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Linear quantile mixed models

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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.

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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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Authors and Affiliations

  1. Centre for Paediatric Epidemiology and Biostatistics, MRC Centre of Epidemiology for Child Health, Institute of Child Health, University College London, 30 Guilford Street, London, WC1N 1EH, UK

    Marco Geraci

  2. Unit of Biostatistics, Institute of Environmental Medicine, Karolinska Institutet, Nobels väg 13, 17177, Stockholm, Sweden

    Matteo Bottai

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  1. 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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