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Linear quantile mixed models

Abstract

Dependent data arise in many studies. Frequently adopted sampling designs, such as cluster, multilevel, spatial, and repeated measures, may induce this dependence, which the analysis of the data needs to take into due account. In a previous publication (Geraci and Bottai in Biostatistics 8:140–154, 2007), we proposed a conditional quantile regression model for continuous responses where subject-specific random intercepts were included to account for within-subject dependence in the context of longitudinal data analysis. The approach hinged upon the link existing between the minimization of weighted absolute deviations, typically used in quantile regression, and the maximization of a Laplace likelihood. Here, we consider an extension of those models to more complex dependence structures in the data, which are modeled by including multiple random effects in the linear conditional quantile functions. We also discuss estimation strategies to reduce the computational burden and inefficiency associated with the Monte Carlo EM algorithm we have proposed previously. In particular, the estimation of the fixed regression coefficients and of the random effects’ covariance matrix is based on a combination of Gaussian quadrature approximations and non-smooth optimization algorithms. Finally, a simulation study and a number of applications of our models are presented.

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Notes

  1. For this scenario, we increased the number of quadrature nodes K from 11 to 17. The relative bias decreased from 0.135 to 0.051 (τ=0.75) and from 0.459 to 0.075 (τ=0.9).

  2. All computations were performed on a 64-bit operating system machine with 16 Gb of RAM and quad-core processor at 2.93 GHz.

References

  • Alhamzawi, R., Yu, K., Pan, J.: Prior elicitation in Bayesian quantile regression for longitudinal data. J. Biometr. Biostat. 2, 1–7 (2011)

    Article  Google Scholar 

  • Azzalini, A., Capitanio, A.: Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J. R. Stat. Soc., Ser. B, Stat. Methodol. 65, 367–389 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  • Barrodale, I., Roberts, F.D.K.: An efficient algorithm for discrete l 1 linear approximation with linear constraints. SIAM J. Numer. Anal. 15, 603–611 (1978)

    MATH  Article  MathSciNet  Google Scholar 

  • Bassett, G., Koenker, R.: Asymptotic theory of least absolute error regression. J. Am. Stat. Assoc. 73, 618–622 (1978)

    MATH  Article  MathSciNet  Google Scholar 

  • Boscovich, R.J.: De Litteraria Expeditione per Pontificiam Ditionem, et Synopsis Amplioris Operis, Ac Habentur Plura Ejus Ex Exemplaria Etiam Sensorum Impressa. Bononiesi Scientiarum et Artum Instituto Atque Academia Commentarii, vol. IV (1757)

    Google Scholar 

  • Bose, A., Chatterjee, S.: Generalized bootstrap for estimators of minimizers of convex functions. J. Stat. Plan. Inference 117, 225–239 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  • Bottai, M., Orsini, N.: A command for Laplace regression. Stata J. (2012, in press)

  • Bottai, M., Zhang, J.: Laplace regression with censored data. Biom. J. 52, 487–503 (2010)

    MATH  Article  MathSciNet  Google Scholar 

  • Buchinsky, M.: Estimating the asymptotic covariance matrix for quantile regression models. A Monte Carlo study. J. Econom. 68, 303–338 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  • Canay, I.A.: A simple approach to quantile regression for panel data. Econom. J. 14, 368–386 (2011)

    MATH  Article  MathSciNet  Google Scholar 

  • Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)

    MATH  Book  Google Scholar 

  • Demidenko, E.: Mixed Models. Theory and Applications. Wiley, Hoboken (2004)

    MATH  Book  Google Scholar 

  • DerSimonian, R., Laird, N.: Meta-analysis in clinical trials. Control. Clin. Trials 7, 177–188 (1986)

    Article  Google Scholar 

  • Doksum, K.: Empirical probability plots and statistical inference for nonlinear models in the two-sample case. Ann. Stat. 2, 267–277 (1974)

    MATH  Article  MathSciNet  Google Scholar 

  • Eltoft, T., Kim, T., Lee, T.-W.: On the multivariate Laplace distribution. IEEE Signal Process. Lett. 13, 300–303 (2006)

    Article  Google Scholar 

  • Farcomeni, A.: Quantile regression for longitudinal data based on latent Markov subject-specific parameters. Stat. Comput. 22, 141–152 (2012)

    Article  MathSciNet  Google Scholar 

  • Feng, X., He, X., Hu, J.: Wild bootstrap for quantile regression. Biometrika 98, 995–999 (2011)

    MATH  Article  MathSciNet  Google Scholar 

  • Fielding, A., Yang, M., Goldstein, H.: Multilevel ordinal models for examination grades. Stat. Model. 3, 127–153 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  • Fu, L., Wang, Y.-G.: Quantile regression for longitudinal data with a working correlation model. Comput. Stat. Data Anal. 56, 2526–2538 (2012)

    MATH  Article  MathSciNet  Google Scholar 

  • Galvao, A.F.: Quantile regression for dynamic panel data with fixed effects. J. Econom. 164, 142–157 (2011)

