Time-varying coefficients in a multivariate frailty model: Application to breast cancer recurrences of several types and death

Abstract

During their follow-up, patients with cancer can experience several types of recurrent events and can also die. Over the last decades, several joint models have been proposed to deal with recurrent events with dependent terminal event. Most of them require the proportional hazard assumption. In the case of long follow-up, this assumption could be violated. We propose a joint frailty model for two types of recurrent events and a dependent terminal event to account for potential dependencies between events with potentially time-varying coefficients. For that, regression splines are used to model the time-varying coefficients. Baseline hazard functions (BHF) are estimated with piecewise constant functions or with cubic M-Splines functions. The maximum likelihood estimation method provides parameter estimates. Likelihood ratio tests are performed to test the time dependency and the statistical association of the covariates. This model was driven by breast cancer data where the maximum follow-up was close to 20 years.

Appendices

Appendix 1: Log-likelihood calculus

The conditional likelihood can be written as:

$$\begin{aligned} L_i\big (\varPhi |u_i,v_i\big )= \Big (Pevent_i^{(1)} Pevent_i^{(2)}\Big ) Pdeath_i \end{aligned}$$

$Pevent_i^{(l)}$ represents the contribution to the likelihood of the recurrent event $l$, and $Pdeath_i$ the contribution to the likelihood of death for the individual $i$:

$$\begin{aligned} Pevent_i^{(1)}&\!\!{=} \displaystyle \left\{ \prod _{j=1}^{n_i^{(1)}+1}r_{i}^{(1)}\Big (T_{ij}^{(1)}\Big )^{\delta _{ij}^{(1)}}\exp (u_i)\right\} \exp \left[ \!\!-\exp (u_i)\displaystyle \sum _{j=1}^{n_i^{(1)}+1}\int _{T_{i(j-1)}^{(1)}}^{T_{ij}^{(1)}}r_{i}^{(1)}(t)dt\right] \\ Pevent_i^{(2)}&\!\! {=} \displaystyle \left\{ \prod _{j=1}^{n_i^{(2)}+1}r_{i}^{(2)}\Big (T_{ij}^{(2)}\Big )^{\delta _{ij}^{(2)}}\exp (v_i)\right\} \exp \left[ \!\!-\exp (v_i)\displaystyle \sum _{j=1}^{n_i^{(2)}+1}\int _{T_{i(j-1)}^{(2)}}^{T_{ij}^{(2)}}r_{i}^{(2)}(t)dt\right] \\ Pdeath_i&\!\! {=} \left\{ \exp (\alpha _1u_i+\alpha _2v_i){\uplambda }\big (T_i^D\big )\right\} ^{\delta _{i}^{D}} \exp \left[ \!\!-\exp \left( \alpha _1 u_i+\alpha _2 v_i\right) \displaystyle \int _0^{T_i^D}{\uplambda }_{i}(t)dt\right] \end{aligned}$$

The conditional individual contribution to the likelihood is:

$$\begin{aligned}&L_i\big (\varPhi |u_i,v_i\big ) = \displaystyle \left\{ \prod _{j=1}^{n_i^{(1)}}r_{i}^{(1)}\Big (T_{ij}^{(1)}\Big )\right\} \exp \left[ n_i^{(1)}u_i-\exp (u_i)\displaystyle \sum _{j=1}^{n_i^{(1)}+1}\int _{T_{i(j-1)}^{(1)}}^{T_{ij}^{(1)}}r_{i}^{(1)}(t)dt\right] \\&\quad \times \left\{ \prod _{j=1}^{n_i^{(2)}}r_{i}^{(2)}\Big (T_{ij}^{(2)}\Big )\right\} \exp \left[ n_i^{(2)}v_i-\exp (v_i)\displaystyle \sum _{j=1}^{n_i^{(2)}+1}\int _{T_{i(j-1)}^{(2)}}^{T_{ij}^{(2)}}r_{i}^{(2)}(t)dt\right] \\&\quad \times \left\{ {\uplambda }\big (T_i^D\big )\right\} ^{\delta _{i}^{D}} \exp \left[ \delta _{i}^{D}(\alpha _1u_i+\alpha _2v_i)-\exp \left( \alpha _1 u_i+\alpha _2 v_i\right) \displaystyle \int _0^{T_i^D}{\uplambda }_{i}(t)dt\right] \end{aligned}$$

Then, we integrate $L_i(\varPhi |u_i,v_i)$ through the random effects $u_i,v_i$ to obtain the likelihood.

