Linear quantile regression models for longitudinal experiments: an overview
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Linear quantile regression models for longitudinal experiments: an overview Maria Francesca Marino1
· Alessio Farcomeni2
Received: 23 October 2014 / Accepted: 1 August 2015 © Sapienza Università di Roma 2015
Abstract We provide an overview of linear quantile regression models for continuous responses repeatedly measured over time. We distinguish between marginal approaches, that explicitly model the data association structure, and conditional approaches, that consider individual-specific parameters to describe dependence among data and overdispersion. General estimation schemes are discussed and available software options are listed. We also mention methods to deal with non-ignorable missing values, with spatially dependent observations and nonparametric and semiparametric models. The paper is concluded by an overview of open issues in longitudinal quantile regression. Keywords Quantile regression · Longitudinal data · Marginal models · Conditional models · Random effects · Fixed effects · Generalized estimating equations
1 Introduction Quantile regression has been introduced by Koenker and Bassett [64] as an extension of standard mean regression to model the conditional quantiles of a continuous response and to provide a thorough overview of its distribution. It has become a very popular and consolidated approach in the statistical literature and is nowadays applied in a wide range of fields, including econometrics, finance, biomedicine, ecology. Comprehensive reviews can be found, among others, in [54,63,67,118]. Specific examples are provided by Machado and Mata [88] in econometrics, by Austin and Schull [4] in epidemiology, by Cade et al. [15] in ecology. The literature on quantile regression methods is now extremely vast. In this work we will review a specific area of application; in particular, we will focus on linear quantile regression
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Maria Francesca Marino [email protected]
1
University of Perugia, Perugia, Italy
2
Sapienza, University of Rome, Rome, Italy
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M. F. Marino, A. Farcomeni
models for longitudinal observations. In the last two decades, longitudinal study designs have raised considerable attention. Obtaining additional information from a unit already in the study is cheaper than adding a new one; also, longitudinal studies allow to monitor the evolution of individual trajectories over time. Weiss [113] lists a number of other benefits of collecting longitudinal in place of cross-sectional data. Additional issues have to be faced when dealing with longitudinal studies, though. Observations from the same individual are naturally dependent and this has to be taken into consideration to avoid potential bias in parameter estimates; moreover, individuals may leave the study before the planned end, thus, presenting incomplete data records. In such a context, standard regression models can not be directly used, and need to be extended to avoid misleading inferential conclusions. We will not propose a unifying framework, but rather try to discuss the available options in a
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