A note on modeling sparse exponential-family functional response curves

Abstract

Non-Gaussian functional data are considered and modeling through functional principal components analysis (FPCA) is discussed. The direct extension of popular FPCA techniques to the generalized case incorrectly uses a marginal mean estimate for a model that has an inherently conditional interpretation, and thus leads to biased estimates of population and subject-level effects. The methods proposed address this shortcoming by using either a two-stage or joint estimation strategy. The performance of all methods is compared numerically in simulations. An application to ambulatory heart rate monitoring is used to further illustrate the distinctions between approaches.

Keywords: Binomial Data; Functional Principal Components; Longitudinal Data; Mixed Models; Smoothing; Sparse Sampling Design.

Figure 1

Left panel shows two samples of latent curves on the logit scale with more and less variance, in red and blue respectively, around a common latent mean shown in black. The right panel shows these curves transformed to the probability scale; marginal means are shown in red and blue, while the transformation of the true mean is shown in black.

Figure 2

Simulation results for Setting 1 (top row) and Setting 2 (bottom row). In each row, the left panel shows the average estimated mean function across all simulated datasets, the middle panel shows the IMSE for the mean function, and the right panel shows the average IMSE of latent processes across subjects. In case of Setting 1 for Gamm.FPCA and Gamm.HMY one extreme outlier is not shown.

Figure 3

Results of the analysis of the ambulatory blood pressure dataset. The left panel shows estimated mean functions from the marginal approach, the two-step GAMM-based approach, and the Bayesian approach. The middle and right panels show observed data and estimated latent means for two subjects.