Bayesian Functional Mixed Models for Survival Responses with Application to Prostate Cancer

Abstract

In this chapter, we propose a flexible approach to model functional measurements for survival outcomes. Often the class of models for functional observations are assumed to be linear, which may be too restrictive in some cases. We propose an alternative model, in which the simple linear mixed model has been modified by a more flexible semiparametric spline-based functional mixed model, wherein the usage of splines simplifies parameterizations and the joint modeling framework allows synergistic benefit between the regression of functional predictors and the modeling of survival data. We explicitly model the number and location of change points such that our formulation allows for an unknown set of basis functions characterizing the population-averaged and patient-specific trajectories. In addition, we propose a novel auxiliary variable scheme for a fully Bayesian estimation of our model, which not only allows dimension reduction of the parameter space but also allows efficient sampling from the conditional distributions. We illustrate our approach with a recent prostate cancer clinical trial study.

Appendices

Acknowledgments

We thank Dr. Randall Millikan from the University of Texas M.D. Anderson Cancer Center for supplying the prostate cancer data set as an application example of our proposed methodology. V. Baladandayuthapani was partially supported by NIH grant R01 CA160736. Both K-A Do and V. Baladandayuthapani were partially supported by the Cancer Center Support Grant (CCSG) (P30 CA016672). Dr. K-A Do was also partially supported by the University of Texas SPORE in prostate cancer National Institute of Health grant (P50 CA140388). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Cancer Institute or the National Institutes of Health.

Appendix

2.1 The Model Summary with Specified Prior Distributions

To summarize the hierarchical model setup, we define

$$\begin{aligned}& &\\ {\rm Random \;function} \;{\mathbf{Y}}_{i} &\sim{\rm MVN}({\mathbf{X}}_{i,{\boldsymbol{\gamma}}}{\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}},\sigma_{\boldsymbol{\epsilon}}^2 {\mathbf{I}}_{p_i}),\\ \nonumber \sigma_{\boldsymbol{\epsilon}}^2 & \sim IG(a_{\sigma}, b_{\sigma}),\\ \nonumber{\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}} &\sim{\rm MVN}({\boldsymbol{\beta}}_{{\boldsymbol{\gamma}}_i}, {\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}), \\ \nonumber{\boldsymbol{\beta}}_{{\boldsymbol{\gamma}}_i} & = {\mathbf{J}}_i {\boldsymbol{\beta}}_{\boldsymbol{\gamma}}, \\ \nonumber{\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}} &= {\mathbf{J}}_i {\boldsymbol{\Omega}}_{\boldsymbol{\gamma}} {\mathbf{J}}_i^{\prime} \;{\rm where} {\boldsymbol{\Omega}}_{\boldsymbol{\gamma}} = {\rm diag}({\boldsymbol{Upsigma}}, \sigma^2 {\mathbf{I}}_{K_{\boldsymbol{\gamma}}^*}), \\ \nonumber{\boldsymbol{\beta}}_{\boldsymbol{\gamma}} & \sim{\rm MVN} (0, c{\mathbf{I}}_{K_{\boldsymbol{\gamma}}}), \\ \nonumber{\boldsymbol{\Sigma}} &\sim{\rm IW}({\mathbf{A}}, b), \\ \nonumber \sigma^2 & \sim{\rm IG}(c_{\sigma}, d_{\sigma}), \\ \nonumber \gamma_k & \sim \mathit{\rm Bernoulli} (\pi_k), \;\mathit{\rm where} \pi_k=\pi {\rm for} \;{\rm all} \;k, \\ \nonumber \pi & \sim \mathit{\rm Beta}(a_{\pi}, b_{\pi}), \\ \nonumber \mathit{\rm Linear} \;\mathit{\rm predictor} \;w_{i} &\sim{\rm N}({\mathbf{B}}_{i,{\boldsymbol{\gamma}}}^{\prime}{\boldsymbol{\theta}}_{\boldsymbol{\gamma}},\tau^2), \;\mathit{\rm where} \;{\mathbf{B}}_{i,{\boldsymbol{\gamma}}}^{\prime}=[{\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}}^{\prime}, {\mathbf{L}}_i^{\prime}],\\ \nonumber{\boldsymbol{\theta}}_{\boldsymbol{\gamma}},\tau ^{2}| {\boldsymbol{\gamma}}, {\mathbf{V}}_{\boldsymbol{\gamma}} & \sim{\rm NIG}(0,{\mathbf{V}}_{\boldsymbol{\gamma}},a_{\tau},b_{\tau}), \;\mathit{\rm where} \;{\mathbf{V}}_{\boldsymbol{\gamma}}={\rm diag}({\mathbf{h}}),\\ \nonumber h_\ell &\sim{\rm IG}(c_\ell,d_\ell), \\ \nonumber \mathit{\rm Hazard} \;\mathit{\rm function} \;h(t \mid{\mathbf{Y}}_i) & = h_0 (t) {\rm exp}(w_i), \\ \nonumber h_0(t)&= \lambda_j \;(s_{j-1} \le t<s_j), \\ \nonumber \lambda_j & \sim{\rm IG}(a_j,b_j),\end{aligned}$$

