Fast penalized generalized estimating equations for large longitudinal functional datasets

Longitudinal binary or count functional data are common in neuroscience, but are often too large to analyze with existing functional regression methods. We propose one-step penalized generalized estimating equations that support generalized functional outcomes (e.g., count, binary, proportion, continuous-valued) and is fast even when datasets have a large number of clusters and large cluster sizes. The method applies to functional and scalar covariates and the one-step estimation framework enables efficient smoothing parameter selection and joint confidence interval construction. Importantly, this semi-parametric approach yields coefficient confidence intervals that are provably valid asymptotically even under working correlation misspecification. By developing a general theory for adaptive one-step M-estimation, we prove that the coefficient estimates are asymptotically normal and as efficient as the fully-iterated estimator; we verify these theoretical properties in simulations. We illustrate the benefits of our approach for analyzing large-scale neural recordings by applying it to a recent calcium imaging dataset published in Nature. We show that our method reveals important timing effects obscured in non-functional analyses. In doing so, we also demonstrate scaling to common neuroscience dataset sizes: the one-step estimator fits to a dataset with 150,000 (binary) functional outcomes, each observed at 120 functional domain points, in only $\sim 6.5$ minutes on a laptop without parallelization. We release our methods in the $\tt {R}$ package $\tt {fastFGEE}$, which supports a wide range of link functions and working covariances.