Skip to main content
SpringerLink
Log in

A Bayesian quantile regression approach to multivariate semi-continuous longitudinal data

  • Original paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

Quantile regression is a powerful tool for modeling non-Gaussian data, and also for modeling different quantiles of the probability distributions of the responses. We propose a Bayesian approach of estimating the quantiles of multivariate longitudinal data where the responses contain excess zeros. We consider a Tobit regression approach, where the latent responses are estimated using a linear mixed model. The longitudinal dependence and the correlations among different (latent) responses are modeled by the subject-specific vector of random effects. We consider a mixture representation of the Asymmetric Laplace Distribution (ALD), and develop an efficient MCMC algorithm for estimating the model parameters. The proposed approach is used for analyzing data from the health and retirement study (HRS) conducted by the University of Michigan, USA; where we jointly model (i) out-of-pocket medical expenditures, (ii) total financial assets, and (iii) total financial debt for the aged subjects, and estimate the effects of different covariates on these responses across different quantiles. Simulation studies are performed for assessing the operating characteristics of the proposed approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Albert JH, Chib S (1993) Bayesian analysis of binary and polychotomous response data. J Am Stat Assoc 88:669–679

    Article  MathSciNet  Google Scholar 

  • Alfo M et al. (2016) M-quantile regression for multivariate longitudinal data: analysis of the Millennium Cohort Study data. arXiv:1612.08114

  • Barndorff-Nielsen OE, Shephard N (2001) Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics. J Roy Stat Soc B 63:167–241

    Article  MathSciNet  Google Scholar 

  • Bhuyan P, Biswas J, Ghosh P, Das K (2019) A Bayesian two-stage regression approach of analyzing longitudinal outcomes with endogeneity and incompleteness. Stat Modell 19:157–173

    Article  Google Scholar 

  • Biswas J, Das K (2020) A Bayesian approach of analyzing semi-continuous longitudinal data with monotone missingness. Stat Modell 20:148–170

    Article  Google Scholar 

  • Biswas J, Ghosh P, Das K (2020) A semi-parametric quantile regression approach to zero-inflated and incomplete longitudinal outcomes. Adv Stat Anal 104:261–283

    Article  MathSciNet  Google Scholar 

  • Brooks SP, Gelman A (1998) General methods for monitoring convergence of iterative simulations. J Comput Gr Stat 7:434–455

    MathSciNet  Google Scholar 

  • Brown S, Ghosh P, Taylor K (2015) Modelling household finances: A Bayesian approach to a multivariate two-part model. J Empir Finance 33:190–207

    Article  Google Scholar 

  • Buchinsky M, Hahn J (1998) An alternative estimator for the censored quantile regression model. Econometrica 66:653–671

    Article  MathSciNet  Google Scholar 

  • Chaudhuri P (1996) On a geometric notion of quantiles for multivariate data. J Am Stat Assoc 91:862–872

    Article  MathSciNet  Google Scholar 

  • Chatterjee A, Biswas J, Das K (2020) An automated patient monitoring using discrete-time wireless sensor networks. Int J Commun Syst 50:160. https://doi.org/10.1002/dac.4390

    Article  Google Scholar 

  • Cole TJ, Green PJ (1992) Smoothing reference centile curves: the LMS method and penalized likelihood. Stat Med 11:1305–1319

    Article  Google Scholar 

  • Das K, Daniels MJ (2014) A semiparametric approach to simultaneous covariance estimation for bivariate sparse longitudinal data. Biometrics 70:33–43

    Article  MathSciNet  Google Scholar 

  • Delattre M, Lavielle M, Poursat M-A (2014) A note on BIC in mixed effects models. Electron J Stat 8:456–475

    Article  MathSciNet  Google Scholar 

  • Drovandi C, Pettitt A (2011) Likelihood-free bayesian estimation of multivariate quantile distributions. Comput Stat Data Anal 55:2541–2556

    Article  MathSciNet  Google Scholar 

  • Duan N, Manning W, Morris C, Newhouse JP (1983) A comparison of alternative models for the demand for medical care (Corr: V2 P413). J Bus Econ Stat 1:115–126

