Semiparametric analysis of multivariate panel count data with nonlinear interactions

Abstract

Multivariate panel count data frequently arise in follow up studies involving several related types of recurrent events. For univariate panel count data, several varying coefficient models have been developed. However, varying coefficient models for multivariate panel count data remain to be studied. In this paper, we propose a varying coefficient mean model for multivariate panel count data to describe the possible nonlinear interact effects between the covariates and the local logarithm partial likelihood procedure is considered to estimate the unknown covariate effects. Furthermore, a Breslow-type estimator is constructed for the baseline mean functions. The consistency and asymptotic normality of the proposed estimators are established under some mild conditions. The utility of the proposed approach is evaluated by some numerical simulations and an application to a dataset of skin cancer study.

Appendices

Appendix I: tables and figures

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

Table 1 BIAS, SSE, BSE and 95% CP for the nonparametric parts $\beta (\cdot )$, $g'(\cdot )$ and $g(\cdot )$ on pre-specified grid-points of Case I

Table 2 BIAS, SSE, BSE and 95% CP for the nonparametric parts $\beta (\cdot )$, $g'(\cdot )$ and $g(\cdot )$ on pre-specified grid-points of Case II

Table 3 BIAS, SSE, BSE and 95% CP for the nonparametric parts $\beta (\cdot )$, $g'(\cdot )$ and $g(\cdot )$ on pre-specified grid-points of Case III

Appendix II: proofs of asymptotic properties

The following lemma is needed in the proofs of the theorems, which is similar as Fan et al. (1997) and Cai et al. (2007). The detail proof of this lemma can be found in the paper of Cai et al. (2007).

Lemma 1

Define

$$\begin{aligned} c_{nk}(s,v)=\frac{1}{n}\sum _{i=1}^n Y_i(s)\Psi \{Z_i,V_i,(V_i-v)/h\}K_h(V_i-v)o_{ik}(s), \end{aligned}$$

and

$$\begin{aligned} c_k(s,v)=f(v)\int E\{Y(s)\Psi (Z,V,w)o_{k}(s)|V=v\}K(w) dw, \end{aligned}$$

where $\Psi (\cdot ,\cdot ,\cdot )$ is continuous for its three arguments and $E\{\Psi (Z,V,w)|V=v\}$ is continuous at the point v. Suppose conditions (C1) and (C6) hold and $h\rightarrow 0$, $nh/\log n\rightarrow \infty$, then we have

$$\begin{aligned} \mathop {sup}\limits _{0\le s\le \tau }\sum _{k=1}^K|c_{nk}(s,v)-c_k(s,v)|\rightarrow _p 0. \end{aligned}$$

Furthermore, we can have

$$\begin{aligned} \mathop {sup}\limits _{0\le s\le \tau }\mathop {sup}\limits _{v\in B}\sum _{k=1}^K|c_{nk}(s,v)-c_k(s,v)|\rightarrow _p 0. \end{aligned}$$

where B is a compact set satisfying $inf_{v\in B} f(v)>0$.

:

Proof of Theorem 4.1

By the definition of ${{\tilde{N}}_{ik}}(t)$, we can have

$$\begin{aligned} M_{ik}(t)=\int _0^t Y_i(s)\left[ d{\tilde{N}_{ik}}(s)-\exp \{\beta _0(V_i)Z_i+g(V_i)\}\mu _{0k}(s)o_{ik}(s)ds\right] ,0\le t\le \tau , \end{aligned}$$

is a $\bigcup _{i=1}^n \mathcal {F}_{t,ik}$ martingale, where $\mathcal {F}_{t,ik}=\sigma \{{{\tilde{N}}_i}(s), Z_i,V_i,Y_i(s),0\le s\le t\}$, $i=1,\cdots ,n$, $k=1,\cdots ,K$. Define $\gamma _0(v)$ be the true values of $\gamma (v)$. Let $\zeta (v)=H\{\gamma (v)-\gamma _0(v)\}$, then we have

$$\begin{aligned}&\ell \{\gamma _0(v)+H^{-1}\zeta (v),t\}=\\&\quad \frac{1}{n}\sum _{k=1}^K\sum _{i=1}^n\int _0^t K_h(V_i-v)Y_i(s)\left[ \gamma _0(v)^\top X_i+\zeta (v)^\top X_i^*-\log S_{kh,0}\{s,\zeta (v)\}\right] d{{\tilde{N}}_{ik}}(s), \end{aligned}$$

where $X_i^*=H^{-1}X_i$ and

$$\begin{aligned} S_{kh,j}(s,\zeta ,v)&=\frac{1}{n}\sum _{i=1}^n K_h(V_i-v) Y_i(s)X_i^{*\otimes j}\exp \{\gamma _0(v)^\top X_i+\zeta (v)^\top X_i^*\}o_{ki}(s),\\&\quad j=0,1,2. \end{aligned}$$

