Abstract
We propose quantile regression (QR) in the Bayesian framework for a class of nonlinear mixed effects models with a known, parametric model form for longitudinal data. Estimation of the regression quantiles is based on a likelihood-based approach using the asymmetric Laplace density. Posterior computations are carried out via Gibbs sampling and the adaptive rejection Metropolis algorithm. To assess the performance of the Bayesian QR estimator, we compare it with the mean regression estimator using real and simulated data. Results show that the Bayesian QR estimator provides a fuller examination of the shape of the conditional distribution of the response variable. Our approach is proposed for parametric nonlinear mixed effects models, and therefore may not be generalized to models without a given model form.
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References
Davidian M, Giltinan M (1995) Nonlinear models for repeated measurement data. Chapman & Hall, London
Gelfand AE et al (1990) Illustration of Bayesian inference in normal data models using Gibbs sampling. J Am Stat Assoc 85: 972–985
Geraci M, Bottai M (2007) Quantile regression for longitudinal data using the asymmetric Laplace distribution. Biostatistics 8: 140–154
Geweke J (1992) Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In: Bernardo J, Berger J, Dawid A, Smith M (eds) Bayesian statistics. Oxford University Press, Oxford, pp 169–193
Gilks WR, Best NG, Tan KKC (1995) Adaptive rejection sampling for Gibbs sampling. Appl Stat 44: 455–472
Karlsson A (2008) Nonlinear quantile regression estimation of longitudinal data. Commun Stat Simul Comput 37: 114–131
Koenker R (1984) A note on L-estimators for linear models. Stat and Prob Lett 2: 323–325
Koenker R (2004) Quantile regression for longitudinal data. J Multivar Anal 91: 74–89
Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46: 33–50
Koenker R, Machado J (1999) Goodness of fit and related inference processes for quantile regression. J Am Stat Assoc 94: 1296–1310
Nelder A, Mead R (1965) A simplex method for function minimization. Comput J 7: 308–313
O’Neill R (1971) Function minimization using a simplex procedure, Algorithm AS47. Appl Stat 20: 338–345
Potvin C, Lechowicz MJ, Tardif S (1990) The statistical analysis of ecophysiological response curves obtained from experiments involving repeated measures. Ecology 71: 1389–1400
R Development Core Team (2011) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org.
SAS Institute Inc (2008) SAS/STAT 9.2 User’s Guide. SAS Institute Inc, Cary, NC
Wakefield JC, Smith AF, Racine-Poon A, Gelfand AE (1994) Bayesian analysis of linear and nonlinear population models using the Gibbs sampler. Appl Stat 43: 201–221
Yu K, Moyeed A (2001) Bayesian quantile regression. Stat Probab Lett 54: 437–447
Yu K, Stander J (2007) Bayesian analysis of a Tobit quantile regression model. J Econ 137: 260–276
Yuan Y, Yin G (2010) Bayesian quantile regression for longitudinal studies with nonignorable missing data. Biometrics 66: 105–114
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Wang, J. Bayesian quantile regression for parametric nonlinear mixed effects models. Stat Methods Appl 21, 279–295 (2012). https://doi.org/10.1007/s10260-012-0190-7
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DOI: https://doi.org/10.1007/s10260-012-0190-7