Statistical Inference in a Growth Curve Quantile Regression Model for Longitudinal Data

This article describes a polynomial growth curve quantile regression model that provides a comprehensive assessment about the treatment effects on the changes of the distribution of outcomes over time. The proposed model has the flexibility, as it allows the degree of a polynomial to vary across quantiles. A high degree polynomial model fits the data adequately, yet it is not desirable due to the complexity of the model. We propose the model selection criterion based on an empirical loglikelihood that consistently identifies the optimal degree of a polynomial at each quantile. After the parsimonious model is fitted to the data, the hypothesis test is further developed to evaluate the treatment effects by comparing the growth curves. It is shown that the proposed empirical loglikelihood ratio test statistic follows a chi-square distribution asymptotically under the null hypothesis. Various simulation studies confirm that the proposed test successfully detects the difference between the curves across quantiles. When the empirical loglikelihood is employed, we incorporate the within-subject correlation commonly existing in longitudinal data and gain estimation efficiency of the quantile regression parameters in the growth curve model. The proposed process is illustrated through the analysis of randomized controlled longitudinal depression data.