Semiparametric Bayesian analysis of censored linear regression with errors-in-covariates

Abstract

The accelerated failure time (AFT) model is a well-known alternative to the Cox proportional hazard model for analyzing time-to-event data. In this paper we consider fitting an AFT model to right censored data when a predictor variable is subject to measurement errors. First, without measurement errors, estimation of the model parameters in the AFT model is a challenging task due to the presence of censoring, especially when no specific assumption is made regarding the distribution of the logarithm of the time-to-event. The model complexity increases when a predictor is measured with error. We propose a non-parametric Bayesian method for analyzing such data. The novel component of our approach is to model (1) the distribution of the time-to-event, (2) the distribution of the unobserved true predictor, and (3) the distribution of the measurement errors all non-parametrically using mixtures of the Dirichlet process priors. Along with the parameter estimation we also prescribe how to estimate survival probabilities of the time-to-event. Some operating characteristics of the proposed approach are judged via finite sample simulation studies. We illustrate the proposed method by analyzing a data set from an AIDS clinical trial study.

Keywords

Buckley–James estimator Dirichlet process prior functional approach measurement errors mixture distributions posterior inference

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References

Carroll

. Measurement error in epidemiologic studies. Encyclopedia of Biostatistics, New York: John Wiley & Sons, 1997.

Buckley

James

. Linear regression with censored data. Biometrika 1979; 66: 429–436.

Lai

Ying

. Large sample theory of a modified Buckley–James estimator for regression analysis with censored data. Ann Statist 1991; 19: 1370–1402.

Zeng

Lin

. Efficient estimation for the accelerated failure time model. J Am Statist Assoc 2007; 102: 1387–1396.

Jin

Lin

Wei

Ying

. Rank-based inference for the accelerated failure time model. Biometrika 2003; 90: 341–353.

Zhou

. Empirical likelihood analysis of the Buckley–James estimator. J Multivar Anal 2008; 99: 649–664.

Cox

Chu

Schneider

Muñoz

. Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Statist Med 2007; 26: 4252–4374.

Christensen

Johnson

. Modelling accelerated failure time with a Dirichlet process. Biometrika 1988; 75: 693–704.

Kuo

Mallick

. Bayesian semiparametric inference for the accelerated failure-time model. Can J Statist 1997; 25: 457–472.

10.

Walker

Mallick

. A Bayesian semiparametric accelerated failure time model. Biometrics 1999; 55: 477–483.

11.

Xiong

. Accelerated failure time models with covariates subject to measurement Error. Statist Med 2007; 26: 4817–4832.

12.

Yin

. Censored quantile regression with covariate measurement errors. Statistica Sinica 2011; 21: 949–971.

13.

Ishwaran

James

. Gibbs sampling methods for stick breaking priors. J Am Statist Assoc 2001; 96: 161–173.

14.

Ghosh

Ramamoorthi

. Bayesian Nonparametrics, New York: Springer, 2003.

15.

Müller

Roeder

. A Bayesian semiparametric model for case–control studies with errors in variables. Biometrika 1997; 84: 523–537.

16.

Gustafson

Vallee

. A Bayesian approach to case–control studies with errors in covariables. Biostatistics 2002; 3: 229–243.

17.

Sinha

Mallick

Kipnis

Carroll

. Semiparametric Bayesian analysis of nutritional epidemiology data in the presence of measurement error. Biometrics 2010; 66: 444–454.

18.

Delaigle

Hall

Meister

. On deconvolution with repeated measurements. Ann Statist 2008; 36: 665–685.

19.

Carroll

Ruppert

Stefanski

Crainiceanu

. Measurement Error in Nonlinear Models: A Modern Perspective, 2nd edn. Boca Raton, FL: Chapman and Hall/CRC Press, 2006.

20.

. Variable selection in measurement error models. Bernoulli 2010; 16: 274–300.

21.

Carroll

. Locally efficient estimators for semiparametric models with measurement error. J Am Statist Assoc 2006; 101: 1465–1474.

22.

Ishwaran

Zarepour

. Exact and approximate sum representations for the Dirichlet process. Can J Statist 2002; 30: 269–283.

23.

Hall

. Measurement error models with unknown error structure. J R Statist Soc Ser B 2007; 69: 429–446.

24.

Vuong

. Nonparametric estimation of the measurement error model using multiple indicators. J Multivar Anal 1998; 65: 139–165.

25.

Yang

Dunson

Baird

. Semiparametric Bayes hierarchical models with mean and variance constraints. Computat Statist Data Anal 2010; 54: 2172–2186.

26.

Escobar

West

. Bayesian density estimation and inference using mixtures. J Am Statist Assoc 1995; 90: 577–588.

27.

Cowles

Carlin

. Markov chain monte carlo convergence diagnostics: A comparative review. J Am Statist Assoc 1996; 91: 883–904.

28.

Newton

Raftery

. Approximate Bayesian inference with the weighted likelihood bootstrap. J R Statist Soc Ser B 1994; 56: 3–26.

29.

Hammer

Katzenstein

Hughes

. A trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults with CD4 cell counts from 200 to 500 per cubic millimeter. N Engl J Med 1996; 335: 1081–1090.

30.

Huang J and Wellner J. Interval censored survival data: a review of recent progress. In Proceedings of the first Seattle symposium in biostatistics: survival analysis (Lecture Notes in Statistics, vol. 123). Berlin: Springer, 1997; 123–169.

31.

Veronesi

Ferrario

Chambless

. Comparing measurement error correction methods for rate-of-change exposure variables in survival analysis. Statist Meth Med Res 2011; 22: 583–597.

32.

Crowther

Lambert

Abrams

. Adjusting for measurement error in baseline prognostic biomarkers included in a time-to-event analysis: a joint modelling approach. BMC Med Res Methodol 2013; 13: 146–146.

33.

Tsiatis

. Semiparametric Theory and Missing Data, New York: Springer, 2006.

34.

Gill

Levit

. Applications of the van Trees inequality: a Bayesian Cramér–Rao bound. Bernoulli 1995; 1: 59–79.

35.

Gelfand

Hills

Racine-Poon

Smith

. Illustration of Bayesian inference in normal data models using Gibbs sampling. J Am Statist Assoc 1990; 85: 972–985.