Kernel regression for cause-specific hazard models with time-dependent coefficients

Abstract

Competing risk data appear widely in modern biomedical research. In the past two decades, cause-specific hazard models are often used to deal with competing risk data. There is no current study on the kernel likelihood method for the cause-specific hazard model with time-varying coefficients. We propose to use the local partial log-likelihood approach for nonparametric time-varying coefficient estimation. Simulation studies demonstrate that our proposed nonparametric kernel estimator performs well under assumed finite sample settings. And we also compare the local kernel estimator with the penalized spline estimator. Finally, we apply the proposed method to analyze a diabetes dialysis study with competing death causes.

A cross-validation score

In this appendix A, we first construct the log-likelihood of the ith subject. For failure type $j=1,\dots ,m$, omitting the n in the log-likelihood formula (2), we can obtain the local partial log-likelihood as follows:

$$\begin{aligned}&l_j(b_j)=\sum _{i=1}^n\int _0^\tau K_h(u-t)\left\{ \widetilde{Z}_i(u,t)^Tb_j-\log \left[ \sum _{l=1}^nY_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) \right] \right\} \nonumber \\&\quad dN_{ji}(u),~j=1,\dots ,m. \end{aligned}$$

(10)

It is equivalent to the local partial log-likelihood defined in (2). Similarly, when the ith subject is left out, the local partial log-likelihood can be written as

$$\begin{aligned}&l_j^{(-i)}(b_j)=\sum _{s\ne i}\int _0^\tau K_h(u-t)\left\{ \widetilde{Z}_s(u,t)^Tb_j-\log \left[ \sum _{l\ne i}Y_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) \right] \right\} \nonumber \\&\quad dN_{js}(u),~j=1,\dots ,m. \end{aligned}$$

(11)

From (10) and (11) yields the contribution of individual i to the likelihood

$$\begin{aligned} l^i_j(b_j)= & {} l_j(b_j)-l^{(-i)}_j(b_j)\nonumber \\= & {} \int _0^\tau K_h(u-t)\left\{ \widetilde{Z}_i(u,t)^Tb_j-\log \left[ \sum _{l=1}^nY_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) \right] \right\} dN_{ji}(u)\nonumber \\&+\sum _{s\ne i}\int _0^\tau K_h(u-t)\log \left[ \frac{\sum _{l\ne i}Y_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) }{\sum _{l=1}^nY_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) }\right]\,\,dN_{js}(u),~j=1,\dots ,m. \nonumber \end{aligned}$$

For the second term on the right-hand side, the term

$$\begin{aligned}&\log \left[ \frac{\sum _{l\ne i}Y_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) }{\sum _{l=1}^nY_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) }\right] \nonumber \\&\quad =\log \left[ 1-\frac{Y_i(u)\exp \left( \widetilde{Z}_i(u,t)^Tb_j\right) }{\sum _{l=1}^nY_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) }\right] \approx 0,~j=1,\dots ,m. \end{aligned}$$

Therefore, for failure type $j=1,\dots ,m$, we derive an alternative expression for $l_j^i(b_j)$ by

$$\begin{aligned}&l_j^i(b_j)=\int _0^\tau K_h(u-t)\left\{ \widetilde{Z}_i(u,t)^Tb_j-\log \left[ \sum _{l=1}^nY_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) \right] \right\} \nonumber \\&\quad dN_{ji}(u),~j=1,\dots ,m.\nonumber \end{aligned}$$

It is equivalent to the likelihood as follows:

$$\begin{aligned}&l_j^i(b_j)=\frac{1}{n}\int _0^\tau K_h(u-t)\left\{ \widetilde{Z}_i(u,t)^Tb_j-\log \left[ \sum _{l=1}^nY_l(u)\exp \left( \widetilde{Z}_l(u,t)^Tb_j\right) \right] \right\} \nonumber \\&\quad dN_{ji}(u),~j=1,\dots ,m. \end{aligned}$$

(12)

Next, we give the derived process of the approximations for $\widehat{\mathbf {b}}^{(-i)}_{j}$ and the cross-validated score $CV_s(h)$. For failure type $j=1,\dots ,m$, we apply a Taylor expansion to approximate $\widehat{\mathbf {b}}^{(-i)}_{j}$. From (6), we have $l^{(-i)}_j(b_j)=l_j(b_j)-l^i_j(b_j),$ taking the derivative of both sides of this equation with respect to $b_j$

$$\begin{aligned} \frac{\partial l^{(-i)}_j}{\partial b_j}(b_{j})= & {} \frac{\partial l_j}{\partial b_j}(b_j)-\frac{\partial l^i_j}{\partial b_j}(b_j),~j=1,\dots ,m, \end{aligned}$$

(13)

$$\begin{aligned} \frac{\partial ^2 l^{(-i)}_j}{\partial b_j^2}(b_{j})= & {} \frac{\partial ^2 l_j}{\partial b_j^2}(b_{j})-\frac{\partial ^2 l^i_j}{\partial b_j^2}(b_{j}),~j=1,\dots ,m, \end{aligned}$$

