Efficient quantile marginal regression for longitudinal data with dropouts

Robust Bayesian growth curve modelling using conditional medians

Growth curve models have been widely used to analyse longitudinal data in social and behavioural sciences. Although growth curve models with normality assumptions are relatively easy to estimate, practical data are rarely normal. Failing to account for non-normal data may lead to unreliable model estimation and misleading statistical inference. In this work, we propose a robust approach for growth curve modelling using conditional medians that are less sensitive to outlying observations. Bayesian methods are applied for model estimation and inference. Based on the existing work on Bayesian quantile regression using asymmetric Laplace distributions, we use asymmetric Laplace distributions to convert the problem of estimating a median growth curve model into a problem of obtaining the maximum likelihood estimator for a transformed model. Monte Carlo simulation studies have been conducted to evaluate the numerical performance of the proposed approach with data containing outliers or leverage observations. The results show that the proposed approach yields more accurate and efficient parameter estimates than traditional growth curve modelling. We illustrate the application of our robust approach using conditional medians based on a real data set from the Virginia Cognitive Aging Project.

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.

A Randomized Comparison of Medication and Cognitive Behavioral Therapy for Treating Depression in Low-Income Young Minority Women

BACKGROUND Longitudinal data arise frequently in biomedical science and health studies where each subject is repeatedly measured over time. We compared the effectiveness of medication and cognitive behavioral therapy on depression in predominantly low-income young minority women. MATERIAL AND METHODS The treatment effects on patients with low-level depression may differ from the treatment effects on patients with high-level depression. We used a quantile regression model for longitudinal data analysis to determine which treatment is most beneficial for patients at different stress levels over time. RESULTS The results confirm that both treatments are effective in reducing the depression score over time, regardless of the depression level. CONCLUSIONS Compared to cognitive behavioral therapy, treatment with medication more often effective, although the size of the effect differs. Thus, no matter how severe a patient's depression symptoms are, antidepressant medication is effective in decreasing depression symptoms.