    Google Scholar 

  • Ghosh P, Hanson TA (2010) Semiparametric Bayesian approach to multivariate longitudinal data. Aust New Zealand J Stat 5:275–288

    Article  MathSciNet  Google Scholar 

  • Geraci M, Bottai M (2007) Quantile regression for longitudinal data using the asymmetric Laplace distribution. Biostatistics 8:140–154

    Article  Google Scholar 

  • Greven S, Kneib T (2010) On the behaviour of marginal and conditional AIC in linear mixed models. Biometrika 97:773–789

    Article  MathSciNet  Google Scholar 

  • Hahn J (1995) Bootstrapping quantile regression estimators. Econ Theory 11:105–121

    Article  MathSciNet  Google Scholar 

  • Hallin M, Paindaveine D, Siman M (2010) Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth. Ann Stat 38:635–669

    Article  Google Scholar 

  • Heagerty P, Pepe M (1999) Semiparametric estimation of regression quantiles with application to standardizing weight for height and age in U.S. children. J R Stat Soc Ser C 48:533–551

    Article  Google Scholar 

  • Jung SH (1996) Quasi-likelihood for median regression models. J Am Stat Assoc 91:251–257

    Article  MathSciNet  Google Scholar 

  • Koenker R, Bassett G (1978) Regression Quantiles. Econometrica 46:33–50

    Article  MathSciNet  Google Scholar 

  • Koenker R (2004) Quantile regression for longitudinal data. J Multivar Anal 91:74–89

    Article  MathSciNet  Google Scholar 

  • Kozumi H, Kobayashi G (2011) Gibbs sampling methods for Bayesian quantile regression. J Stat Comput Simul 81:1565–1578

    Article  MathSciNet  Google Scholar 

  • Kulkarni H, Biswas J, Das K (2019) A joint quantile regression model for multiple longitudinal outcomes. Adv Stat Anal 103:453–473

    Article  MathSciNet  Google Scholar 

  • Min Y, Agresti A (2002) Modeling nonnegative data with clumping at zero: a survey. J Iranian Stat Soc 1:7–33

    MATH  Google Scholar 

  • Mukherji A, Roychoudhury S, Ghosh P, Brown S (2016) Estimating health demand for an aging population: a flexible and robust Bayesian joint model. J Appl Econ 31:1140–1158

    Article  MathSciNet  Google Scholar 

  • Park T, Casella G (2008) The bayesian lasso. J Am Stat Assoc 103:681–686

    Article  MathSciNet  Google Scholar 

  • Powell JL (1986) Censored regression quantiles. J Econ 32:143–155

    Article  MathSciNet  Google Scholar 

  • R Core Team (2020) R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/

  • Reich B, Fuentes M, Dunson D (2011) Bayesian spatial quantile regression. J Am Stat Assoc 106:6–20

    Article  MathSciNet  Google Scholar 

  • Tobin J (1958) Estimation of relationships for limited dependent variables. Econometrica 26:24–36

    Article  MathSciNet  Google Scholar 

  • Waldmann E, Kneib T (2015) Bayesian bivariate quantile regression. Stat Modell 15:326–344

    Article  MathSciNet  Google Scholar 

  • Yu K, Moyeed R (2001) Bayesian quantile regression. Stat Probab Lett 54:437–447

    Article  MathSciNet  Google Scholar 

  • Yu K, Stander J (2007) Bayesian analysis of a Tobit quantile regression model. J Econ 137:260–276

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank Prof. Pulak Ghosh, Indian Institute of Management, Bangalore, India; for sharing the HRS dataset.

Author information

Authors and Affiliations

  1. Interdisciplinary Statistical Research Unit, Applied Statistics Division, Indian Statistical Institute, 203 B.T. Road, Kolkata, 700108, India

    Jayabrata Biswas & Kiranmoy Das

  2. Department of Statistics, University of Burdwan, Burdwan, West Bengal, India

    Jayabrata Biswas

Authors
  1. Jayabrata Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kiranmoy Das
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kiranmoy Das.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (xls 16138 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biswas, J., Das, K. A Bayesian quantile regression approach to multivariate semi-continuous longitudinal data. Comput Stat 36, 241–260 (2021). https://doi.org/10.1007/s00180-020-01002-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-020-01002-1

Keywords

Search

Navigation