Bayesian Nonlinear Quantile Regression Approach for Longitudinal Ordinal Data

Abstract

Longitudinal data with ordinal outcomes commonly arise in clinical and social studies, where the purpose of interest is usually quantile curves rather than a simple reference range. In this paper we consider Bayesian nonlinear quantile regression for longitudinal ordinal data through a latent variable. An efficient Metropolis–Hastings within Gibbs algorithm was developed for model fitting. Simulation studies and a real data example are conducted to assess the performance of the proposed method. Results show that the proposed approach performs well.

Keywords

Ordinal longitudinal data / Bayesian approach / Quantile regression / MCMC / Metropolis–Hastings algorithm

References

[1] Alhamzawi R. Bayesian model selection in ordinal quantile regression. Comput. Stat. Data Anal.. 2016, 103 68-78

[2] Alhamzawi R, Ali HTM. Bayesian quantile regression for longitudinal ordinal data. J. Appl. Stat.. 2018, 45 5 815-828

[3] Alhamzawi R, Yu K, Vinciotti V, Tucker A. Prior elicitation for mixed quantile regression with an allometric model. Environmetrics. 2011, 22 7 911-920

[4] Chopra A, Cavalieri TA, Libon DJ. Dementia screening tolls for the primary care physician. Clin. Geriatr.. 2007, 15 1 38-45

[5] Cole TJ, Green PJ. Smoothing reference centile curves: the LMS method and penalized likelihood. Stat. Med.. 1992, 11 10 1305-1319

[6] Commenges D, Scotet V, Renaud S, Jacqmin-Gadda H, Barberger-Gateau P, Dartigues JF. Intake of flavonoids and risk of dementia. Eur. J. Epidemiol.. 2000, 16 4 357-363

[7] Dartigues JF, Gagnon M, Barberger-Gateau P, Letenneur L, Commenges D, Sauvel C . The Paquid epidemiological program on brain ageing. Neuroepidemiology. 1992, 11 Suppl. 1 14-18

[8] Geraci M, Bottai M. Quantile regression for longitudinal data using the asymmetric Laplace distribution. Biostatistics. 2006, 8 1 140-154

[9] Gilks, W.R., Richardson, S., Spiegelhalter, D. (eds.): Markov Chain Monte Carlo in Practice. CRC Press, Boca Raton (1995)

[10] Hans C. Bayesian lasso regression. Biometrika. 2009, 96 835845

[11] Helmer C, Peres K, Letenneur L, Guttirez-Robledo LM, Ramaroson H, Barberger-Gateau P . Dementia in subjects aged 75 years or over within the Paquid cohort: prevalence and burden by severity. Dementia Geriatr. Cogn. Disord.. 2006, 22 1 87-94

[12] Hong HG, He X. Prediction of functional status for the elderly based on a new ordinal regression model. J. Am. Stat. Assoc.. 2010, 105 491 930-941

[13] Hong HG, Zhou J. A multi-index model for quantile regression with ordinal data. J. Appl. Stat.. 2013, 40 6 1231-1245

[14] Karlsson A. Nonlinear quantile regression estimation of longitudinal data. Commun. Stat. Simul. Comput.. 2007, 37 1 114-131

[15] Katzman R. Education and the prevalence of dementia and Alzheimer’s disease. Neurology. 1993, 43 1 13-20

[16] Koenker R. Quantile regression for longitudinal data. J. Multivar. Anal.. 2004, 91 1 74-89

[17] Koenker R, Bassett G Jr. Regression quantiles. Econ. J. Econ. Soc.. 1978, 46 1 33-50

[18] Kotz S, Kozubowski T, Podgorski K. The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance. 2012 Berlin: Springer

[19] Kozumi H, Kobayashi G. Gibbs sampling methods for Bayesian quantile regression. J. Stat. Comput. Simul.. 2011, 81 11 1565-1578

[20] Lemeshow S, Letenneur L, Dartigues JF, Lafont S, Orgogozo JM, Commenges D. Illustration of analysis taking into account complex survey considerations: the association between wine consumption and dementia in the Paquid study. Am. J. Epidemiol.. 1998, 148 3 298-306

[21] Letenneur L, Gilleron V, Commenges D, Helmer C, Orgogozo JM, Dartigues JF. Are sex and educational level independent predictors of dementia and Alzheimers disease? incidence data from the Paquid project. J. Neurol. Neurosurg. Psychiatry. 1999, 66 2 177-183

[22] Luo Y, Lian H, Tian M. Bayesian quantile regression for longitudinal data models. J. Stat. Comput. Simul.. 2012, 82 11 1635-1649

[23] Montesinos-Lpez OA, Montesinos-Lpez A, Crossa J, Burgueo J, Eskridge K. Genomic-enabled prediction of ordinal data with Bayesian logistic ordinal regression. G3 Genes Genom. Genet.. 2015, 5 10 2113-2126

[24] Prince MJ, Beekman AT, Deeg DJ, Fuhrer R, Kivela SL, Lawlor BA . Depression symptoms in late life assessed using the EURO-D scale. Effect of age, gender and marital status in 14 European centres. Br. J. Psychiatry. 1999, 174 4 339-345

[25] Ramaroson H, Helmer C, Barberger-Gateau P, Letenneur L, Dartigues JF. Prevalence of dementia and Alzheimer’s disease among subjects aged 75 years or over: updated results of the Paquid cohort. Revue Neurol.. 2003, 159 4 405-411

[26] Royston P, Altman DG. Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling. Appl. Stat.. 1994, 43 3 429-467

[27] Sorensen DA, Andersen S, Gianola D, Korsgaard I. Bayesian inference in threshold models using Gibbs sampling. Genet. Sel. Evol.. 1995, 27 3 229

[28] Spiegelhalter DJ, Best NG, Carlin BP, Van Der Linde A. Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B (Stat. Methodol.). 2002, 64 4 583-639

[29] Yu K, Lu Z, Stander J. Quantile regression: applications and current research areas. J. R. Stat. Soc. Ser. D (Stat.). 2003, 52 3 331-350

[30] Yu K, Moyeed RA. Bayesian quantile regression. Stat. Probab. Lett.. 2001, 54 4 437-447

[31] Yu K, Zhang J. A three-parameter asymmetric Laplace distribution and its extension. Commun. Stat. Theory Methods. 2005, 34 9–10 1867-1879

[32] Zhou, L.: Conditional quantile estimation with ordinal data. (Doctoral Dissertation) (2010). https://scholarcommons.sc.edu/etd/301. Accessed 3 Oct 2018

Funding

the National Key Research and Development Plan(2016YFC0800100)

National Natural Science Foundation of China(11671374)