Statistics and Computing

, Volume 24, Issue 3, pp 461-479

First online:

Linear quantile mixed models

  • Marco GeraciAffiliated withCentre for Paediatric Epidemiology and Biostatistics, MRC Centre of Epidemiology for Child Health, Institute of Child Health, University College London Email author 
  • , Matteo BottaiAffiliated withUnit of Biostatistics, Institute of Environmental Medicine, Karolinska Institutet

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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.


Best linear predictor Clarke’s derivative Hierarchical models Gaussian quadrature