Journal of Advanced Statistics

Download PDF (674 KB) PP. 109 - 121 Pub. Date: September 1, 2016

DOI: 10.22606/jas.2016.13001

• Yangxin Huang^*
  School of Mathematics and Computer, Wuhan Textile University, Wuhan 430073, P.R.China and College of Public Health, University of South Florida, Tampa, Florida 33612, USA
• Jiaqing Chen
  Department of Statistics, Wuhan University of Technology, Wuhan, Hubei, 430070, P.R.China
• Xiaosun Lu
  Department of Biostatistics, Medpace Inc., Cincinnati, OH 45227, USA

In longitudinal studies, measurements of the same individuals are taken repeatedly through time, but it often happens that some collected data are observed with the following issues. (i) Often, the primary goal is to characterize the change in response over time. Compared with conventional mean regression, quantile regression (QR) can characterize the entire conditional distribution of the outcome variable, and may be more robust to outliers and mis-specification of error distribution. (ii) longitudinal outcomes may suffer from a serious departure of normality in which normality assumption may cause lack of robustness and subsequently lead to invalid inference; and (iii) the response observations may be subject to left-censoring due to a limit of detection. Inferential procedures will become very complicated when one analyzes data with these features together. In this article, Bayesian modeling approach to nonlinear mixed-effects quantile regression models for longitudinal data is developed to study simultaneous impact of multiple data features (non-normality, left-censoring, non-linearity, outliers and heavy-tails). Simulation studies are conducted to assess the performance of the proposed models and methods. A real data example is analyzed to demonstrate the proposed methodology through comparing potential models with different distribution specifications of random-effects.

Asymmetric Laplace distribution, Bayesian inference, Left-censoring, Nonlinear mixedeffects quantile regression, Skew-normal distribution.

[1] Hammer S.M., Vaida F., Bennett K.K. et al., "Dual vs single protease inhibitor therapy following antiretroviral treatment failure: a randomized trial", The Journal of the American Medical Association, vol. 288, pp. 169-180, 2002.

[2] Perelson, A.S., Essunger, P., Cao, Y., Vesanen, M., Hurley, A., Saksela, K., Markowitz, M., Ho, D.D., "Decay characteristics of HIV-1-infected compartments during combination therapy", Nature, vol. 387, pp. 188–191, 1997.

[3] Wu, H., Ding, A.A., "Population HIV-1 dynamics in vivo: Applicable models and inferential tools for virological data from AIDS clinical trials",Biometrics, vol. 55, pp. 410–418, 1999.

[4] Wu, L., Liu, W., Hu, X.J., "Joint inference on HIV viral dynamics and immune suppression in presence of measurement errors",Biometrics, vol. 66, pp. 327–335, 2010.

[5] Hughes J.P., "Mixed effects models with censored data with applications to HIV RNA levels",Biometrics, vol. 55, pp. 625–629, 1999.

[6] Liu, W., Wu, L., "Simultaneous inference for semiparametric nonlinear mixed-effects models with covariate measurement errors and missing responses",Biometrics, vol. 63, pp. 342–350, 2007.

[7] Wu, L., "A joint model for nonlinear mixed-effects models with censoring and covariates measured with error, with application to AIDS studies",Journal of the American Statistical Association, vol. 97, pp. 955–964, 2002.

[8] Yi, G.Y., Liu, W., Wu L., "Simultaneous inference and bias analysis for longitudinal data with covariate measurement error and missing responses",Biometrics, vol. 67, pp. 67–75, 2011.

[9] Arellano-Valle R.B., Bolfarine H. and Lachos V.H., "Bayesian inference for skew-normal linear mixed models", Journal of Applied Statistics, vol. 34, pp. 663-682.

[10] Ho HJ, Lin TI. "Robust linear mixed models using the skew-t distribution with application to schizophrenia data",Biometrical Journal, vol. 52, pp. 449–469, 2010.

[11] A. Jara, F. Quintana and E.S. Martin, "Linear mixed models with skew-elliptical distributions: A Bayesian approach",Computational Statistics and Data Analysis , vol. 52, pp. 5033-5045, 2008.

[12] S.K. Sahu, D.K. Dey and M.D. Branco, "A new class of multivariate skew distributions with applications to Bayesian regression models",The Canadian Journal of Statistics, vol. 31, pp. 129-150, 2003.

[13] Arellano-Valle R.B. and Genton, M., "On fundamental skew distributions",Journal of Multivariate Analysis, vol. 96, pp. 93-116, 2005.

[14] Azzalini A. and Capitanio A., "Statistical applications of the multivariate skew normal distribution",Journal of Royal Statistical Society, Series B, vol. 67, pp. 579-602, 1999.

[15] Tobin, J., "Estimation of relationships for limited dependent variables", Econometrica, vol. 26, pp. 24–36, 1958.

