# Estimation of the hazard function in a semiparametric model with covariate measurement error

Marie-Laure Martin-Magniette; Marie-Luce Taupin

ESAIM: Probability and Statistics (2009)

- Volume: 13, page 87-114
- ISSN: 1292-8100

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topMartin-Magniette, Marie-Laure, and Taupin, Marie-Luce. "Estimation of the hazard function in a semiparametric model with covariate measurement error." ESAIM: Probability and Statistics 13 (2009): 87-114. <http://eudml.org/doc/250633>.

@article{Martin2009,

abstract = {
We consider a failure hazard function,
conditional on a time-independent covariate Z,
given by $\eta_\{\gamma^0\}(t)f_\{\beta^0\}(Z)$. The baseline hazard
function $\eta_\{\gamma^0\}$ and the relative risk $f_\{\beta^0\}$ both belong to parametric
families with
$\theta^0=(\beta^0,\gamma^0)^\top\in \mathbb\{R\}^\{m+p\}$. The covariate Z has an unknown density and is measured with an error through an
additive error model U = Z + ε where ε is a random variable, independent from Z, with
known density $f_\varepsilon$.
We observe a n-sample (Xi, Di, Ui), i = 1, ..., n, where Xi is
the minimum between the failure time and the censoring time,
and Di is the censoring indicator.
Using least square criterion and deconvolution methods, we propose a consistent estimator of θ0
using the observations
n-sample (Xi, Di, Ui), i = 1, ..., n.
We give an upper bound for its risk
which depends on the smoothness properties of $f_\varepsilon$ and $f_\beta(z)$ as a
function of z, and we derive sufficient conditions
for the $\sqrt\{n\}$-consistency.
We give detailed examples considering
various type of relative risks $f_\{\beta\}$ and various types of error
density $f_\varepsilon$. In particular, in the Cox model and in
the excess risk model, the estimator of θ0 is
$\sqrt\{n\}$-consistent and asymptotically Gaussian
regardless of the form of $f_\varepsilon$.
},

author = {Martin-Magniette, Marie-Laure, Taupin, Marie-Luce},

journal = {ESAIM: Probability and Statistics},

keywords = {Semiparametric estimation; errors-in-variables model; measurement error; nonparametric estimation; excess risk model; Cox model; censoring; survival analysis; density deconvolution; least square criterion; semiparametric estimation; least squares criterion},

language = {eng},

month = {3},

pages = {87-114},

publisher = {EDP Sciences},

title = {Estimation of the hazard function in a semiparametric model with covariate measurement error},

url = {http://eudml.org/doc/250633},

volume = {13},

year = {2009},

}

TY - JOUR

AU - Martin-Magniette, Marie-Laure

AU - Taupin, Marie-Luce

TI - Estimation of the hazard function in a semiparametric model with covariate measurement error

JO - ESAIM: Probability and Statistics

DA - 2009/3//

PB - EDP Sciences

VL - 13

SP - 87

EP - 114

AB -
We consider a failure hazard function,
conditional on a time-independent covariate Z,
given by $\eta_{\gamma^0}(t)f_{\beta^0}(Z)$. The baseline hazard
function $\eta_{\gamma^0}$ and the relative risk $f_{\beta^0}$ both belong to parametric
families with
$\theta^0=(\beta^0,\gamma^0)^\top\in \mathbb{R}^{m+p}$. The covariate Z has an unknown density and is measured with an error through an
additive error model U = Z + ε where ε is a random variable, independent from Z, with
known density $f_\varepsilon$.
We observe a n-sample (Xi, Di, Ui), i = 1, ..., n, where Xi is
the minimum between the failure time and the censoring time,
and Di is the censoring indicator.
Using least square criterion and deconvolution methods, we propose a consistent estimator of θ0
using the observations
n-sample (Xi, Di, Ui), i = 1, ..., n.
We give an upper bound for its risk
which depends on the smoothness properties of $f_\varepsilon$ and $f_\beta(z)$ as a
function of z, and we derive sufficient conditions
for the $\sqrt{n}$-consistency.
We give detailed examples considering
various type of relative risks $f_{\beta}$ and various types of error
density $f_\varepsilon$. In particular, in the Cox model and in
the excess risk model, the estimator of θ0 is
$\sqrt{n}$-consistent and asymptotically Gaussian
regardless of the form of $f_\varepsilon$.

