Efficient estimation of the partly linear additive hazards model with current status data. (English) Zbl 1364.62195
Summary: This paper focuses on efficient estimation, optimal rates of convergence and effective algorithms in the partly linear additive hazards regression model with current status data. We use polynomial splines to estimate both cumulative baseline hazard function with monotonicity constraint and nonparametric regression functions with no such constraint. We propose a simultaneous sieve maximum likelihood estimation for regression parameters and nuisance parameters and show that the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound. In addition, we show that rates of convergence for the estimators of nonparametric functions are optimal. We implement the proposed estimation through a backfitting algorithm on generalized linear models. We conduct simulation studies to examine the finite-sample performance of the proposed estimation method and present an analysis of renal function recovery data for illustration.
MSC:
| 62J12 | Generalized linear models (logistic models) |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
Software:
ElemStatLearnReferences:
| [1] | Bickel, P. J., Klaassen, C. A., Ritov, Y. & Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models, The Johns Hopkins University Press, Baltimore. · Zbl 0786.62001 |
| [2] | Claeskens, G., Krivobokova, T. & Opsomer, J. D. (2009). Asymptotic properties of penalized spline estimators. Biometrika96, 529-544. · Zbl 1170.62031 |
| [3] | Cox, D. R. (1972). Regression models and life‐tables. J. Roy. Stat. Soc. B Met.34, 187-220. · Zbl 0243.62041 |
| [4] | Gray, R. J. (1992). Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. J. Am. Stat. Assoc.87, 942-951. |
| [5] | Groeneboom, P. & Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation, Verlag Basel, Birkhäuser. · Zbl 0757.62017 |
| [6] | Hastie, T., Tibshirani, R. & Friedman, J. (2008). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, (2nd edn)., Springer‐Verlag, New York. |
| [7] | He, X., Zhu, Z. Y. & Fung, W. K. (2002). Estimation in a semiparametric model for longitudinal data with unspecified dependence structure. Biometrika89, 579-590. · Zbl 1036.62035 |
| [8] | Heung, M., Wolfgram, D. F., Kommareddi, M., Hu, Y., Song, P. X. & Ojo, A. O. (2012). Fluid overload at initiation of renal replacement therapy is associated with lack of renal recovery in patients with acute kidney injury. Nephrol. Dial. Transpl.27, 956-961. |
| [9] | Huang, J. (1996). Efficient estimation for the proportional hazards model with interval censoring. Ann. Stat.24, 540-568. · Zbl 0859.62032 |
| [10] | Huang, J. (1999). Efficient estimation of the partly linear additive Cox. model. Ann. Stat.27, 1536-1563. · Zbl 0977.62035 |
| [11] | Huang, J. & Rossini, A. J. (1997). Sieve estimation for the proportional‐odds failure‐time regression model with interval censoring. J. Am. Stat. Assoc.92, 960-967. · Zbl 0889.62082 |
| [12] | Kalbfleisch, J. D. & Prentice, R. L. (2002). The Statistical Analysis of Failure Time Data, (2nd edn)., John Wiley & Sons, Inc., New York. · Zbl 1012.62104 |
| [13] | Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference, Springer, New York. · Zbl 1180.62137 |
| [14] | Lian, H., Chen, X. & Yang, J. Y. (2012). Identification of partially linear structure in additive models with an application to gene expression prediction from sequences. Biometrics68, 437-445. · Zbl 1251.62013 |
| [15] | Lin, D. Y., Oakes, D. & Ying, Z. (1998). Additive hazards regression with current status data. Biometrika85, 289-298. · Zbl 0938.62121 |
| [16] | Lin, D. Y. & Ying, Z. (1994). Semiparametric analysis of the additive risk model. Biometrika81, 61-71. · Zbl 0796.62099 |
| [17] | Lu, M., Zhang, Y. & Huang, J. (2007). Estimation of the mean function with panel count data using monotone polynomial splines. Biometrika94, 705-718. · Zbl 1135.62069 |
| [18] | Ma, S. (2011). Additive risk model for current status data with a cured subgroup. Ann. I. Stat. Math.63, 117-134. · Zbl 1432.62230 |
| [19] | Ma, S. & Kosorok, M. R. (2005a). Penalized log‐likelihood estimation for partly linear transformation models with current status data. Ann. Stat.33, 2256-2290. · Zbl 1086.62056 |
| [20] | Ma, S. & Kosorok, M. R. (2005b). Robust semiparametric m‐estimation and the weighted bootstrap. J. Multivariate Anal.96, 190-217. · Zbl 1073.62030 |
| [21] | Martinussen, T. & Scheike, T. H. (2002). Efficient estimation in additive hazards regression with current status data. Biometrika89, 649-658. · Zbl 1036.62109 |
| [22] | McKeague, I. W. & Sasieni, P. D. (1994). A partly parametric additive risk model. Biometrika81, 501-514. · Zbl 0812.62041 |
| [23] | Rabinowitz, D., Tsiatis, A. & Aragon, J. (1995). Regression with interval‐censored data. Biometrika82, 501-513. · Zbl 0830.62093 |
| [24] | Robertson, T., Wright, F. T., Dykstra, R. L. & Robertson, T. (1988). Order Restricted Statistical Inference, Wiley, New York. · Zbl 0645.62028 |
| [25] | Sasieni, P. (1992). Non‐orthogonal projections and their application to calculating the information in a partly linear Cox. model. Scand. J. Stat.9, 215-233. · Zbl 0753.62025 |
| [26] | Schumaker, L. (1981). Spline Functions: Basic Theory, Wiley, New York. · Zbl 0449.41004 |
| [27] | Schwarz, G. (1978). Estimating the dimension of a model. Ann. Stat.6, 461-464. · Zbl 0379.62005 |
| [28] | Shiboski, S. C. (1998). Generalized additive models for current status data. Lifetime Data Anal.4, 29-50. · Zbl 0894.62116 |
| [29] | Shiboski, S. C. & Jewell, N. P. (1992). Statistical analysis of the time dependence of HIV infectivity based on partner study data. J. Am. Stat. Assoc.87, 360-372. |
| [30] | Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Stat.10, 1040-1053. · Zbl 0511.62048 |
| [31] | Stone, C. J. (1985). Additive regression and other nonparametric models. Ann. Stat.13, 689-705. · Zbl 0605.62065 |
| [32] | Thisted, R. A. (1988). Elements of Statistical Computing: Numerical Computation, Chapman & Hall/CRC, New York. · Zbl 0663.62001 |
| [33] | Van der Vaart, A. & Wellner, J. A. (1996). Weak Convergence and Empirical Processes: with Applications to Statistics, Springer, New York. · Zbl 0862.60002 |
| [34] | Xue, H., Lam, K. F. & Li, G. (2004). Sieve maximum likelihood estimator for semiparametric regression models with current status data. J. Am. Stat. Assoc.99, 346-356. · Zbl 1117.62449 |
| [35] | Zeng, D., Cai, J. & Shen, Y. (2006). Semiparametric additive risks model for interval‐censored data. Stat. Sinica16, 287-302. · Zbl 1087.62106 |
| [36] | Zhang, H. H., Cheng, G. & Liu, Y. (2011). Linear or nonlinear? automatic structure discovery for partially linear models. J. Am. Stat. Assoc.106, 1099-1109. · Zbl 1229.62051 |
| [37] | Zucker, D. M. & Karr, A. F. (1990). Nonparametric survival analysis with time‐dependent covariate effects: a penalized partial likelihood approach. Ann. Stat.18, 329-353. · Zbl 0708.62035 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
