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A joint quantile regression model for multiple longitudinal outcomes

  • Hemant Kulkarni
  • Jayabrata Biswas
  • Kiranmoy Das
Original Paper
  • 39 Downloads

Abstract

Complexity of longitudinal data lies in the inherent dependence among measurements from same subject over different time points. For multiple longitudinal responses, the problem is challenging due to inter-trait and intra-trait dependence. While linear mixed models are popularly used for analysing such data, appropriate inference on the shape of the population cannot be drawn for non-normal data sets. We propose a linear mixed model for joint quantile regression of multiple longitudinal responses. We consider an asymmetric Laplace distribution for quantile regression and estimate model parameters by Monte Carlo EM algorithm. Nonparametric bootstrap resampling method is used for estimating confidence intervals of parameter estimates. Through extensive simulation studies, we investigate the operating characteristics of our proposed model and compare the performance to other traditional quantile regression models. We apply proposed model for analysing data from nutrition education programme on hypercholesterolemic children of the USA.

Keywords

Asymmetric Laplace Distribution EM algorithm Longitudinal data MCMC Quantile regression 

Notes

Acknowledgements

The authors sincerely thank Prof. Vern Chinchilli, the Pennsylvania State University, for sharing data from one-year nutrition education.

References

  1. Alfo, M. et al.: M-quantile regression for multivariate longitudinal data: analysis of the Millennium Cohort study data. arXiv:1612.08114 (2016)
  2. Bandyopadhyay, D., Lachos, V.H., Abanto-Valle, C.A., Ghosh, P.: Linear mixed models for skew-normal/independent bivariate responses with an application to periodontal disease. Stat. Med. 29, 2643–2655 (2010)MathSciNetCrossRefGoogle Scholar
  3. Booth, J., Hobert, J.P.: Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 61, 265–285 (1999)CrossRefzbMATHGoogle Scholar
  4. Cai, Y.: Multivariate quantile function models. Stat. Sin. 20, 481–496 (2010)MathSciNetzbMATHGoogle Scholar
  5. Chakraborty, B.: On multivariate quantile regression. J. Stat. Plan. Inference 110, 101–132 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Chaudhuri, P.: On a geometric notion of quantiles for multivariate data. J. Am. Stat. Assoc. 91, 862–872 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cole, T.J., Green, P.J.: Smoothing reference centile curves: the LMS method and penalized likelihood. Stat. Med. 11, 1305–1319 (1992)CrossRefGoogle Scholar
  8. Das, K., Li, J., Fu, G., Wang, Z., Wu, R.: Genome-wide association studies for bivariate sparse longitudinal data. Hum. Hered. 72, 110–120 (2011)CrossRefGoogle Scholar
  9. Das, K., Li, R., Sengupta, S., Wu, R.: A Bayesian semi-parametric model for bivariate sparse longitudinal data. Stat. Med. 32, 3899–3910 (2013)MathSciNetCrossRefGoogle Scholar
  10. Das, K., Daniels, M.J.: A semiparametric approach to simultaneous covariance estimation for bivariate sparse longitudinal data. Biometrics. 70, 33–43 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Das, K., Afriyie, P., Spirko, L.: A semiparametric Bayesian approach for analyzing longitudinal data from multiple related groups. Int. J. Biostat. 11, 273–284 (2015)MathSciNetCrossRefGoogle Scholar
  12. Delattre, M., Lavielle, M., Poursat, M.-A.: A note on BIC in mixed effects models. Electron. J. Stat. 8, 456–475 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Drovandi, C., Pettitt, A.: Likelihood-free bayesian estimation of multivariate quantile distributions. Comput. Stat. Data Anal. 55, 2541–2556 (2011)MathSciNetCrossRefGoogle Scholar
  14. Ghosh, P., Hanson, T.A.: Semiparametric Bayesian approach to multivariate longitudinal data. Aust. N. Z. J. Stat. 52, 275–288 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Geraci, M., Bottai, M.: Quantile regression for longitudinal data using the asymmetric Laplace distribution. Biostatistics 8, 140–154 (2007)CrossRefzbMATHGoogle Scholar
  16. Greven, S., Kneib, T.: On the behaviour of marginal and conditional AIC in linear mixed models. Biometrika 97, 773–789 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Guggisberg, M.: A bayesian approach to multiple-output quantile regression. In: Technical Report (2016)Google Scholar
  18. Hallin, M., Paindaveine, D., Siman, M.: Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth. Ann. Stat. 38, 635–669 (2010)CrossRefzbMATHGoogle Scholar
  19. Heagerty, P., Pepe, M.: 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 (1999)CrossRefzbMATHGoogle Scholar
  20. Jung, S.H.: Quasi-likelihood for median regression models. J. Am. Stat. Assoc. 91, 251–257 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Jang, W., Wang, H.: A semiparametric Bayesian approach for joint-quantile regression with clustered data. Comput. Stat. Data Anal. 84, 99–115 (2015)MathSciNetCrossRefGoogle Scholar
  22. Koenker, R., Bassett, G.: Regression quantiles. Econometrica 46, 33–50 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Koenker, R., Machado, J.: Goodness of fit and related inference processes for quantile regression. J. Am. Stat. Assoc. 94, 1296–1310 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Koenker, R.: Quantile regression for longitudinal data. J. Multivar. Anal. 91, 74–89 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Kozumi, H., Kobayashi, G.: Gibbs sampling methods for Bayesian quantile regression. J. Stat. Comput. Simul. 81, 1565–1578 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Meng, L., van Dyk, D.: Fast EMtype implementations for mixed effects models. J. R. Stat. Soc. Ser. B. 60, 559–578 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Reich, B., Fuentes, M., Dunson, D.: Bayesian spatial quantile regression. J. Am. Stat. Assoc. 106, 6–20 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Sithole, J.S., Jones, P.W.: Bivariate longitudinal model for detecting prescribing change in two drugs simultaneously with correlated errors. J. Appl. Stat. 34, 339–352 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Sy, J.P., Taylor, J., Cumberland, W.G.: A stochastic model for the analysis of bivariate longitudinal AIDS data. Biometrics 53, 542–555 (1997)CrossRefzbMATHGoogle Scholar
  30. Tershakovec, A.: One-year follow-up of nutrition education for hypercholesterolemic children. Am. J. Public Health 88, 258–261 (1998)CrossRefGoogle Scholar
  31. Thiebaut, R., Jacqmin-Gadda, H., Chene, G., Leport, C., Commenges, D.: Bivariate linear mixed models using SAS PROC MIXED. Comput. Methods Progr. Biomed. 69, 249–256 (2002)CrossRefGoogle Scholar
  32. Waldmann, E., Kneib, T.: Bayesian bivariate quantile regression. Stat. Model. 15, 326–344 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Yu, K., Moyeed, R.: Bayesian quantile regression. Stat. Probab. Lett. 54, 437–447 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Yu, K., Zhang, J.: A three-parameter asymmetric Laplace distribution and its extension. Commun. Stat. Theory Methods. 34, 1867–1879 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Yuan, Y., Yin, G.: Bayesian quantile regression for longitudinal studies with nonignorable missing data. Biometrics 66, 105–114 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Hemant Kulkarni
    • 1
  • Jayabrata Biswas
    • 2
  • Kiranmoy Das
    • 2
  1. 1.Human Genetics UnitIndian Statistical InstituteKolkataIndia
  2. 2.Interdisciplinary Statistical Research UnitIndian Statistical InstituteKolkataIndia

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