[16] Huang Y., Dagne G.A., "Bayesian semiparametric nonlinear mixed-effects joint models for data with skewness, missing responses and measurement errors in covariates", Biometrics, vol. 68(3), pp. 943–953, 2012.

[17] Koenker, R., Bassett, G., "Regression quantiles", Econometrica, vol. 46, pp. 33–50, 1978.

[18] Koenker, R., Quantile Regression. Cambridge University Press, New York, 2005.

[19] Kozumi, H., Kobayashi G., "Gibbs sampling methods for Bayesian quantile regression", Journal of Statistical Computation and Simulation, vol. 81(11), pp. 1565–1578, 2011.

[20] Yuan, Y., Yin, G., "Bayesian quantile regression for longitudinal studies with nonignorable missing data",Biometrics, vol. 66, pp. 105–114, 2010.

[21] Koenker, R., "Quantile regression for longitudinal data", Journal of Multivariate Analysis, vol. 91(1), pp. 74–89, 2004.

[22] Alhamzawi R. and Yu K., "Bayesian Lasso-mixed quantile regression", Journal of Statistical Computation and Simulation, vol. 84(4), pp. 868-880, 2014.

[23] Farcomeni A., "Quantile regression for longitudinal data based on latent Markov subjectspecific parameters", Journal Statistics and Computing, vol. 22, pp. 141–152, 2012.

[24] Geraci M. and Bottai M., "Quantile regression for longitudinal data using the asymmetric Laplace distribution", Biostatistics, vol. 8, pp. 140–154, 2007.

[25] Kim, M.O., Yang, Y., "Semiparametric approach to a random effects quantile regression model", Journal of the American Statistical Association, vol. 106(496), pp. 1405–1417, 2011.

[26] Lipsitz, S.R., Fitzmaurice, G.M., Molenberghs, G., Zhao, L.P., "Quantile regression methods for longitudinal data with drop-outs: Application to CD4 cell counts of patients infected with the human immunodeficiency virus", Journal of the Royal Statistical Society, Series C, vol. 46, pp. 463–476, 1997.

[27] Liu, Y., Bottai, M., "Mixed-effects models for conditional quantiles with longitudinal data", The International Journal of Biostatistics, vol. 5(1), Article 28, 2009.

[28] Luo, Y., Lain, H., Tian, M., "Bayesain quantile regression for longitudinal data models", Journal of Statistical Computation and Simulation, vol. 82(11), pp. 1635–1649, 2012.

[29] Wang, H.J., Fygenson, M., “Inference for censored quantile regression models in longitudinal studies,” The Annals of Statistics, vol. 37(2), pp. 756–781, 2009.

[30] Yu, K., Zhang, J., "A three-parametric asymmetric Laplace distribution and its extension", Communication in Statistics–Theory and Methods, vol. 34, pp. 1867–1879, 2005.

[31] Yu, K., Moyeed, R.A., "Bayesian quantile regression", Statistics and Probability Letters, vol. 54, pp. 437–447, 2001.

[32] Koenker, R., Machado, J., "Goodness of fit and related inference process for quantile regression", Journal of the American Statistical Association, vol. 94, pp. 1296–1310, 1999.

[33] Yu, K., Stander, J., "Bayesian analysis of a Tobit quantile regression model", Journal of Econometrics, vol. 137, pp. 260–276, 2007.

[34] Kotz, S., Kozabowski, T.J., Podgorski K., The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance, Birkhauser, Boston, 2001.

[35] Gelman, A. and Rubin, D., "Inference from iterative simulation using multiple sequences", Statistical Science, vol. 7, pp. 457-511, 1992.

[36] Lunn, D.J., Thomas, A., Best, N., Spiegelhalter, D., "WinBUGS - a Bayesian modelling framework: concepts, structure, and extensibility", Statistics and Computing, vol. 10, pp. 325–337, 2000.

[37] Nowak, MA., May, RM., Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford, 2000.

[38] Huang, Y., Liu, D., Wu, H., "Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system", Biometrics, vol. 62, pp. 413–423, 2006.

[39] Huang Y., Chen, J., Yan C., "Mixed-effects joint models with skew-normal distribution for HIV dynamic response with missing and mismeasured time-varying covariate", International Journal of Biostatistics, vol. 8(1), Article 34, 1–30, 2012.

[40] Spiegelhalter, D. J., Best, N. G., Carlin, B. P., Van der Linde, A., "Bayesian measures of model complexity and fit", Journal of the Royal Statistical Society, Series B, vol. 64, pp. 583–639, 2002.

[41] Gelman, A. Carlin, J.B., Stern, H.S., Rubin. D.B., Bayesian Data Analysis. Chapman and Hall, London, 2003.

[42] Huang Y, Dagne GA, Wu L., "Bayesian inference on joint models of HIV dynamics for time-to-event and longitudinal data with skewness and covariate measurement errors", Statistics in Medicine, vol. 30, pp. 2930–2946, 2011.