Polynomial spline estimation of panel count data model with an unknown link function

Abstract

Panel count data are frequently encountered in follow-up studies such as clinical trials, reliability researches, and insurance studies. Models about this type data usually assume the linearity form of the covariate variables on the log conditional mean function. However, the linearity assumption cannot be always guaranteed in practical applications, especially when high-dimensional covariates exist under investigation. In this paper, we propose a more flexible conditional mean regression model of panel count data with an unknown link function to describe the possible nonlinearity of the covariate effects. The partial likelihood procedure is developed to estimate the unknown link function and the regression parameters simultaneously by first approximating the unknown link function by polynomial splines, and then a two-step iterative algorithm is developed for computing implementation. Finally, the Breslow-type estimator is constructed for the baseline mean function. Asymptotic results of the proposed estimators are discussed under some regularity conditions. In addition, penalized spline estimation procedure is also introduced as an extension. Extensive numerical studies are carried out and indicate that the proposed procedure works well. Finally, two applications of bladder cancer study and skin cancer study are also presented for illustration.

Appendices

Appendix I: Tables and figures

See Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and Figs. 1, 2, 3, 4, 5, 6.

Table 1 Summary statistics for angles of ${\hat{\beta }}$ and $\beta$ based on known link function, unknown link function and identity link function of Case I

Table 2 Results of the coefficients $\beta$ based on known link function, unknown link function and identity link function of Case I

Table 3 Summary statistics for angles of the ${\hat{\beta }}$ and $\beta$ based on known link function, unknown link function and identity link function of Case II

Table 4 Results of the coefficients $\beta$ based on known link function, unknown link function and identity link function of Case II

Table 5 Summary statistics for angles of the ${\hat{\beta }}$ and $\beta$ based on known link function, unknown link function and identity link function of Case III

Table 6 Results of the coefficients $\beta$ based on unknown link function and identity link function of Case III

Table 7 Comparisons between polynomial spline estimation, penalized spline estimation and kernel estimation of Case IV

Table 8 Variable selection results of $\beta$ for Case V

Table 9 Sparse estimation results of the nonzero coefficients of $\beta$ of Case V

Table 10 Estimated values (Est), standard errors (SE) and $95\%$ confidence intervals (CI) of $\beta$ for the bladder tumor data

Table 11 Estimated values (Est), standard errors (SE) and $95\%$ confidence intervals (CI) of $\beta$ for the skin cancer data

Fig. 1

Estimated curves for baseline function $\mu _0$ and link function $\psi$ of Case I

Fig. 2

Estimated curves for baseline function $\mu _0$ and link function $\psi$ of Case II

Fig. 3

Estimated curves for baseline function $\mu _0$ and link function $\psi$ of Case III

Fig. 4

Boxplots of RASEs for $\mu _0$ and $\psi$ of Case I–III

Fig. 5

Estimated curve of the link function $\psi$ for the bladder cancer data

Fig. 6

Estimated curves of the link function $\psi$ for the skin cancer data

Appendix II: Formulas for $H(\alpha ,\gamma )$ and $\mathrm{var}({\hat{\beta }},{\hat{\gamma }})$

The Hessian matrix of $\alpha$ and $\gamma$ is defined as

$$\begin{aligned} H(\alpha ,\gamma )=\left( \begin{array}{l} H_{\alpha ,\alpha },H_{\alpha ,\gamma } \\ H_{\alpha ,\gamma }^\top ,H_{\gamma ,\gamma }\\ \end{array}\right) . \end{aligned}$$

Let ${\tilde{Z}}_i=(Z_{i2},\ldots ,Z_{ip})^\top$ and $\xi _i=-Z_{i1}\alpha /(1-\Vert \alpha \Vert _2^2)^{1/2}+{\tilde{Z}}_i$, $i=1,\ldots ,n$. Denote $A=(a_{ij})$ as a $(p-1)\times (p-1)$ matrix with entries $a_{ii}=1+\alpha _i^2/(1-\Vert \alpha \Vert _2^2)$ and $a_{ij}=\alpha _i\alpha _j/(1-\Vert \alpha \Vert _2^2)$ if $i\ne j$. Then we can have

$$\begin{aligned}H_{\alpha ,\alpha }=H_9+H_{10}+H_{11}+H_{12}+H_{13},\end{aligned}$$

