Generalized accelerated failure time spatial frailty model for arbitrarily censored data

Abstract

Flexible incorporation of both geographical patterning and risk effects in cancer survival models is becoming increasingly important, due in part to the recent availability of large cancer registries. Most spatial survival models stochastically order survival curves from different subpopulations. However, it is common for survival curves from two subpopulations to cross in epidemiological cancer studies and thus interpretable standard survival models can not be used without some modification. Common fixes are the inclusion of time-varying regression effects in the proportional hazards model or fully nonparametric modeling, either of which destroys any easy interpretability from the fitted model. To address this issue, we develop a generalized accelerated failure time model which allows stratification on continuous or categorical covariates, as well as providing per-variable tests for whether stratification is necessary via novel approximate Bayes factors. The model is interpretable in terms of how median survival changes and is able to capture crossing survival curves in the presence of spatial correlation. A detailed Markov chain Monte Carlo algorithm is presented for posterior inference and a freely available function frailtyGAFT is provided to fit the model in the R package spBayesSurv. We apply our approach to a subset of the prostate cancer data gathered for Louisiana by the surveillance, epidemiology, and end results program of the National Cancer Institute.

Appendix

Proposition 1

Assume that ${\varvec{\gamma }}_{l,k}|\alpha \overset{ind.}{\sim } N_{q+1}\left( {\mathbf {0}}, \frac{2n}{\alpha \rho (l+1)} ({\mathbf {X}}'{\mathbf {X}})^{-1}\right)$ under $H_1$ and ${\varvec{\gamma }}_{l,k,-j}|\alpha \overset{ind.}{\sim } N_{q}\left( {\mathbf {0}}, \frac{2n}{\alpha \rho (l+1)} ({\mathbf {X}}_{-j}'{\mathbf {X}}_{-j})^{-1}\right)$ under $H_0$, where $\alpha$ is fixed and ${\mathbf {X}}_{-j}$ is the design matrix ${\mathbf {X}}$ excluding the $(j+1)$th column. Then the Assumption (2.12) holds, and

$$\begin{aligned} p({\varvec{\varUpsilon }}_j={\mathbf {0}}|\alpha ) = \prod _{l=1}^{L-1}\prod _{k=1}^{2^l} \phi \left( 0\bigg |0, \frac{2n}{\alpha \rho (l+1)} ({\mathbf {X}}'{\mathbf {X}})_{jj}^{-1}\right) . \end{aligned}$$

where $({\mathbf {X}}'{\mathbf {X}})_{jj}^{-1}$ is the $(j+1,j+1)$th element of $({\mathbf {X}}'{\mathbf {X}})^{-1}$, and $\phi (\cdot |\mu , \sigma ^2)$ denotes the normal density with mean $\mu$ and variance $\sigma ^2$.

Proof

Since ${\varvec{\gamma }}_{l,k}|\alpha$ follows a multivariate normal, $({\varvec{\gamma }}_{l,k,-j}|{\varvec{\gamma }}_{l,k,j}=0,\alpha )$ still follows a multivariate normal distribution

$$\begin{aligned} p({\varvec{\gamma }}_{l,k,-j}|{\varvec{\gamma }}_{l,k,j}=0,\alpha )&\propto \exp \left\{ -\frac{\alpha \rho (l+1)}{4n} {\varvec{\gamma }}_{l,k}'({\mathbf {X}}'{\mathbf {X}}){\varvec{\gamma }}_{l,k} \right\} \nonumber \\&\propto \exp \left\{ -\frac{\alpha \rho (l+1)}{4n} {\varvec{\gamma }}_{l,k,-j}'({\mathbf {X}}_{-j}'{\mathbf {X}}_{-j}){\varvec{\gamma }}_{l,k,-j} \right\} \nonumber \\&\propto N_{q}\left( {\mathbf {0}}, \frac{2n}{\alpha \rho (l+1)} ({\mathbf {X}}_{-j}'{\mathbf {X}}_{-j})^{-1}\right) . \end{aligned}$$

This implies that $p({\varvec{\gamma }}_{l,k,-j}|{\varvec{\gamma }}_{l,k,j}=0,\alpha )=p_0({\varvec{\gamma }}_{l,k,-j}|\alpha )$ and by independence $p({\varvec{\varUpsilon }}_{-j}|{\varvec{\varUpsilon }}_j=0,\alpha )=p_0({\varvec{\varUpsilon }}_{-j}|\alpha )$. In addition, $\alpha$ is fixed and ${\varvec{\varUpsilon }}_{-j}$ is independent of all other parameters in ${\varvec{\psi }}$, thus the Assumption (2.12) holds. It is easy to evaluate $p({\varvec{\varUpsilon }}_j=0)$ by noting the properties of multivariate normal. $\square$

Proposition 2

Assume the same priors on ${\varvec{\gamma }}_{l,k}$ as Proposition 1 and additional prior on $\alpha$ as $\pi (\alpha )={\varGamma }(\alpha |a_0, b_0)$ under both $H_1$ and $H_0$. Then given existence of all involved expectations, $BF_{10}$ can be written as

$$\begin{aligned} BF_{10} = \left\{ p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\mathcal {D}})\right\} ^{-1} \left\{ E\left[ \frac{1}{p({\varvec{\varUpsilon }}_j={\mathbf {0}}|\alpha )} \right] \right\} ^{-1}, \end{aligned}$$

where the expectation is with respect to $p(\alpha |{\varvec{\varUpsilon }}_j={\mathbf {0}}, {\mathcal {D}})$.

