M-Quantile and expectile random effects regression for multilevel data

The analysis of hierarchically structured data is usually carried out by using random effects models. Theprimary goal of random effects regression is to model the expected value of the conditional distributionof an outcome variable given a set of explanatory variables while accounting for the dependence structureof hierarchical data. The expected value, however, may not offer a complete picture of this conditionaldistribution. In this paper we propose using linear M-quantile regression, to model other parts of theconditional distribution of the outcome variable given the covariates. The proposed random effectsregression model extends M-quantile regression and can be viewed as an alternative to the quantilerandom effects model. Inference for estimators of the fixed and random effects parameters is discussed.The performance of the proposed methods is evaluated in a series of simulation studies. Finally, wepresent a case study where M-quantile and expectile random effects regression is employed for analyzingrepeated measures data collected from a rotary pursuit tracking experiment