\lambda_2x_{1i}x_{2i}} \cos(u_i)\right)\right] \\ &\preceq&\frac{\lambda_1\lambda_2}{(1-\rho)^2} \exp\left[ -\frac{1}{(1-\rho)} \sum\limits_{i=1}^2 \left( \lambda_1x_{1i} + \lambda_2 x_{2i}-2\sqrt{\lambda_1\lambda_2x_{1i}x_{2i}} \right)\right] \\ &=&\frac{\lambda_1\lambda_2}{(1-\rho)^2} \exp\left[ -\frac{1}{(1-\rho)} \sum\limits_{i=1}^2 \left( \sqrt{\lambda_1x_{1i}} - \sqrt{\lambda_2 x_{2i}} \right)^2\right], \end{array} $$

where $0 < \rho < 1$, $\lambda_1, \lambda_2 > 0$ and $0 < u_1, u_2 < 2\pi$ and $a(x) \preceq b(x)$ means b(x) is bigger than or equal to a constant multiple of a(x). Integrating out u ₁ and u ₂, we get

$$ \pi(\lambda_1, \lambda_2,\rho|D) \preceq \frac{\lambda_1\lambda_2}{(1-\rho)^2} \exp\left[ -\frac{1}{(1-\rho)} Q_0(\lambda_1,\lambda_2) \right], $$

where $Q_0(\lambda_1,\lambda_2) = \sum_{i=1}^2 \left(\sqrt{\lambda_1x_{1i}} - \sqrt{\lambda_2 x_{2i}} \right)^2$. With the transformation $t = 1/(1-\rho)$,

$$ \begin{array}{lll} &&\pi(\lambda_1,\lambda_2|D) \\ &\preceq & \int_0^1 \frac{\lambda_1\lambda_2}{(1-\rho)^2} \exp\left[ -\frac{1}{(1-\rho)} Q_0(\lambda_1,\lambda_2) \right] \hbox{d}\rho \\ &=& \lambda_1\lambda_2 \int_1^\infty \exp\left[-t Q_0(\lambda_1,\lambda_2)\right] \hbox{d}\rho \\ &=& \frac{\lambda_1\lambda_2}{Q_0(\lambda_1,\lambda_2)} \exp[-Q_0(\lambda_1,\lambda_2)]. \end{array} $$

With $\mu_j =\sqrt{\lambda_j}$, for j = 1,2,

$$ \int_0^\infty\int_0^\infty \pi(\lambda_1,\lambda_2|D) \hbox{d}\lambda_1\hbox{d}\lambda_2\preceq \int_0^\infty\int_0^\infty \frac{\mu_1^3\mu_2^3}{Q(\mu_1,\mu_2)} \exp\left[-Q(\mu_1,\mu_2)\right]\hbox{d}\mu_1\,\hbox{d}\mu_2,$$

(8)

where

$$ \begin{array}{lll} Q(\mu_1,\mu_2)&=&\left(\sqrt{x_{11}}\mu_1-\sqrt{x_{21}}\mu_2\right)^2 +\left(\sqrt{x_{12}}\mu_1- \sqrt{x_{22}}\mu_2\right)^2 \\ &=&\mu^T A(D)\mu \end{array} $$

and

$$ A(D) = \left( \begin{array}{ll} x_{11}+x_{12} & -\left(\sqrt{x_{11}x_{21}} + \sqrt{x_{12}x_{22}}\right) \\ -\left(\sqrt{x_{11}x_{21}} + \sqrt{x_{12}x_{22}}\right) & x_{21}+x_{22} \end{array} \right).$$

With the assumption of Proposition 1, the matrix A(D) is a positive definite matrix. Hence, the RHS of (8) defines an integration with respect to a nonsingular bivariate normal density. Since every polynomial order moment exists for a nonsingular bivariate normal distribution, we conclude (8) is finite. $\square$