M‐quantile regression for multivariate longitudinal data with an application to the Millennium Cohort Study

Motivated by the analysis of data from the UK Millennium Cohort Study on emotional and behavioural disorders, we develop an M-quantile regression model for multi- variate longitudinal responses. M-quantile regression is an appealing alternative to standard regression models; it combines features of quantile and expectile regression and it may produce a detailed picture of the conditional response variable distribution, while ensuring robustness to outlying data. As we deal with multivariate data, we need to specify what it is meant by M-quantile in this context, and how the structure of dependence between uni- variate profiles may be accounted for. Here, we consider univariate (conditional) M-quantile regression models with outcome-specific random effects for each outcome. Dependence between outcomes is introduced by assuming that the random effects in the univariate models are dependent. The multivariate distribution of the random effects is left unspecified and estimated from the observed data. Adopting this approach, we are able to model dependence both within and between outcomes. We further discuss a suitable model parameterisation to account for potential endogeneity of the observed covariates. An ex- tended EM algorithm is defined to derive estimates under a maximum likelihood approach.