for $i=1,\ldots,n$, $j=1,\ldots,J$, $k=1, \ldots, K$, and $\ell=1,\ldots,(K_{\gamma}+m)$.

The fourth and fifth lines in the above model need special attention. Based on the fact that the ith curve may not span the complete set of selected change points, ${\boldsymbol{\beta}}_{{\boldsymbol{\gamma}}_i}$ and ${\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}$ are the subject-specific realizations of parameters ${\boldsymbol{\beta}}_{\boldsymbol{\gamma}}$ and ${\boldsymbol{\Omega}}_{\boldsymbol{\gamma}}$, where they respectively represent the population curve and its covariance corresponding to the latent variable ${\boldsymbol{\gamma}}$. The relationship can be expressed via a rectangular indicator matrix ${\mathbf{J}}_i$ as ${\boldsymbol{\beta}}_{{\boldsymbol{\gamma}}_i} = {\mathbf{J}}_i {\boldsymbol{\beta}}_{\boldsymbol{\gamma}}$ and ${\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}} = {\mathbf{J}}_i {\boldsymbol{\Omega}}_{\boldsymbol{\gamma}} {\mathbf{J}}_i^{\prime}$ with ${\boldsymbol{\Omega}}_{\boldsymbol{\gamma}} = {\rm diag}({\boldsymbol{\Sigma}}, \sigma^2 {\mathbf{I}}_{K_{\boldsymbol{\gamma}}^*})$. For example, suppose there are five change points for the population curve, and the ith individual only spans the first two change points (i.e., does not have measurements beyond the third change point). Because the basis has the quadratic polynomial segment and the change points segment, the dimensions of ${\boldsymbol{\beta}}_{\boldsymbol{\gamma}}$ and ${\boldsymbol{\Omega}}_{\boldsymbol{\gamma}}$ will be 8 and 8 by 8. However, for the ith individual, the dimensions of ${\boldsymbol{\beta}}_{{\boldsymbol{\gamma}}_i}$ and ${\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}$ are 5 and 5 by 5. Therefore, ${\boldsymbol{\beta}}_{{\boldsymbol{\gamma}}_i}$ is linked to ${\boldsymbol{\beta}}_{\boldsymbol{\gamma}}$ via a 5 by 8 rectangular index matrix:

$${\mathbf{J}}_i=\left[\begin{array}{cccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\end{array} \right].$$

(3.7)

The same ${\mathbf{J}}_i$ is used to link ${\boldsymbol{\Omega}}_{\boldsymbol{\gamma}}$ and ${\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}$. These expressions with ${\mathbf{J}}_i$ enable the derivation of the posterior distributions below.

2.2 Posterior Distributions

The conditional distribution for the ith regressed covariates vector ${\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}}$ is updated using regression likelihood

$$ \nonumber{\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}} \mid{\mathbf{X}}_{i,{\boldsymbol{\gamma}}}, {\mathbf{Y}}_i, \sigma_{\boldsymbol{\epsilon}}^2, {\boldsymbol{\Omega}}_{\boldsymbol{\gamma}}, w_i, \tau^2,{\boldsymbol{\theta}}_{\boldsymbol{\gamma}}, {\boldsymbol{\gamma}} \sim{\rm MVN}({\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}}^*,\tau^2 {\boldsymbol{\Omega}}_{i, {\boldsymbol{\gamma}}}^*),$$