(14)

at $b_j=\widehat{\mathbf {b}}_j$, using first-order Taylor expansion to approximate

$$\begin{aligned} \frac{\partial l^{(-i)}_j}{\partial b_j}(b_{j})= & {} \frac{\partial l^{(-i)}_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})+\frac{\partial ^2 l_j^{(-i)}}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})(b_j-\widehat{\mathbf {b}}_{j}),~j=1,\dots ,m.\nonumber \end{aligned}$$

Combining $\frac{\partial l^{(-i)}_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})=\frac{\partial l_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})-\frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})$, $\frac{\partial l_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})=0$, and substituting $\widehat{\mathbf {b}}_{j}^{(-i)}$ for $b_j$ in the above formula, we have

$$\begin{aligned}&0=\frac{\partial l^{(-i)}_j}{\partial b_j}\left( \widehat{\mathbf {b}}_{j}^{(-i)}\right) = -\frac{\partial l^{i}_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})+\frac{\partial ^2 l_j^{(-i)}}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})\left( \widehat{\mathbf {b}}_{j}^{(-i)}-\widehat{\mathbf {b}}_{j}\right) ,~j=1,\dots ,m.\nonumber \end{aligned}$$

Solving the above equation with respect to $\widehat{\mathbf {b}}^{(-i)}_{j}$, we infer

$$\begin{aligned} \widehat{\mathbf {b}}^{(-i)}_{j}\,\,= \,\,& {} \widehat{\mathbf {b}}_{j}+\left\{ \frac{\partial ^2 l_j^{(-i)}}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})\right\} ^{-1}\frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j}),~j=1,\dots ,m. \end{aligned}$$

(15)

Substituting $\widehat{\mathbf {b}}_{j}$ for $b_j$ in (14), and combined with (15), for failure type $j=1,\dots ,m$, we establish the approximations for $\widehat{\mathbf {b}}^{(-i)}_{j}$ as

$$\begin{aligned} \widehat{\mathbf {b}}^{(-i)}_{j}\,\,=\,\, & {} \widehat{\mathbf {b}}_{j}+\left\{ \frac{\partial ^2 l_j}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})\right\} ^{-1}\frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j}),~j=1,\dots ,m, \end{aligned}$$

(16)

where we ignore the second derivation of $l_j^i(b_j)$ because its calculation will consume much computer storage and time, and the approximation of the estimator $\widehat{\mathbf {b}}^{(-i)}_{j}$ is the function of $\widehat{\mathbf {b}}_{j}$.

Then we are going to approximate the cross-validated score $CV_s(h)=\sum _{i=1}^n l^i_j\left( \widehat{\mathbf {b}}^{(-i)}_{j}\right)$ for failure type $j=1,\dots ,m$. Associating with (12) and (16), for $l_j^i(b_j)$ using a first-order Taylor approximation, we obtain

$$\begin{aligned} l_j^i\left( \widehat{\mathbf {b}}_{j}^{(-i)}\right)\,=\, & {} l_j^i\left( \widehat{\mathbf {b}}_{j}+\left\{ \frac{\partial ^2 l_j}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})\right\} ^{-1}\frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\right) \nonumber \\ \,=\, & {} l_j^i(\widehat{\mathbf {b}}_{j})+\left\{ \frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\right\} ^T\left[ \widehat{\mathbf {b}}_{j}+\left\{ \frac{\partial ^2 l_j}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})\right\} ^{-1}\frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})-\widehat{\mathbf {b}}_{j}\right] \nonumber \\ \,=\, & {} l_j^i(\widehat{\mathbf {b}}_{j})+\left\{ \frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\right\} ^T\left\{ \frac{\partial ^2 l_j}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})\right\} ^{-1}\frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\nonumber \\\,=\, & {} l_j^i(\widehat{\mathbf {b}}_{j})+tr\left[ \left\{ \frac{\partial ^2 l_j}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})\right\} ^{-1}\frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\left\{ \frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\right\} ^T\right] ,~j=1,\dots ,m, \end{aligned}$$

(17)

where for the first term of the right-hand side, using a Taylor expansion at $\widehat{\mathbf {b}}_{j}^{(-i)}=\widehat{\mathbf {b}}_{j}$, and for the third term of the right-hand side, utilizing the trace’s basic properties. Then, from (17), we establish the approximation of $CV_s$ for failure type $j=1,\dots ,m$

$$\begin{aligned} CV_s(h)= & {} \sum _{i=1}^{n}l_j^i(\widehat{\mathbf {b}}_{j})+tr\,\,\left[ \left\{ \frac{\partial ^2 l_j}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})\right\} ^{-1}\sum _{i=1}^{n}\frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\left\{ \frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\right\} ^T\right] \nonumber \\ \,=\, & {} l_j(\widehat{\mathbf {b}}_{j})+tr\,\,\left[ \left\{ \frac{\partial ^2 l_j}{\partial b^2_j}(\widehat{\mathbf {b}}_{j})\right\} ^{-1}\sum _{i=1}^{n}\frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\left\{ \frac{\partial l^i_j}{\partial b_j}(\widehat{\mathbf {b}}_{j})\right\} ^T\right] ,~j=1,\dots ,m.\nonumber \end{aligned}$$