  Linear quantile regression models aim at providing a detailed and robust
picture of the (conditional) response distribution as function of a set of
observed covariates. Longitudinal data represent an interesting field of
application of such models; due to their peculiar features, they represent a
substantial challenge, in that the standard, cross-sectional, model
representation needs to be extended for dealing with such kind of data. In
fact, repeated observations from the same statistical unit poses a problem of
dependence; in a conditional perspective, this dependence could be ascribed to
sources of unobserved, individual-specific, heterogeneity. Along these lines,
quantile regression models have recently been extended to the analysis of
longitudinal, continuous, responses, by modelling dependence via time-constant
or time-varying random effects. In this manuscript, we introduce a general
quantile regression model for longitudinal, continuous, responses where
time-varying and time-constant random parameters are jointly taken into
account. A further feature of longitudinal designs is the presence of partially
incomplete sequences, due to some individuals leaving the study before its
designed end. The missing data process may produce a selection of units which
can be informative with respect to the parameters of the longitudinal data
model. To deal with the case of irretrievable drop-out, we introduce a pattern
mixture version of the linear quantile hidden Markov model, where we account
for time-varying heterogeneity and for changes in the fixed effect vector due
to differential propensities to stay in the study. The proposed models are
illustrated using a well known benchmark dataset on longitudinal dynamics of
CD4 cells and by means of a large scale simulation study, entailing different
quantiles and both complete and partially complete (ie subject to drop-out)
individual sequences.