where ${\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^*=(\tau^2({\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1}+{\mathbf{X}}_{i,{\boldsymbol{\gamma}}}^{\prime} {\mathbf{X}}_{i,{\boldsymbol{\gamma}}}/\sigma_{\boldsymbol{\epsilon}}^2)+{\boldsymbol{\theta}}_{1i,{\boldsymbol{\gamma}}}{\boldsymbol{\theta}}_{1i,{\boldsymbol{\gamma}}}^{\prime})^{-1}$ and ${\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}}^*={\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^*\times$ $(\tau^2({\mathbf{X}}_{i,{\boldsymbol{\gamma}}}^{\prime}{\mathbf{Y}}_i/\sigma_{\boldsymbol{\epsilon}}^2+{\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1}{\boldsymbol{\mu}}_{i,{\boldsymbol{\gamma}}})+(w_i-{\mathbf{L}}_i^{\prime}{\boldsymbol{\theta}}_2){\boldsymbol{\theta}}_{1i,{\boldsymbol{\gamma}}})$. The notation ${\boldsymbol{\theta}}_{1i,{\boldsymbol{\gamma}}}$ is the part of the coefficients corresponding to time-dependent covariates ${\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}}$, while ${\boldsymbol{\theta}}_2$ is the part of the coefficients corresponding to time-independent covariates ${\mathbf{L}}_i$ in later posteriors. The model variance $\sigma_{\boldsymbol{\epsilon}}^2$ is updated by

$$\nonumber \sigma_{\boldsymbol{\epsilon}}^2 | {\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}},{\mathbf{Y}}_i,{\mathbf{X}}_{i,{\boldsymbol{\gamma}}} \sim{\rm IG}(a_{\sigma}^*, b_{\sigma}^*),$$

where $a_{\sigma}^*=a_{\sigma}+(\sum_{i=1}^n p_i)/2$ and $b_{\sigma}^*=b_{\sigma}+[\sum_{i=1}^n({\mathbf{Y}}_i - {\mathbf{X}}_{i,{\boldsymbol{\gamma}}} {\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}})^{\prime}({\mathbf{Y}}_i - {\mathbf{X}}_{i,{\boldsymbol{\gamma}}} {\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}})]/2$. The indicator vector ${\boldsymbol{\gamma}}$ can be updated elementwise using the Metropolis–Hastings algorithm with marginal posterior $\gamma_k \mid{\boldsymbol{\gamma}}_{-k},{\mathbf{Y}}_i, {\mathbf{X}}_{i,{\boldsymbol{\gamma}}}, \sigma_{\boldsymbol{\epsilon}}^2,$ ${\boldsymbol{Upsigma}}, \sigma^2, {\boldsymbol{\theta}}_{\boldsymbol{\gamma}}, {\mathbf{V}}_{\boldsymbol{\gamma}}$ proportional to

$$\begin{aligned} & \pi(\gamma_k)\pi({\boldsymbol{\theta}}_{\boldsymbol{\gamma}})\pi({\mathbf{V}}_{\boldsymbol{\gamma}})\left[\frac{|{\boldsymbol{\Upphi}}_{\boldsymbol{\gamma}}^{-1}|}{|c{\mathbf{I}}_{\boldsymbol{\gamma}}|} \prod\nolimits_{i=1}^n \frac{|\tau^2 {\mathbf{M}}_{i,{\boldsymbol{\gamma}}}^{-1}|}{|{\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}|} \right]^{1/2} {\rm exp} \left\{ \frac{1}{2 \tau^2} \sum\nolimits_{i=1}^n {\boldsymbol{\alpha}}_{i,{\boldsymbol{\gamma}}}^{\prime} {\mathbf{M}}_{i,{\boldsymbol{\gamma}}}^{-1} {\boldsymbol{\alpha}}_{i,{\boldsymbol{\gamma}}} \right\} \\ \nonumber & \times {\rm exp} \left\{ \frac{1}{2} \left( \sum\nolimits_{i=1}^n {\boldsymbol{\alpha}}_{i,{\boldsymbol{\gamma}}}^{\prime} {\mathbf{M}}_{i,{\boldsymbol{\gamma}}}^{-1} {\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1} {\mathbf{J}}_i\right) {\boldsymbol{\Phi}}_{\boldsymbol{\gamma}}^{-1} \left( \sum\nolimits_{i=1}^n {\mathbf{J}}_i {\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1} {\mathbf{M}}_{i,{\boldsymbol{\gamma}}}^{-1} {\boldsymbol{\alpha}}_{i,{\boldsymbol{\gamma}}} \right) \right\},\end{aligned}$$