$$\begin{aligned} L_i(\varPhi )= & {} \displaystyle \int _{0}^{\infty }\int _{0}^{\infty }\left\{ \prod _{j=1}^{n_i^{(1)}+1}r_{i}^{(1)}\Big (T_{ij}^{(1)}\Big )^{\delta _{ij}^{(1)}} \exp \left[ n_i^{(1)}u_i-\exp (u_i)\displaystyle \sum _{j=1}^{n_i^{(1)}+1}\int _{T_{i(j-1)}^{(1)}}^{T_{ij}^{(1)}}r_{i}^{(1)}(t)dt\right] \right. \\&\left. \prod _{j=1}^{n_i^{(2)}+1}r_{i}^{(2)}\Big (T_{ij}^{(2)}\Big )^{\delta _{ij}^{(2)}} \exp \left[ n_i^{(2)}v_i-\exp (v_i)\displaystyle \sum _{j=1}^{n_i^{(2)}+1}\int _{T_{i(j-1)}^{(2)}}^{T_{ij}^{(2)}}r_{i}^{(2)}(t)dt\right] \right\} ^{1-\delta _i^D}\\&\left[ \exp (\alpha _1u_i+\alpha _2v_i){\uplambda }\big (T_i^D\big )\right] ^{\delta _{i}^{D}} \exp \left[ -\exp \left( \alpha _1 u_i+\alpha _2 v_i\right) \displaystyle \int _0^{T_i^D}{\uplambda }_{i}(t)dt\right] f(u_i,v_i)du_idv_i \end{aligned}$$

Appendix 2: Estimation of the individual random effects $u_i,v_i$ for the martingale residuals

Let $T_i=\{T_{ij}^{(l)}, j=1,\ldots ,n_i^{(l)+1}\}$. The posterior probability density function is

$$\begin{aligned} f\big (u_i,v_i|T_i,\widehat{\varPhi }\big )= \displaystyle \frac{f\big (T_i|u_i,v_i,\widehat{\varPhi }\big )\times f\big (u_i,v_i|\widehat{\varPhi }\big )}{f\big (T_i|\widehat{\varPhi }\big )} \end{aligned}$$

and

$$\begin{aligned} f\big (u_i,v_i|T_i,\widehat{\varPhi }\big )\propto f\big (T_i|u_i,v_i,\widehat{\varPhi }\big )\times f\big (u_i,v_i|\widehat{\varPhi }\big ). \end{aligned}$$

Here $f(T_i|u_i,v_i,\widehat{\varPhi })$ corresponds to the likelihood of the individual $i$ given $\widehat{\varPhi }$ and given the random effects $u_i,v_i$.

The mode of the posterior probability density function is obtained by maximizing it using the Marquardt algorithm:

$$\begin{aligned}&f\big (u_i,v_i|T_i,\widehat{\varPhi }\big ) \propto \exp \left[ \Bigg (-\exp (u_i)\int _{0}^{T_i^D}\widehat{r_{i}^{(1)}}(u)du-\exp (v_i)\int _{0}^{T_i^D}\widehat{r_{i}^{(2)}}(u)du\right. \\&\quad - \,\exp \Big (\widehat{\alpha _1} u_i+\widehat{\alpha _2} v_i\Big )\int _{0}^{T_i^D}\widehat{{\uplambda }_{i}}(u)du \Bigg )\\&\quad + \, \left. \left( \frac{-u_i^2/\widehat{\theta }+2\widehat{\rho }u_iv_i/\sqrt{\widehat{\theta }\widehat{\eta }}-v_i^2/\widehat{\eta }}{2\big (1-\widehat{\rho }^2\big )} \Big (n_i^{(1)} +\delta _i^D\widehat{\alpha _1}\Big ) u_i+ \Big (n_i^{(2)} +\delta _i^D\widehat{\alpha _2}\Big ) v_i \right) \right] \end{aligned}$$

where,

$$\begin{aligned}&\displaystyle r_{i}^{(l)}(t)=r_{0}^{(l)}(t)\exp \left[ \displaystyle \sum _{j=-q+1}^{m}\zeta ^{(l)}_j B_{j,q}(t)Z_i(t)\right] \, \hbox {and} \\&\displaystyle {\uplambda }_{i}(t)={\uplambda }_{0}(t)\exp \left[ \displaystyle \sum _{j=-q+1}^{m}\zeta ^{(D)}_j B_{j,q}(t)Z_i(t)\right] \end{aligned}$$