Modelling unobserved heterogeneity in quantile regression for longitudinal data

Quantile regression has become a standard tool in the analysis of longitudinal data as it offers a thorough overview on the conditional distribution of a response variable given a set of covariates. When dealing with longitudinal studies, observations coming from the same individual are naturally dependent because of the presence of unobserved sources of heterogeneity. If such a dependence is not properly taken into consideration in the data analysis, misleading inferential conclusions can be easily drawn. A mixed hidden Markov quantile model for continuous longitudinal data is proposed. Time-constant and time-varying random parameters are considered in the model specification to jointly account for time-invariant and dynamic unobserved factors affecting the response variable distribution. The resulting model offers great flexibility being a generalization of the basic linear mixed quantile regression model and the standard hmm for quantiles frequently used in the quantile regression framework for longitudinal data. In order to face the numerical integration problem typically arising in the mixed hmm context, a non parametric maximum likelihood approach is applied. Parameter estimates are obtained via an em algorithm and their computation is greatly simplified by exploiting the the forward and backward variables defined in the so called Baum-Welch algorithm.