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.
KeywordsBest linear predictor Clarke’s derivative Hierarchical models Gaussian quadrature
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.
- 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
- Bottai, M., Orsini, N.: A command for Laplace regression. Stata J. (2012, in press) Google Scholar
- Geraci, M.: lqmm: Linear quantile mixed models. R package version 1.02 (2012) 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
- 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) 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) zbMATHCrossRefGoogle 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
- 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
- Reich, B.J., Fuentes, M., Dunson, D.B.: Bayesian spatial quantile regression. J. Am. Stat. Assoc. (2010b) 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) CrossRefGoogle Scholar
- Wang, J.: Bayesian quantile regression for parametric nonlinear mixed effects models. Stat. Methods Appl. (2012) Google Scholar