SummaryIn clinical studies, longitudinal biomarkers are often used to monitor disease progression and failure time. Joint modeling of longitudinal and survival data has certain advantages and has emerged as an effective way to mutually enhance information. Typically, a parametric longitudinal model is assumed to facilitate the likelihood approach. However, the choice of a proper parametric model turns out to be more elusive than models for standard longitudinal studies in which no survival endpoint occurs. In this article, we propose a nonparametric multiplicative random effects model for the longitudinal process, which has many applications and leads to a flexible yet parsimonious nonparametric random effects model. A proportional hazards model is then used to link the biomarkers and event time. We use B-splines to represent the nonparametric longitudinal process, and select the number of knots and degrees based on a version of the Akaike information criterion (AIC). Unknown model parameters are estimated through maximizing the observed joint likelihood, which is iteratively maximized by the Monte Carlo Expectation Maximization (MCEM) algorithm. Due to the simplicity of the model structure, the proposed approach has good numerical stability and compares well with the competing parametric longitudinal approaches. The new approach is illustrated with primary biliary cirrhosis (PBC) data, aiming to capture nonlinear patterns of serum bilirubin time courses and their relationship with survival time of PBC patients.

BackgroundActigraphy provides a way to objectively measure activity in human subjects. This paper describes a novel family of statistical methods that can be used to analyze this data in a more comprehensive way.MethodsA statistical method for testing differences in activity patterns measured by actigraphy across subgroups using functional data analysis is described. For illustration this method is used to statistically assess the impact of apnea-hypopnea index (apnea) and body mass index (BMI) on circadian activity patterns measured using actigraphy in 395 participants from 18 to 80 years old, referred to the Washington University Sleep Medicine Center for general sleep medicine care. Mathematical descriptions of the methods and results from their application to real data are presented.ResultsActivity patterns were recorded by an Actical device (Philips Respironics Inc.) every minute for at least seven days. Functional linear modeling was used to detect the association between circadian activity patterns and apnea and BMI. Results indicate that participants in high apnea group have statistically lower activity during the day, and that BMI in our study population does not significantly impact circadian patterns.ConclusionsCompared with analysis using summary measures (e.g., average activity over 24 hours, total sleep time), Functional Data Analysis (FDA) is a novel statistical framework that more efficiently analyzes information from actigraphy data. FDA has the potential to reposition the focus of actigraphy data from general sleep assessment to rigorous analyses of circadian activity rhythms.

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