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.
Similar content being viewed by others
References
Cai J, Fan J, Zhou H, Zhou Y (2007) Hazard models with varying coefficients for multivariate failure time data. Ann Statist 35(1):324–354
Carroll RJ, Fan J, Gijbels I, Wand MP (1997) Generalized partially linear single-index models. J Am Statist Assoc 92(438):477–489
Fan J, Gijbels I, King M (1997) Local likelihood and local partial likelihood in hazard regression. Ann Statist 25(4):1661–1690
He X, Feng X, Tong X, Zhao X (2017) Semiparametric partially linear varying coefficient models with panel count data. Lifetime Data Anal 23(3):439–466
He X, Tong X, Sun J (2009) Semiparametric analysis of panel count data with correlated observation and follow-up times. Lifetime Data Anal 15(2):177
He X, Tong X, Sun J, Cook RJ (2008) Regression analysis of multivariate panel count data. Biostatistics 9(2):234–248
Hu XJ, Sun J, WEI LJ (2003) Regression parameter estimation from panel counts. Scandinavian J Statist 30(1):25–43
Li N, Park D, Sun J, Kim K (2011) Semiparametric transformation models for multivariate panel count data with dependent observation process. Can J Statist 39(3):458–474
Li Y, He X, Wang H, Zhang B, Sun J (2015) Semiparametric regression of multivariate panel count data with informative observation times. J Multivariate Anal 140:209–219
Lu M, Zhang Y (2007) Estimation of the mean function with panel count data using monotone polynomial splines. Biometrika 94(3):705–718
Lu M, Zhang Y, Huang J (2009) Semiparametric estimation methods for panel count data using monotone b-splines. J Am Statist Assoc 104(487):1060–1070
Sun J, Zhao X (2013) Statistical Analysis of Panel Count Data. Springer, New York
Tibshirani R, Hastie T (1987) Local likelihood estimation. J Am Statist Assoc 82(398):559–567
Wang Y, Yu Z (2019) A kernel regression model for panel count data with time-varying coefficients. arXiv: Statistics Theory
Wang Y, Yu Z (2021) A kernel regression model for panel count data with nonparametric covariate functions. Biometrics. https://doi.org/10.1111/biom.13440
Zhang H, Zhao H, Sun J, Wang D, Kim K (2013) Regression analysis of multivariate panel count data with an informative observation process. J Multivariate Anal 119:71–80
Zhao H, Li Y, Sun J (2013a) Analyzing panel count data with a dependent observation process and a terminal event. Can J Statist 41(1):174–191
Zhao H, Li Y, Sun J (2013b) Semiparametric analysis of multivariate panel count data with dependent observation processes and a terminal event. J Nonparametr Statist 25(2):379–394
Zhao H, Tu W, Yu Z (2018) A nonparametric time-varying coefficient model for panel count data. J Nonparametr Statist 30(3):640–661
Zhao H, Virkler K, Sun J (2014) Nonparametric comparison for multivariate panel count data. Commun Statist Theory Methods 43(3):644–655
Zhao X, Tong X (2011) Semiparametric regression analysis of panel count data with informative observation times. Comput Statist Data Anal 55(1):291–300
Zhao X, Tong X, Sun J (2013) Robust estimation for panel count data with informative observation times. Comput Statist Data Anal 57(1):33–40
Acknowledgements
This paper was partially supported by the National Natural Science Foundation of China under Grand No. 12001485 and 12101549, the Ministry of Education Humanities and Social Sciences Research Youth Project of China under Grand No. 21YJCZH153, the National Bureau of Statistics of China under Grand No. 2020LY073 and the Characteristic & Preponderant Discipline of Key Construction Universities in Zhejiang Province (Zhejiang Gongshang University-Statistics).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix I: tables and figures
See Tables 1, 2, 3 and Figs. 1, 2, 3, 4, 5, 6, 7 and 8.
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
and
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
Furthermore, we can have
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
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
where \(X_i^*=H^{-1}X_i\) and
Furthermore, we define
and
where
Then,
By Lemma 1, we can easily get that
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
and based the Lemma 1, we can have
Thus, it implies that \(I_{1k}\{\zeta (v),t\}\rightarrow _p 0\) for \(k=1,\cdots ,K\). Hence,
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
Similarly, we can get
This completes the proof of Theorem 4.1
\(\square \)
:
Proof of Theorem 4.2
Denote
By the Taylor expansion and Lemma 1, we can have
where \(b=\int x^{d+1}K(x)dx\). Besides, we can have
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
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
As \({\hat{\zeta }}\) is the maximizer of function \(\ell \{\gamma _0(v)+H^{-1}\zeta (v),t\}\), we can have
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
By Slutsky’s theorem, we can have
This completes the proof of Theorem 4.2. \(\square \)
Rights and permissions
About this article
Cite this article
Wang, W., Wang, Y. & Zhao, X. Semiparametric analysis of multivariate panel count data with nonlinear interactions. Lifetime Data Anal 28, 89–115 (2022). https://doi.org/10.1007/s10985-021-09537-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10985-021-09537-1