  Nonparametric varying coefficient (NVC) models are useful for modeling
time-varying effects on responses that are measured repeatedly for the same
subjects. When the number of covariates is moderate or large, it is desirable
to perform variable selection from the varying coefficient functions. However,
existing methods for variable selection in NVC models either fail to account
for within-subject correlations or require the practitioner to specify a
parametric form for the correlation structure. In this paper, we introduce the
nonparametric varying coefficient spike-and-slab lasso (NVC-SSL) for Bayesian
high-dimensional NVC models. Through the introduction of functional random
effects, our method allows for flexible modeling of within-subject correlations
without needing to specify a parametric covariance function. We further propose
several scalable optimization and Markov chain Monte Carlo (MCMC) algorithms.
For variable selection, we propose an Expectation Conditional Maximization
(ECM) algorithm to rapidly obtain maximum a posteriori (MAP) estimates. Our ECM
algorithm scales linearly in the total number of observations $N$ and the
number of covariates $p$. For uncertainty quantification, we introduce an
approximate MCMC algorithm that also scales linearly in both $N$ and $p$. We
demonstrate the scalability, variable selection performance, and inferential
capabilities of our method through simulations and a real data application.
These algorithms are implemented in the publicly available R package NVCSSL on
the Comprehensive R Archive Network.