    Article  MathSciNet  Google Scholar 

  • Galvao, A.F., Montes-Rojas, G.V.: Penalized quantile regression for dynamic panel data. J. Stat. Plan. Inference 140, 3476–3497 (2010)

    MATH  Article  MathSciNet  Google Scholar 

  • Genz, A., Keister, B.D.: Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. J. Comput. Appl. Math. 71, 299–309 (1996)

    MATH  Article  MathSciNet  Google Scholar 

  • Geraci, M.: lqmm: Linear quantile mixed models. R package version 1.02 (2012)

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

    MATH  Article  Google Scholar 

  • Geraci, M., Salvati, N.: The geographical distribution of the consumption expenditure in Ecuador: estimation and mapping of the regression quantiles. Stat. Appl. 19, 167–183 (2007)

    Google Scholar 

  • He, X.: Quantile curves without crossing. Am. Stat. 51, 186–192 (1997)

    Google Scholar 

  • He, X., Hu, F.: Markov chain marginal bootstrap. J. Am. Stat. Assoc. 97, 783–795 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  • He, X.M., Ng, P., Portnoy, S.: Bivariate quantile smoothing splines. J. R. Stat. Soc., Ser. B, Stat. Methodol. 60, 537–550 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  • He, X.M., Portnoy, S.: Some asymptotic results on bivariate quantile splines. J. Stat. Plan. Inference 91, 341–349 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  • Heiss, F., Winschel, V.: Likelihood approximation by numerical integration on sparse grids. J. Econom. 144, 62–80 (2008)

    Article  MathSciNet  Google Scholar 

  • Higham, N.: Computing the nearest correlation matrix—a problem from finance. IMA J. Numer. Anal. 22, 329–343 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  • Hinkley, D.V., Revankar, N.S.: Estimation of the Pareto law from underreported data: a further analysis. J. Econom. 5, 1–11 (1977)

    MATH  Article  MathSciNet  Google Scholar 

  • Karlsson, A.: Nonlinear quantile regression estimation of longitudinal data. Commun. Stat., Simul. Comput. 37, 114–131 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  • Kim, M.-O., Yang, Y.: Semiparametric approach to a random effects quantile regression model. J. Am. Stat. Assoc. 106, 1405–1417 (2011)

    MATH  Article  MathSciNet  Google Scholar 

  • Kocherginsky, M., He, X., Mu, Y.: Practical confidence intervals for regression quantiles. J. Comput. Graph. Stat. 14, 41–55 (2005)

    Article  MathSciNet  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  • Koenker, R.: Quantile Regression. Cambridge University Press, New York (2005)

    MATH  Book  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  • Koenker, R., Machado, J.A.F.: Goodness of fit and related inference processes for quantile regression. J. Am. Stat. Assoc. 94, 1296–1310 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  • Koenker, R., Mizera, I.: Penalized triograms: total variation regularization for bivariate smoothing. J. R. Stat. Soc., Ser. B, Stat. Methodol. 66, 145–163 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  • Koenker, R., Ng, P., Portnoy, S.: Quantile smoothing splines. Biometrika 81, 673–680 (1994)

    MATH  Article  MathSciNet  Google Scholar 

  • Koenker, R., Xiao, Z.J.: Inference on the quantile regression process. Econometrica 70, 1583–1612 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  • Kotz, S., Kozubowski, T.J., Podgórski, K.: An asymmetric multivariate Laplace distribution. Tech. Rep. 367, Department of Statistics and Applied Probability, University of California at Santa Barbara (2000)

  • Kozubowski, T.J., Nadarajah, S.: Multitude of Laplace distributions. Stat. Pap. 51, 127–148 (2010)

    MATH  Article  MathSciNet  Google Scholar 

  • Lamarche, C.: Robust penalized quantile regression estimation for panel data. J. Econom. 157, 396–498 (2010)

    Article  MathSciNet  Google Scholar 

  • Lee, D., Neocleous, T.: Bayesian quantile regression for count data with application to environmental epidemiology. J. R. Stat. Soc., Ser. C, Appl. Stat. 59, 905–920 (2010)

    Article  MathSciNet  Google Scholar 

  • Lee, Y., Nelder, J.A.: Conditional and marginal models: another view. Stat. Sci. 19, 219–228 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  • Lehmann, E.L.: Nonparametrics: Statistical Methods Based on Ranks. Holden-Day, San Francisco (1975)

    MATH  Google Scholar 

  • Li, Q., Xi, R., Lin, N.: Bayesian regularized quantile regression. Bayesian Anal. 5, 533–556 (2010)

    Article  MathSciNet  Google Scholar 

  • 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. J. R. Stat. Soc., Ser. C, Appl. Stat. 46, 463–476 (1997)

    MATH  Article  Google Scholar 

  • Liu, Y., Bottai, M.: Mixed-effects models for conditional quantiles with longitudinal data. Int. J. Biostat. 5, 1–22 (2009)

    MATH  MathSciNet  Google Scholar 

  • Lum, K., Gelfand, A.: Spatial quantile multiple regression using the asymmetric Laplace process. Bayesian Anal. 7, 235–258 (2012)