where ${\boldsymbol{\alpha}}_{i,{\boldsymbol{\gamma}}} = \tau^2 {\mathbf{X}}_{i,{\boldsymbol{\gamma}}}^{\prime} {\mathbf{Y}}_i/\sigma_{\boldsymbol{\epsilon}}^2 + (w_i-{\mathbf{L}}_i^{\prime} {\boldsymbol{\theta}}_2){\boldsymbol{\theta}}_{1i,{\boldsymbol{\gamma}}}$, ${\mathbf{M}}_{i,{\boldsymbol{\gamma}}}=\tau^2({\mathbf{X}}_{i,{\boldsymbol{\gamma}}}^{\prime} {\mathbf{X}}_{i,{\boldsymbol{\gamma}}} /\sigma_{\boldsymbol{\epsilon}}^2 +{\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1}) + {\boldsymbol{\theta}}_{1i,{\boldsymbol{\gamma}}} {\boldsymbol{\theta}}_{1i,{\boldsymbol{\gamma}}}^{\prime}$ and ${\boldsymbol{\Upphi}}_{\boldsymbol{\gamma}}=(\sum_{i=1}^n {\mathbf{J}}_i^{\prime} {\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1} {\mathbf{J}}_i) - \tau^2 (\sum_{i=1}^n {\mathbf{J}}_i^{\prime} {\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1}{\mathbf{M}}_{i,{\boldsymbol{\gamma}}}^{-1}{\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1} {\mathbf{J}}_i)+(1/c){\mathbf{I}}_{K_{\boldsymbol{\gamma}}}$. It is worth to point out that the generation of a candidate ${\boldsymbol{\gamma}}$ is done by changing one element at a time in fixed sequencing order within each iteration. The conditional distribution for the informative scalar w _i follows the combination of information from both the regression and proportional hazards models. The likelihood of the PH model leads to its nonstandard form

$$\begin{aligned} w_i \mid T_i, \delta_i, h_0(t), {\mathbf{B}}_{i, {\boldsymbol{\gamma}}}, {\boldsymbol{\theta}}_{\boldsymbol{\gamma}}, \tau^2 \quad\propto &{\rm exp} \left\{-\frac{(w_i^2-2w_i{\mathbf{B}}_{i,{\boldsymbol{\gamma}}}^{\prime} {\boldsymbol{\theta}}_{i,{\boldsymbol{\gamma}}})}{2\tau^2} \right\} \times \\ & \Bigg[ h_0(T_i) {\rm exp}(w_i) \Bigg]^{\delta_i} {\rm exp}\left\{ - \int_0^{T_i} {\rm exp}(w_i)h_0(t) du \right\},\end{aligned}$$

which can be updated by a Metropolis step.

The following layer includes the regression coefficient as population mean ${\boldsymbol{\beta}}_{\boldsymbol{\gamma}}$, which can be updated as

$$\nonumber{\boldsymbol{\beta}}_{\boldsymbol{\gamma}} \mid{\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}}, {\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}} \sim{\rm MVN}({\boldsymbol{\beta}}_{\boldsymbol{\gamma}}^*, c{\mathbf{M}}),$$

where ${\mathbf{M}}=(c \sum_{i=1}^n {\mathbf{J}}_i^{\prime} {\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1} {\mathbf{J}}_i+{\mathbf{I}}_{K_{\boldsymbol{\gamma}}})^{-1}$ and ${\boldsymbol{\beta}}_{\boldsymbol{\gamma}}^*=c{\mathbf{M}}(\sum_{i=1}^n {\mathbf{J}}_i^{\prime}{\boldsymbol{\Omega}}_{i,{\boldsymbol{\gamma}}}^{-1}{\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}})$. The unstructured covariance matrix of the polynomial part for quadratic spline coefficients, ${\boldsymbol{Upsigma}}$, is updated as

$$\nonumber{\boldsymbol{Upsigma}} | {\boldsymbol{\beta}}_{\boldsymbol{\gamma}}, {\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}} \sim IW({\mathbf{A}}^*, b^*),$$

where ${\mathbf{A}}^*=[{\mathbf{A}}^{-1}+ \sum_{i=1}^n ({\boldsymbol{\alpha}}_{i1} {\boldsymbol{\alpha}}_{i1}^{\prime})]^{-1}$, ${\boldsymbol{\alpha}}_i={\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}}-{\boldsymbol{\beta}}_{{\boldsymbol{\gamma}}_i}=[{\boldsymbol{\alpha}}_{i1}^{\prime}, {\boldsymbol{\alpha}}_{i2}^{\prime}]^{\prime}$, and $b^*=b+n$. Here, the dimensions of ${\boldsymbol{\alpha}}_{i1}$ and ${\boldsymbol{\alpha}}_{i2}$ are 3 × 1 and $K_{{\boldsymbol{\gamma}}_i}^* \times 1$. Linking to the covariance of the change points part for the quadratic spline coefficients, σ², is updated as

