Marginally Interpretable Models and Multilevel Models For Quantile Regression With Random-Effects

The quantile regression model is an active area of statistical research that has received a lot of attention. This complements the most widely used statistical tool, that is, mean regression analysis. Quantile regression analysis It has become more flexible because of its properties that include no assumption on the distribution of the response variable, equivalent to monotone transformations, and robustness to outliers. However, regression analysis offers methodological challenges if the observations are not independent. Cluster, multilevel, and repeated measures (longitudinal data) designs introduce such dependence. The correlation between observations on the same units or clusters should be accounted for to obtain correct inferences. The mixed-effects model has been used to analyze such complex sampling designs. As an increased research interest in quantile regression, in the last few years, several approaches to quantile regression for dependent data have been proposed. These approaches consist of distribution-free and likelihood-based methods. The mixed-effects model for quantile regression in existing literature can be applied to 2-levels. However, in some situations, researchers may need to apply the quantile regression for the mixed-effects model where more than 2-levels are of interest (frequentist approach). Conditional modeling of quantiles may not be ideal when the focus is on the marginal quantile effects. A way to make inferences on marginal effects is to adjust estimates from a conditional model when the latter does not naturally lead to marginal interpretations. This is the case in quantile regression with random effects and interest often lies in marginal or population-averaged effects rather than conditional effects. Given the limitations in the existing body of the multilevel linear mixed-effects model, this research work aims to present a mixed model for quantile regression with the potentials of filling the noted literature gaps. In Chapter 2 of this dissertation, we presented a marginally interpretable model for quantile regression with random effects. We discussed the derivative free numerical integral algorithm with Gauss-Hermite quadrature to the proposed model parameter estimations. The performance of the parameter estimation methods is studied through statistical simulations for the proposed model. Given the limitation in literature, we proposed a 3-level mixed-effects model for quantile regression in Chapter 3. The proposed model is an extension of 2-level mixed effects model available in literature. Simulation studies were performed to evaluate the proposed model. In addition to the extensive simulation studies, we have demonstrated applications of all proposed models using Millennium Cohort Study (MCS) data