Semiparametric mixture regression for asynchronous longitudinal data using multivariate functional principal component analysis

The transitional phase of menopause induces significant hormonal fluctuations, exerting a profound influence on the long-term well-being of women. In an extensive longitudinal investigation of women's health during mid-life and beyond, known as the Study of Women's Health Across the Nation (SWAN), hormonal biomarkers are repeatedly assessed, following an asynchronous schedule compared to other error-prone covariates, such as physical and cardiovascular measurements. We conduct a subgroup analysis of the SWAN data employing a semiparametric mixture regression model, which allows us to explore how the relationship between hormonal responses and other time-varying or time-invariant covariates varies across subgroups. To address the challenges posed by asynchronous scheduling and measurement errors, we model the time-varying covariate trajectories as functional data with reduced-rank Karhunen-Lo & eacute;ve expansions, where splines are employed to capture the mean and eigenfunctions. Treating the latent subgroup membership and the functional principal component (FPC) scores as missing data, we propose an Expectation-Maximization algorithm to effectively fit the joint model, combining the mixture regression for the hormonal response and the FPC model for the asynchronous, time-varying covariates. In addition, we explore data-driven methods to determine the optimal number of subgroups within the population. Through our comprehensive analysis of the SWAN data, we unveil a crucial subgroup structure within the aging female population, shedding light on important distinctions and patterns among women undergoing menopause.