LA - eng

KW - Semiparametric estimation; errors-in-variables model; measurement error; nonparametric estimation; excess risk model; Cox model; censoring; survival analysis; density deconvolution; least square criterion; semiparametric estimation; least squares criterion

UR - http://eudml.org/doc/250633

ER -

## References

top- M. Aitkin and D. Clayton, The fitting of exponential, Weibull and extreme value distributions to complex censored survival data using GLIM. J. R. Stat. Soc., Ser. C29 (1980) 156–163. Zbl0437.62092
- P.K. Andersen, O. Borgan, R.D. Gill and N. Keiding, Statistical models based on counting processes. Springer Series in Statistics (1993). Zbl0769.62061
- T. Augustin, An exact corrected log-likelihood function for Cox's proportional hazards model under measurement error and some extensions. Scand. J. Stat.31 (2004) 43–50. Zbl1053.62108
- Ø. Borgan, Correction to: Maximum likelihood estimation in parametric counting process models, with applications to censored failure time data. Scand. J. Statist.11 (1984) 275. Zbl0565.62015
- Ø. Borgan, Maximum likelihood estimation in parametric counting process models, with applications to censored failure time data. Scand. J. Stat., Theory Appl.11 (1984) 1–16. Zbl0546.62010
- C. Butucea and M.-L. Taupin, New M-estimators in semiparametric regression with errors in variables. Ann. Inst. Henri Poincaré: Probab. Stat. (to appear). Zbl1206.62068
- J.S. Buzas, Unbiased scores in proportional hazards regression with covariate measurement error. J. Statist. Plann. Inference, 67 (1998) 247–257. Zbl0932.62109
- R.J. Carroll, D. Ruppert, and L.A. Stefanski, Measurement error in nonlinear models. Chapman and Hall (1995). Zbl0853.62048
- F. Comte and M.-L. Taupin, Nonparametric estimation of the regression function in an errors-in-variables model. Statistica Sinica17 (2007) 1065–1090. Zbl1133.62027
- D.R. Cox and D. Oakes, Analysis of survival data. Monographs on Statistics and Applied Probability. Chapman and Hall (1984).
- J. Fan and Y.K. Truong, Nonparametric regression with errors in variables. Ann. Statist.21 (1993) 1900–1925. Zbl0791.62042
- M.V. Fedoryuk, Asimptotika: integraly i ryady. Asymptotics: Integrals and Series (1987). Zbl0641.41001
- W.A. Fuller, Measurement error models. Wiley Series in Probability and Mathematical Statistics (1987). Zbl0800.62413
- R.D. Gill and P.K. Andersen, Cox's regression model for counting processes: a large sample study. Ann. Statist.10 (1982) 1100–1120. Zbl0526.62026
- G. Gong, A.S. Whittemore and S. Grosser, Censored survival data with misclassified covariates: A case study of breast-cancer mortality. J. Amer. Statist. Assoc.85 (1990) 20–28.
- N.L. Hjort, On inference in parametric survival data models. Int. Stat. Rev.60 (1992) 355–387. Zbl0762.62009
- D.W.J. Hosmer and S. Lemeshow, Applied survival analysis. Regression modeling of time to event data. Wiley Series in Probability and Mathematical Statistics (1999). Zbl0966.62071
- C. Hu and D.Y. Lin, Semiparametric failure time regression with replicates of mismeasured covariates. J. Am. Stat. Assoc.99 (2004) 105–118. Zbl1089.62505
- C. Hu and D.Y. Lin, Cox regression with covariate measurement error. Scand. J. Stat.29 (2002) 637–655. Zbl1035.62102