$$\nonumber \sigma^2 \mid{\boldsymbol{\beta}}_{i,{\boldsymbol{\gamma}}}'s, {\boldsymbol{\beta}}_{\boldsymbol{\gamma}} \sim IG(c_{\sigma}^*, d_{\sigma}^*),$$

where $c_{\sigma}^*=c_{\sigma}+(\sum_{i=1}^n K_{{\boldsymbol{\gamma}}_i}^*)/2$ and $d_{\sigma}^*=d_{\sigma}+(\sum_{i=1}^n {\boldsymbol{\alpha}}_{i2}^{\prime}{\boldsymbol{\alpha}}_{i2})/2$. The probability of being change point π can be updated as

$$\nonumber \pi \mid{\boldsymbol{\gamma}} \sim Beta(a_{\pi}^*, b_{\pi}^*),$$

where $a_{\pi}^*=a_{\pi}+K_{\boldsymbol{\gamma}}$ and $b_{\pi}^*=b_{\pi}+K+K_{\boldsymbol{\gamma}}$. The common coefficient vector ${\boldsymbol{\theta}}$ in the linear predictor model is updated as

$$\nonumber{\boldsymbol{\theta}}_{\boldsymbol{\gamma}} \mid{\mathbf{w}}, {\mathbf{B}}, \tau^2, {\mathbf{V}}_{\boldsymbol{\gamma}} \sim{\rm MVN}({\boldsymbol{\theta}}^*, \tau^2 {\mathbf{V}}^*),$$

where ${\mathbf{V}}^* = ({\mathbf{V}}_{\boldsymbol{\gamma}}^{-1}+\sum_{i=1}^n {\mathbf{J}}_i^{\prime} {\mathbf{B}}_{i,{\boldsymbol{\gamma}}} {\mathbf{B}}_{i,{\boldsymbol{\gamma}}}^{\prime} {\mathbf{J}}_i)^{-1}$ and ${\boldsymbol{\theta}}^* = {\mathbf{V}}^*(\sum_{i=1}^n w_i {\mathbf{J}}_i^{\prime} {\mathbf{B}}_{i,{\boldsymbol{\gamma}}})$. Here the definition of ${\mathbf{J}}_i$ is similar to its definition in (7.2) with a dimension adjustment to match ${\mathbf{B}}_{i,{\boldsymbol{\gamma}}}$. The conjugate inverse gamma prior for variance τ² leads to its conditional distribution:

$$ \tau^2 \mid{\boldsymbol{\theta}}_{\boldsymbol{\gamma}}, {\mathbf{V}}_{\boldsymbol{\gamma}}, {\mathbf{w}}, {\mathbf{B}} \sim IG(a_{\tau}^*,b_{\tau}^*),$$

where $a_{\tau}^*=a_{\tau}+(n+K_{\boldsymbol{\gamma}}+m)/2$ and $b_{\tau}^*=b_{\tau}+\Big[{\boldsymbol{\theta}}_{\boldsymbol{\gamma}}^{\prime} {\mathbf{V}}_{\boldsymbol{\gamma}}^{-1} {\boldsymbol{\theta}}_{\boldsymbol{\gamma}} + \sum_{i=1}^n (w_i-{\mathbf{B}}_{i,{\boldsymbol{\gamma}}}^{\prime} {\boldsymbol{\theta}}_{i,{\boldsymbol{\gamma}}})^2 \Big]/2$.

The next layer includes scale parameters h _k , which is updated by

$$\nonumber h_{\ell} \mid{\boldsymbol{\theta}}_{\boldsymbol{\gamma}}, \tau^2 \sim IG(c_{\ell}^*,d_{\ell}^*),$$

where $c_{\ell}^*=c_{\ell}+1/2$ and $d_{\ell}^*=d_{\ell}+\theta_{\ell}^2 /2 \tau^2$.

The parameters of baseline hazard step function $h_0(t)$, λ _j ’s, can be updated using the proportional hazards model:

$$\nonumber \lambda_j \mid{\mathbf{T}}, {\mathbf{w}} \sim IG(a_j^*, b_j^*),$$

where $a_j^*=a_j+ \sum_{i=1}^n \delta_i I(s_{j-1} \le T_i < s_j)$ and $b_j^*=b_j+\sum_{i=1}^n \Big[ I(T_i> s_{j-1}) \times \Big.$ $\Big. \int_{s_{j-1}}^{{\rm min}(T_i,s_j)} \mathit{\rm exp} (w_i)\mathit{{\rm d}u} \Big]$.