Variable Selection for Fixed and Random Effects in Multilevel Functional Mixed Effects Models

We develop a new method for simultaneously selecting fixed and random effects in a multilevel functional regression model. The proposed method is motivated by accelerometer-derived physical activity data from the 2011 to 2012 cohort of the National Health and Nutrition Examination Survey (NHANES), with the aim of identifying age and race-specific heterogeneity in covariate effects on the diurnal pattern of physical activity across the lifespan. Existing methods for variable selection in function-on-scalar regression have primarily been designed for fixed effect selection and for single-level functional data. In high-dimensional multilevel functional regression, the presence of cluster-specific heterogeneity in covariate effects could be detected through sparsity in fixed and random effects, and for this purpose, we propose a multilevel functional mixed effects selection (MuFuMES) method. The fixed and random functional effects are modelled using splines, with spike-and-slab group lasso (SSGL) priors on the unknown parameters of interest, and a computationally efficient MAP estimation approach is employed for mixed effect selection through an Expectation Conditional Maximization (ECM) algorithm. Numerical analysis using simulation study illustrates the satisfactory selection accuracy of the variable selection method in having a negligible false-positive and false-negative rate. The proposed method is applied to the NHANES 2011-12 accelerometer data, where it effectively identifies age and race-specific heterogeneity in covariate effects on the diurnal pattern of physical activity, recovering biologically meaningful insights.