Proof

First note that ${\varvec{\psi }}$ represents all remaining model parameters but ${\varvec{\varUpsilon }}_j$ and the prior for ${\varvec{\varUpsilon }}_j$ only depend on the precision parameter $\alpha$, so we have $p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\varvec{\psi }})=p({\varvec{\varUpsilon }}_j={\mathbf {0}}|\alpha )$. Also note that ${\mathcal {L}}({\varvec{\varUpsilon }}_j, {\varvec{\psi }})$ is the likelihood function, so we could denote it by $p({\mathcal {D}}|{\varvec{\varUpsilon }}_j, {\varvec{\psi }})$. It follows that

$$\begin{aligned}&p({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}|{\mathcal {D}})\int {\mathcal {L}}({\varvec{\varUpsilon }}_j, {\varvec{\psi }}) p({\varvec{\varUpsilon }}_j, {\varvec{\psi }})d({\varvec{\varUpsilon }}_j,{\varvec{\psi }}) \\&\quad = p({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}|{\mathcal {D}})\int p({\mathcal {D}}|{\varvec{\varUpsilon }}_j, {\varvec{\psi }}) p({\varvec{\varUpsilon }}_j, {\varvec{\psi }})d({\varvec{\varUpsilon }}_j,{\varvec{\psi }})\\&\quad = p({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}|{\mathcal {D}}) p({\mathcal {D}}) \\&\quad = p({\mathcal {D}}|{\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}) p({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }})\\&\quad = {\mathcal {L}}({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}) p({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}). \end{aligned}$$

Then we have

$$\begin{aligned} BF_{10}^{-1}&= p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\mathcal {D}}) \int \frac{ {\mathcal {L}}({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}) p({\varvec{\psi }})}{p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\mathcal {D}})\int {\mathcal {L}}({\varvec{\varUpsilon }}_j, {\varvec{\psi }}) p({\varvec{\varUpsilon }}_j, {\varvec{\psi }})d({\varvec{\varUpsilon }}_j,{\varvec{\psi }})}d{\varvec{\psi }}\\&= p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\mathcal {D}}) \int \frac{ {\mathcal {L}}({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}) p({\varvec{\psi }}) p({\varvec{\psi }}|{\varvec{\varUpsilon }}_j={\mathbf {0}},{\mathcal {D}})}{p({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}|{\mathcal {D}})\int {\mathcal {L}}({\varvec{\varUpsilon }}_j, {\varvec{\psi }}) p({\varvec{\varUpsilon }}_j, {\varvec{\psi }})d({\varvec{\varUpsilon }}_j,{\varvec{\psi }})}d{\varvec{\psi }}\\&= p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\mathcal {D}}) \int \frac{ {\mathcal {L}}({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }}) p({\varvec{\psi }}) p({\varvec{\psi }}|{\varvec{\varUpsilon }}_j={\mathbf {0}},{\mathcal {D}})}{{\mathcal {L}}({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }})p({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }})}d{\varvec{\psi }}\\&= p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\mathcal {D}}) \int \frac{ p({\varvec{\psi }}) p({\varvec{\psi }}|{\varvec{\varUpsilon }}_j={\mathbf {0}},{\mathcal {D}})}{p({\varvec{\varUpsilon }}_j={\mathbf {0}}, {\varvec{\psi }})}d{\varvec{\psi }}\\&= p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\mathcal {D}}) \int \frac{ p({\varvec{\psi }}|{\varvec{\varUpsilon }}_j={\mathbf {0}},{\mathcal {D}})}{p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\varvec{\psi }})}d{\varvec{\psi }}\\&= p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\mathcal {D}}) \int \int \frac{1}{p({\varvec{\varUpsilon }}_j={\mathbf {0}}|\alpha )} p({\varvec{\psi }}_{-\alpha }, \alpha |{\varvec{\varUpsilon }}_j={\mathbf {0}},{\mathcal {D}}) d{\varvec{\psi }}_{-\alpha } d\alpha \\&= p({\varvec{\varUpsilon }}_j={\mathbf {0}}|{\mathcal {D}}) \int \frac{1}{p({\varvec{\varUpsilon }}_j={\mathbf {0}}|\alpha )} p(\alpha |{\varvec{\varUpsilon }}_j={\mathbf {0}},{\mathcal {D}}) d\alpha , \end{aligned}$$

where $({\varvec{\psi }}_{-\alpha }, \alpha )={\varvec{\psi }}$. $\square$