where

$$\begin{aligned} H_{9}=n^{-1}\sum _{i=1}^n\int _0^\tau {Y_i(s)\left\{ \gamma ^\top B''(\beta ^\top Z_i)\xi _i\xi _i^\top +\gamma ^\top B'(\beta ^\top Z_i)\frac{-Z_{i1}}{\sqrt{1-\Vert \alpha \Vert _2^2}}A\right\} d{{\tilde{N}}_i}(s)},\\H_{10}=-n^{-1}\sum _{i=1}^n\int _0^\tau {Y_i(s)\left\{ \sum _{j=1}^n w_{j}(s,\beta ,\gamma )\gamma ^\top B''(\beta ^\top Z_j)\xi _j\xi _j^\top \right\} d{{\tilde{N}}_i}(s)},\\H_{11}=-n^{-1}\sum _{i=1}^n\int _0^\tau {Y_i(s)\left\{ \sum _{j=1}^n w_{j}(s,\beta ,\gamma )\gamma ^\top B'(\beta ^\top Z_j)\frac{-Z_{i1}}{\sqrt{1-\Vert \alpha \Vert _2^2}}A\right\} d{{\tilde{N}}_i}(s)},\\H_{12}=-n^{-1}\sum _{i=1}^n\int _0^\tau {Y_i(s)\left\{ \sum _{j=1}^n w_{j}(s,\beta ,\gamma \{\gamma ^\top B'(\beta ^\top Z_j)\}^2]\xi _j\xi _j^\top \right\} d{{\tilde{N}}_i}(s)},\end{aligned}$$

and

$$\begin{aligned} H_{13}=n^{-1}\sum _{i=1}^n\int _0^\tau {Y_i(s)\left\{ \sum _{j=1}^n w_{j}(s,\beta ,\gamma )\gamma ^\top B'(\beta ^\top Z_j)\xi _j\sum _{j=1}^n w_{j}(s,\beta ,\gamma )\gamma ^\top B'(\beta ^\top Z_j)\xi _j^\top \right\} d{{\tilde{N}}_i}(s)}. \end{aligned}$$

The matrix $H_{\alpha ,\gamma }$ is given as $H_{\alpha ,\gamma }=H_{14}+H_{15}+H_{16}$, where

$$\begin{aligned} H_{14}= & {} n^{-1}\sum _{i=1}^n\int _0^\tau {Y_i(s)\left\{ \xi _i B'^\top (\beta ^\top Z_i)\right\} d{{\tilde{N}}_i}(s)}, \\ H_{15}= & {} -n^{-1}\sum _{i=1}^n\int _0^\tau {Y_i(s)\left\{ \sum _{j=1}^n w_{j}(s,\beta ,\gamma ) [\xi _j B'^\top (\beta ^\top Z_j)+\gamma ^\top B'(\beta ^\top Z_j)\xi _jB^\top (\beta ^\top Z_j)\right\} d{\tilde{N}_i}(s)}, \end{aligned}$$

and

$$\begin{aligned} H_{16}=n^{-1}\sum _{i=1}^n\int _0^\tau {Y_i(s)\left\{ \sum _{j=1}^n w_{j}(s,\beta ,\gamma ) \gamma ^\top B'(\beta ^\top Z_j)\xi _j\sum _{j=1}^n w_{j}(s,\beta ,\gamma ) B^\top (\beta ^\top Z_j)\right\} d{{\tilde{N}}_i}(s)}. \end{aligned}$$

The matrix $H_{\gamma ,\gamma }$ is the same as the one given in Subsection (3.1). From the transformation, $\beta =((1-\Vert \alpha \Vert _2^2)^{1/2}, \alpha _1,\ldots ,\alpha _{p-1})^\top$, we can easily get

$$\begin{aligned} \frac{\partial (\beta ,\gamma )}{\partial (\alpha ,\gamma )}=\left( \begin{array}{l} \frac{{\hat{\beta }}_2}{{\hat{\beta }}_1},\ldots , \frac{{\hat{\beta _p}}}{{\hat{\beta }}_1},\quad 0_{1\times (L)} \\ \quad \quad I_{p-1+L}\\ \end{array}\right) . \end{aligned}$$

Therefore, by the delta method, we can get

$$\begin{aligned} \mathrm{var}({\hat{\beta }},{\hat{\gamma }})=\left( \begin{array}{l} \frac{{\hat{\beta }}_2}{{\hat{\beta }}_1},\ldots , \frac{{\hat{\beta _p}}}{{\hat{\beta }}_1}, \quad 0_{1\times (L)} \\ \quad \quad I_{p-1+L}\\ \end{array}\right) \{-H({\hat{\alpha }},{\hat{\gamma }})\}^{-1} \left( \begin{array}{l} \frac{{\hat{\beta }}_2}{{\hat{\beta }}_1},\ldots , \frac{{\hat{\beta _p}}}{{\hat{\beta }}_1},\quad 0_{1\times (L)} \\ \quad \quad I_{p-1+L}\\ \end{array}\right) ^\top . \end{aligned}$$