    Article  MathSciNet  Google Scholar 

  • Machado, J.A.F., Santos Silva, J.M.C.: Quantiles for counts. J. Am. Stat. Assoc. 100, 1226–1237 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  • Oberhofer, W., Haupt, H.: The asymptotic distribution of the unconditional quantile estimator under dependence. Stat. Probab. Lett. 73, 243–250 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  • Parzen, M., Wei, L., Ying, Z.: A resampling method based on pivotal estimating functions. Biometrika 81, 341–350 (1994)

    MATH  Article  MathSciNet  Google Scholar 

  • Pinheiro, J., Bates, D.: Approximations to the log-likelihood function in the nonlinear mixed-effects model. J. Comput. Graph. Stat. 4, 12–35 (1995)

    Google Scholar 

  • Pinheiro, J.C., Bates, D.M.: Unconstrained parametrizations for variance-covariance matrices. Stat. Comput. 6, 289–296 (1996)

    Article  Google Scholar 

  • Pinheiro, J.C., Chao, E.C.: Efficient Laplacian and adaptive Gaussian quadrature algorithms for multilevel generalized linear mixed models. J. Comput. Graph. Stat. 15, 58–81 (2006)

    Article  MathSciNet  Google Scholar 

  • Pourahmadi, M.: Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation. Biometrika 86, 677–690 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  • Prékopa, A.: Logarithmic concave measures and functions. Acta Sci. Math. 34, 334–343 (1973)

    Google Scholar 

  • R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2012). ISBN 3-900051-07-0

    Google Scholar 

  • Reed, W.: The normal-Laplace distribution and its relatives. In: Balakrishnan, N., Castillo, E., Sarabia Alegria, J.-M. (eds.) Advances in Distribution Theory, Order Statistics, and Inference, pp. 61–74. Birkhäuser Boston, New York (2006)

    Chapter  Google Scholar 

  • Reich, B.J., Bondell, H.D., Wang, H.J.: Flexible Bayesian quantile regression for independent and clustered data. Biostatistics 11, 337–352 (2010a)

    Article  Google Scholar 

  • Reich, B.J., Fuentes, M., Dunson, D.B.: Bayesian spatial quantile regression. J. Am. Stat. Assoc. (2010b)

  • Rigby, R., Stasinopoulos, D.: Generalized additive models for location, scale and shape. J. R. Stat. Soc., Ser. C, Appl. Stat. 54, 507–554 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  • Robinson, G.: That BLUP is a good thing: the estimation of random effects. Stat. Sci. 6, 15–32 (1991)

    MATH  Article  Google Scholar 

  • Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  • Rogan, W.J., Dietrich, K.N., Ware, J.H., Dockery, D.W., Salganik, M., Radcliffe, J., Jones, R.L., Ragan, N.B., Chisolm, J.J., Rhoads, G.G.: The effect of chelation therapy with succimer on neuropsychological development in children exposed to lead. N. Engl. J. Med. 344, 1421–1426 (2001)

    Article  Google Scholar 

  • Ruppert, D., Wand, M., Carroll, R.: Semiparametric Regression. Cambridge University Press, New York (2003)

    MATH  Book  Google Scholar 

  • Sarkar, D.: Lattice: Multivariate Data Visualization with R. Springer, New York (2008)

    Book  Google Scholar 

  • Treatment of Lead-Exposed Children (TLC) Trial Group: Safety and efficacy of succimer in toddlers with blood lead levels of 20–44 μg/dL. Pediatr. Res. 48, 593–599 (2000)

    Article  Google Scholar 

  • Wagner, H.M.: Linear programming techniques for regression analysis. J. Am. Stat. Assoc. 54, 206–212 (1959)

    MATH  Article  Google Scholar 

  • Wang, J.: Bayesian quantile regression for parametric nonlinear mixed effects models. Stat. Methods Appl. (2012)

  • Yu, K., Lu, Z., Stander, J.: Quantile regression: applications and current research areas. Statistician 52, 331–350 (2003)

    MathSciNet  Google Scholar 

  • Yu, K., Zhang, J.: A three-parameter asymmetric Laplace distribution and its extension. Commun. Stat., Theory Methods 34, 1867–1879 (2005)

    MATH  Article  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  • Yuan, Y., Yin, G.: Bayesian quantile regression for longitudinal studies with nonignorable missing data. Biometrics 66, 105–114 (2010)

    MATH  Article  MathSciNet  Google Scholar 

  • Zhao, Q.S.: Restricted regression quantiles. J. Multivar. Anal. 72, 78–99 (2000)

    MATH  Article  Google Scholar 

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Acknowledgements

The Centre for Paediatric Epidemiology and Biostatistics benefits from funding support from the Medical Research Council in its capacity as the MRC Centre of Epidemiology for Child Health (G0400546). The UCL Institute of Child Health receives a proportion of funding from the Department of Health’s NIHR Biomedical Research Centres funding scheme.

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Correspondence to Marco Geraci.

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Geraci, M., Bottai, M. Linear quantile mixed models. Stat Comput 24, 461–479 (2014). https://doi.org/10.1007/s11222-013-9381-9

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Keywords

  • Best linear predictor
  • Clarke’s derivative
  • Hierarchical models
  • Gaussian quadrature