- Y. Huang and C.Y. Wang, Cox regression with accurate covariates unascertainable: A nonparametric-correction approach. J. Am. Stat. Assoc.95 (2000) 1209–1219. Zbl1008.62040
- J. Kiefer and J. Wolfowitz, Consistency of the maximum likelihood estimator in the presence of infinitely many nuisance parameters. Ann. Math. Statist.27 (1956) 887–906. Zbl0073.14701
- F.H. Kong, Adjusting regression attenuation in the Cox proportional hazards model. J. Statist. Plann. Inference79 (1999) 31–44. Zbl0937.62099
- F.H. Kong and M. Gu, Consistent estimation in Cox proportional hazards model with covariate measurement errors. Statistica Sinica9 (1999) 953–969. Zbl0942.62115
- O.V. Lepski and B.Y. Levit, Adaptive minimax estimation of infinitely differentiable functions. Math. Methods Statist.7 (1998) 123–156. Zbl1103.62332
- Y. Li and L. Ryan, Survival analysis with heterogeneous covariate measurement error. J. Amer. Statist. Assoc.99 (2004) 724-735. Zbl1117.62383
- Y. Li and L. Ryan, Inference on survival data with covariate measurement error – An imputation-based approach. Scand. J. Stat.33 (2006) 169–190. Zbl1126.62092
- M.-L. Martin-Magniette, Nonparametric estimation of the hazard function by using a model selection method: estimation of cancer deaths in Hiroshima atomic bomb survivors. J. Roy. Statist. Soc. Ser. C54 (2005) 317–331. Zbl05188688
- T. Nakamura, Corrected score function for errors-in-variables models: methodology and application to generalized linear models. Biometrika77 (1990) 127–137. Zbl0691.62066
- T. Nakamura, Proportional hazards model with covariates subject to measurement error. Biometrics48 (1992) 829-838.
- R.L. Prentice, Covariate measurement errors and parameter estimation in a failure time regression model. Biometrika69 (1982) 331–342. Zbl0523.62083
- R.L. Prentice and S.G. Self, Asymptotic distribution theory for Cox-type regression models with general relative risk form. Ann. Statist.11 (1983) 804–813. Zbl0526.62017
- O. Reiersøl, Identifiability of a linear relation between variables which are subject to error. Econometrica18 (1950) 375-389.
- P. Reynaud-Bouret, Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. Prob. Theory Relat. Fields126 (2003) 103–153. Zbl1019.62079
- L.A. Stefanski, Unbiaised estimation of a nonlinear function of a normal mean with application to measurement error models. Commun. Stat. -Theory Meth.18 (1989) 4335–4358. Zbl0707.62058
- M.-L. Taupin, Semi-parametric estimation in the nonlinear structural errors-in-variables model. Ann. Statist.29 (2001) 66–93. Zbl1029.62039
- T.T. Tsiatis, V. DeGruttola and M.S. Wulfsohn, Modeling the relationship of survival to longitudinal data measured with error. Application to survival and cd4 counts in patients with aids. J. Amer. Statist. Assoc.90 (1995) 27–37. Zbl0818.62102
- Y.N. Tyurin, A. Yakovlev, J. Shi and L. Bass, Testing a model of aging in animal experiments. Biometrics51 (1995) 363–372. Zbl0826.62092
- A.W. van der Vaart and J.A. Wellner, Weak convergences and empirical processes. With applications to statistics. Springer Series in Statistics (1996). Zbl0862.60002
- S.X. Xie, C.Y. Wang and R.L. Prentice, A risk set calibration method for failure time regression by using a covariate reliability sample. J.R. Stat. Soc., Ser. B, Stat. Methodol.63 (2001) 855–870. Zbl0998.62087

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