Furthermore, we define

$$\begin{aligned} S^*_{kh,j}&(s,\beta ,g,v)=\frac{1}{n}\sum _{i=1}^n K_h(V_i-v) Y_i(s)X_i^{*\otimes j}\exp \{\beta (V_i) Z_i+g(V_i)\}o_{ki}(s), \end{aligned}$$

and

$$\begin{aligned} s^*_{kh,0}&(s,\beta ,g,v)=f(v)E[Y(s)\exp \{\beta (V) Z+g(V)\}o_{k}(s)|V=v],\\ s^*_{kh,1}&(s,\beta ,g,v)=f(v)E[Y(s)\exp \{\beta (V) Z+g(V)\}(Z,Zu_1^\top ,u_1^\top )^\top o_{k}(s)|V=v],\\ s^*_{kh,2}&(s,\beta ,g,v)=f(v)E[Y(s)\exp \{\beta (V) Z+g(V)\}\Pi (Z)o_{k}(s)|V=v], \end{aligned}$$

where

$$\begin{aligned}\Pi (Z)= \begin{pmatrix} ZZ^\top &{} ZZ^\top u_1^\top &{} Zu_1^\top \\ ZZ^\top u_1 &{}ZZ^\top u_2 &{} Zu_2\\ Zu_1 &{}Z u_2 &{}u_2\\ \end{pmatrix}. \end{aligned}$$

Then,

$$\begin{aligned}&\ell \{\gamma _0(v)+H^{-1}\zeta (v),t\}-\ell \{\gamma _0(v),t\}\\&\quad =\frac{1}{n}\sum _{k=1}^K\sum _{i=1}^n\int _0^t K_h(V_i-v)\left\{ \zeta (v)^\top X_i^*-\log \frac{S_{kh,0}(s,\zeta ,v)}{S_{kh,0}(s,0,v)}\right\} d{M_{ik}}(s)\\&\qquad +\sum _{k=1}^K\int _0^t \left\{ \zeta (v)^\top S^*_{kh,1}(s,\beta _0,g_0,v) -\log \frac{S_{kh,0}(s,\zeta ,v)}{S_{kh,0}(s,0,v)}S^*_{kh,0}(s,\beta _0,g_0,v)\right\} \mu _{0k}(s)ds\\&\quad =\sum _{k=1}^K I_{1k}\{\zeta (v),t\}+I_2\{\zeta (v),t\}. \end{aligned}$$

By Lemma 1, we can easily get that

$$\begin{aligned}&I_2\{\zeta (v),t\}\\&\quad =\sum _{k=1}^K\int _0^t \left[ \zeta (v)^\top s^*_{kh,1}(s,\beta _0,g_0,v) -\log \frac{s_{kh,0}(s,\zeta ,v)}{s_{kh,0}(s,0,v)}s^*_{kh,0}(s,\beta _0,g_0,v)\right] \\&\qquad \mu _{0k}(s)ds+o_p(1)\\&\quad =I^*_2\{\zeta (v),t\}+o_p(1). \end{aligned}$$

Fig. 1

Estimated curves for $\beta (\cdot )$ and $g(\cdot )$ of Case I under $n=100$ and 200

Fig. 2

Boxplots of RASEs for $\beta (\cdot )$ and $g(\cdot )$ of Case I under $n=100$ and 200

Fig. 3

Estimated curves for $\beta (\cdot )$ and $g(\cdot )$ of Case II under $n=100$ and 200

Fig. 4

Boxplots of RASEs for $\beta (\cdot )$ and $g(\cdot )$ of Case II under $n=100$ and 200

Fig. 5

Estimated curves for $\beta (\cdot )$ and $g(\cdot )$ of Case III under $n=100$ and 200

Fig. 6

Boxplots of RASEs for $\beta (\cdot )$ and $g(\cdot )$ of Case III under $n=100$ and 200

Fig. 7

Estimated curves for $\mu _{01}(\cdot )$ and $\mu _{02}(\cdot )$ of Case I under $n=100$ and 200

Fig. 8

Estimated curves for $\mu _{01}(\cdot )$ and $\mu _{02}(\cdot )$ of Case II under $n=100$ and 200

Fig. 9

Estimated curves for $\mu _{01}(\cdot )$, $\mu _{02}(\cdot )$ and $\mu _{03}(\cdot )$ of Case III under $n=100$ and 200

Fig. 10

Estimated curves for $\beta (\cdot )$ and $g(\cdot )$ of Case IV under $n=100$ and 200

