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Biostatistics Advance Access published online on April 24, 2006
Biostatistics, doi:10.1093/biostatistics/kxj039
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© The Author 2006. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org
Received November 11, 2005
Revised April 18, 2006
Accepted April 21, 2006

Article

Quantile regression for longitudinal data using the asymmetric Laplace distribution

Marco Geraci 1 * and Matteo Bottai 1

1 Department of Biostatistics and Epidemiology, University of South Carolina, 800 Sumter Street, Columbia, SC 29208, USA

* To whom correspondence should be addressed.
Marco Geraci, E-mail: geraci{at}gwm.sc.edu


  Abstract

In longitudinal studies, measurements of the same individuals are taken repeatedly through time. Often, the primary goal is to characterize the change in response over time and the factors that influence change. Factors can affect not only the location, but more generally the shape of the distribution of the response over time. To make inference about the shape of a population distribution, the widely-popular mixed-effects regression, for example, would be inadequate, if the distribution is not approximately Gaussian. We propose a novel linear model for quantile regression that includes random effects in order to account for the dependence between serial observations on the same subject. The notion of quantile regression is synonymous with robust analysis of the conditional distribution of the response variable. We present a likelihood-based approach to the estimation of the regression quantiles that uses the asymmetric Laplace density. In a simulation study, the proposed method had an advantage in terms of mean squared error of the quantile regression estimator, when compared with the approach that considers penalized fixed effects. Following our strategy, a nearly optimal degree of shrinkage of the individual effects is automatically selected by the data and their likelihood. Also, our model appears to be a robust alternative to the mean regression with random effects when the location parameter of the conditional distribution of the response is of interest. We apply our model to a real data set which consists of self-reported amount of labor pain measurements taken on women repeatedly over time, whose distribution is characterized by skewness, and the significance of the parameters is evaluated by using the likelihood ratio statistic.

Keywords: Asymmetric Laplace distribution; Clinical trials; Monte Carlo Markov Chain; Quantile regression; Random effects.
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