Article
Simultaneous estimation in Cox proportional hazards model with measurement errors
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Authors
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Oksana Chernova
- Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
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Alexander Kukush
- Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
| Published: | February 26, 2021 |
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Outline
Text
Background: The Cox proportional hazards (PH) model is a regression model that can be used in medical research, engineering, finance or insurance for investigating the association between the survival time (the so-called lifetime) of an object and predictor variables. In practice, some covariates could be measured with errors, which, if ignored, may lead to biased estimation and erroneous inference, see Wallace [1]. The Cox PH model with measurement errors have been studied, among others, by Kong & Gu [2] and Augustin [3]. Firstly, the vector of regression parameters is estimated and then one gets estimator of the cumulative baseline hazard rate. In our approach the baseline hazard rate belongs to an unbounded set of nonnegative Lipschitz functions, and is estimated itself together with the vector of regression parameters.
Methods: We investigate the Cox proportional hazards model for right-censored data, where the baseline hazard rate belongs to an unbounded set of nonnegative Lipschitz functions, with fixed constant, and the vector of regression parameters belongs to a compact parameter set, and in addition, the time-independent covariates are subject to additive measurement errors. We assume that random errors have known moment generating function. A couple lifetime and regressor, censor, and measurement error are independent.
In Kukush & Chernova [4], we define the simultaneous estimator of baseline hazard rate and regression parameters. Under certain assumptions, the estimator is proven to be strongly consistent.
Furthermore, we show that the estimator of baseline hazard function is a linear spline, whose ordinates are solutions to the constrained optimization problem. We describe how to calculate the estimator and present the simulation results using the programming language Python, in case when true baseline hazard function is linear and measurement errors have normal distribution.
Results: In Kukush & Chernova [4], we derive a simultaneous consistent estimator 
that maximize the corrected log-likelihood function, show that the estimator 
and linear integral functionals of 
are asymptotically normal, in addition, is a linear spline.
Conclusion: The simultaneous estimator in the Cox proportional hazards model for right-censored data, where the baseline hazard rate belongs to an unbounded set of nonnegative Lipschitz functions and the vector of regression parameters belongs to a compact parameter set, in case of time-independent covariates that are subject to measurement errors is constructed. The simulation of the model has been performed.
The authors declare that they have no competing interests.
The authors declare that an ethics committee vote is not required.
References
- 1.
- Wallace M. Analysis in an imperfect world. Significance. 2020;17(1):14-19.
- 2.
- Kong FH, Gu M. Consistent estimation in Cox proportional hazards model with covariate measurement errors. Statistica Sinica. 1999:953-969.
- 3.
- Augustin T. An exact corrected log-likelihood function for Cox's proportional hazards model under measurement error and some extensions. Scandinavian Journal of Statistics. 2004;31(1):43-50.
- 4.
- Kukush A, Chernova O. Consistent estimation in Cox proportional hazards model with measurement errors and unbounded parameter set. Theory of Probability and Mathematical Statistics. 2018;96:101-110.