Fig. 11

Estimated curves for nonparametric parts of skin cancer study

It can easily shown that the $I_2\{\zeta (v),t\}$ is strictly concave with respect to $\zeta (v)$ and it has the maximum value at $\zeta (v)=0$. Next, we can note $I_{1k}\{\zeta (v),t\}$ is a local square integrable martingale with the square variation process being

$$\begin{aligned} \langle I_{1k}\{\zeta (v),t\},I_{1k}\{\zeta (v),t\}\rangle =&\frac{1}{n^2}\sum _{i=1}^n\int _0^t K_h^2(V_i-v)Y_i(s)\\&\left\{ \zeta (v)^\top X_i^*-\log \frac{S_{kh,0}(s,\zeta ,v)}{S_{kh,0}(s,0,v)}\right\} ^2\\&\times \exp \{\beta _0(V_i)Z_i+g(V_i)\}\mu _{0k}(s)o_{ik}(s)ds, \end{aligned}$$

and based the Lemma 1, we can have

$$\begin{aligned} E I_{1k}^2\{\zeta (v),t\}=E\langle I_{1k}\{\zeta (v),t\},I_{1k}\{\zeta (v),t\}\rangle =O\left( \frac{1}{nh}\right) \rightarrow 0. \end{aligned}$$

Thus, it implies that $I_{1k}\{\zeta (v),t\}\rightarrow _p 0$ for $k=1,\cdots ,K$. Hence,

$$\begin{aligned} \ell \{\gamma _0(v)+H^{-1}\zeta (v),t\}-\ell \{\gamma _0(v),t\}=I^*_2\{\zeta (v),t\}+o_p(1). \end{aligned}$$

Then, we can have that $\ell \{\gamma _0(v)+H^{-1}\zeta (v),t\}-\ell \{\gamma _0(v),t\}$ is strictly concave with respect to $\zeta (v)$ and it has the maximum value at $\zeta (v)=0$. By Lemma A.1 of Carroll et al. (1997), ${\hat{\zeta }}(v)\rightarrow _p 0$. So

$$\begin{aligned} H\{{\hat{\gamma }}(v)-\gamma _0(v)\}\rightarrow _p 0. \end{aligned}$$

Similarly, we can get

$$\begin{aligned} \mathop {sup}\limits _{v\in \Phi _v}|H\{{\hat{\gamma }}(v)-\gamma _0(v)\}|\rightarrow _p 0. \end{aligned}$$

This completes the proof of Theorem 4.1

$\square$

:

Proof of Theorem 4.2

Denote

$$\begin{aligned}&\ell '\{\gamma _0(v),\tau \}=\frac{1}{n}\sum _{k=1}^K\sum _{i=1}^n\int _0^\tau K_h(V_i-v)\left\{ X_i^*-\frac{S_{kh,1}(s,0,v)}{S_{kh,0}(s,0,v)}\right\} d{M_{ik}}(s)\\&\qquad +\frac{1}{n}\sum _{k=1}^K\sum _{i=1}^n\int _0^\tau K_h(V_i-v)Y_i(s)\left\{ X_i^*-\frac{S_{kh,1}(s,0,v)}{S_{kh,0}(s,0,v)}\right\} \exp \{\beta _0(V_i)Z_i+g_0(V_i)\}\mu _{0k}(s)o_{ik}(s)ds\\&\quad =I_3(v,\tau )+I_4(v,\tau ). \end{aligned}$$

By the Taylor expansion and Lemma 1, we can have

$$\begin{aligned} I_4(v,\tau )=&\frac{1}{(d+1)!n}\sum _{k=1}^K\sum _{i=1}^n\int _0^\tau K_h(V_i-v)Y_i(s)\left\{ X_i^*-\frac{s^*_{kh,1}(s,\beta _0,g_0,v)}{s^*_{kh,0}(s,\beta _0,g_0,v)}\right\} \\&\times \exp \{\gamma _0(v)^\top X_i^*+g_0(v)\}\{\beta ^{(d+1)}_0(v)Z_i\\&+g^{(d+1)}_0(v)\}(V_i-v)^{d+1}\mu _{0k}(s)o_{ik}(s)ds\{1+O_p(h^d)\}\\ =&\frac{h^{d+1}}{(d+1)!}b[\Gamma (v)^{-1}\beta ^{(d+1)}_0(v), 0,\cdots ,0]^\top \{1+O_p(h^d)\}=A(v,\tau ), \end{aligned}$$

where $b=\int x^{d+1}K(x)dx$. Besides, we can have

$$\begin{aligned}&I_3(v,\tau )=\frac{1}{n}\sum _{i=1}^n\sum _{k=1}^K\int _0^\tau K_h(V_i-v)\left\{ X_i^*-\frac{s_{kh,1}(s,\zeta ,v)}{s_{kh,0}(s,\zeta ,v)}\right\} d{M_{ik}}(s)+o_p(1)\\&\quad =\frac{1}{n}\sum _{i=1}^n\sum _{k=1}^K A_{ik}(v,\tau )+o_p(1) \end{aligned}$$

It is easily note that $\sqrt{nh}I_3(v,\tau )$ is a sum of i.i.d. random vectors $\sum _{k=1}^K A_{ik}(v,\tau )$ with zero mean and finite variance. By calculation, we can get the asymptotic variance is

$$\begin{aligned} \Sigma&=lim_{n\rightarrow \infty } Eh\left\{ \sum _{k=1}^K A_{ik}(v,\tau )\right\} ^{\otimes 2}\\&=\sum _{k_i=1}^K\sum _{k_2=1,k_1\ne k_2}^K lim_{n\rightarrow \infty } Eh A_{1k_1}(v,\tau )A_{1k_2}(v,\tau )^\top +\sum _{k=1}^K lim_{n\rightarrow \infty } EhA_{ik}(v,\tau )^{\otimes 2}\\&= \Sigma _1+ \Sigma _{11}. \end{aligned}$$

As $\sum _{i=1}^n A_{ik}(v,\tau )$ is a local square-integrable martingale, it can be easily obtained that $\Sigma _{11}$ converges to $\Sigma _2$, where $\Sigma _2$ is $diag\{\Gamma ^{-1}(v)\nu _0,Q_2\nu _2\}$.

By Theorem 4.1, we have ${\hat{\zeta }}(v) \rightarrow 0$ in probability. Therefore, based on the mean value theorem, we can obtain that

$$\begin{aligned}&\ell ''\{\gamma _0(v)+H^{-1}{\hat{\zeta }}(v),t\}= \ell ''\{\gamma _0(v),t\}+o_p(1)\\&\quad =\frac{1}{n}\sum _{k=1}^K\sum _{i=1}^n\int _0^t K_h(V_i-v)\\&\qquad \left\{ \frac{s_{kh,2}^*(s,\beta _0,g_0,v)s_{kh,0}^*(s,\beta _0,g_0,v)-s_{kh,1}^*(s,\beta _0,g_0,v)s_{kh,2}^*(s,\beta _0,g_0,v)^\top }{s_{kh,0}^*(s,\beta _0,g_0,v)^2}\right\} \\&\qquad \times d N_{ik}(s)+o_p(1)\\&\quad =\sum _{k=1}^K\int _0^t \left\{ \frac{s_{kh,2}^*(s,\beta _0,g_0,v)s_{kh,0}^*(s,\beta _0,g_0,v)-s_{kh,1}^*(s,\beta _0,g_0,v)s_{kh,2}^*(s,\beta _0,g_0,v)^\top }{s_{kh,0}^*(s,\beta _0,g_0,v)^2}\right\} \\&\qquad \times s_{kh,0}^*(s,\beta _0,g_0,v) d s+o_p(1)\\&\quad =-B(v,\tau )+o_p(1). \end{aligned}$$

As ${\hat{\zeta }}$ is the maximizer of function $\ell \{\gamma _0(v)+H^{-1}\zeta (v),t\}$, we can have

$$\begin{aligned} -\ell '\{\gamma _0(v),t\}=\ell '\{\gamma _0(v)+H^{-1}{\hat{\zeta }}(v),t\}-\ell '\{\gamma _0(v),t\}=\ell ''\{\gamma _0(v)+H^{-1}{\hat{\zeta }}^{*}(v),t\}^\top {\hat{\zeta }}, \end{aligned}$$

where ${\hat{\zeta }}^{*}(v)$ lies between 0 and ${\hat{\zeta }}(v)$ (the second equality is obtained by Taylor expansion of $\ell \{\gamma _0(v)+H^{-1}\zeta (v),t\}$ around 0). Hence, we can have

$$\begin{aligned}&{\hat{\zeta }}(v)-B(v,\tau )^{-1}A(v,\tau )=\\&\quad -\ell ''\{\gamma _0(v)+H^{-1}{\hat{\zeta }}^*(v),t\}^{-1} [\ell '\{\gamma _0(v),t\}-A(v,\tau )]+o_p(1). \end{aligned}$$

By Slutsky’s theorem, we can have

$$\begin{aligned} \sqrt{nh} \{{\hat{\zeta }}(v)-B(v,\tau )^{-1}A(v,\tau )\}\rightarrow _d N\{0, B(v,\tau )^{-1}\Sigma B(v,\tau )^{-1}\}. \end{aligned}$$

This completes the proof of Theorem 4.2. $\square$