Functional Additive Mixed Models

Abstract

We propose an extensive framework for additive regression models for correlated functional responses, allowing for multiple partially nested or crossed functional random effects with flexible correlation structures for, e.g., spatial, temporal, or longitudinal functional data. Additionally, our framework includes linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. It accommodates densely or sparsely observed functional responses and predictors which may be observed with additional error and includes both spline-based and functional principal component-based terms. Estimation and inference in this framework is based on standard additive mixed models, allowing us to take advantage of established methods and robust, flexible algorithms. We provide easy-to-use open source software in the pffr() function for the R-package refund. Simulations show that the proposed method recovers relevant effects reliably, handles small sample sizes well and also scales to larger data sets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach.

Keywords: Functional data analysis; P-splines; Smoothing; Varying coefficient models; functional principal component analysis.

Figure 1

Baseline levels and estimated multiplicative change in rIMSE for the 4 scenarios. The scenarios are depicted with different symbols, and the segments accompanying the symbols correspond to the estimated effect ± 2 standard errors. Effects other than b₀(t) only occur in a subset of scenarios. Horizontal axis on log₂.

Figure 2

Computation times for scenarios 1 to 4 (from left to right). Vertical axis on log₁₀-scale. Horizontal axis for the various combinations of numbers of subjects M and average number of replications per subject n_i. Results for T = 30 in dark grey and in light grey for T = 60. Timings are wall-clock time taken on an 2.2 GHz AMD Opteron 6174.

Figure 3

From left to right: FA profiles along CCA, OPR and CST for MS patients (red) and controls (blue). Solid line: females; dashed: males. FA-OPR and FA-CST are de-trended and smoothed.

Figure 4

Estimated components of model (9) with ±2 pointwise standard errors. Coefficient surfaces are color-coded for sign and approximate pointwise significance (95%): blue if sig. < 0, light blue if < 0, light red if > 0, red if sig. > 0. Left to right: mean FA-CCA for healthy (blue, dotted) versus MS (red, solid) females; mean FA-CCA for female (purple, solid) and male (green, dotted) MS patients; estimated smooth index-varying age effect $\widehat{\nu }(u,t)$.

Figure 5

Left to right: Estimated coefficient surfaces ${\widehat{\beta }}_{1,0}(s,t)$, ${\widehat{\beta }}_{1,1}(s,t)$, ${\widehat{\beta }}_{2,0}(r,t)$, ${\widehat{\beta }}_{2,1}(r,t)$.

Figure 6

Predicted functional intercepts b̂_i0(t) and observed residuals ê_ij(t) for model (9).

Figure 7

Estimated components of model (8) with ±2 pointwise standard errors, using Ivanescu et al. (2012). Coefficient surfaces are color-coded for sign and pointwise significance (95%): blue if sig. < 0, light blue if < 0, light red if > 0, red if sig. > 0. Top row, left to right: mean FA-CCA for healthy (blue, dotted) versus MS (red, solid) females; mean FA-CCA for female (purple, solid) and male (green, dotted) MS patients; estimated effect of age-at-visit $\widehat{\nu }(u,t)$. Bottom row, left to right: Estimated coefficient surfaces ${\widehat{\beta }}_{1,0}(s,t)$, ${\widehat{\beta }}_{1,1}(s,t)$, ${\widehat{\beta }}_{2,0}(r,t)$, ${\widehat{\beta }}_{2,1}(r,t)$.

Figure 8

Top row, left to right: Observed residuals ${\widehat{\epsilon }}_{ij}\left(t\right)$ for model (8); empirical covariance for ${\widehat{\epsilon }}_{ij}\left(t\right)$ for model (8); empirical covariance for ${\widehat{\epsilon }}_{ij}\left(t\right)$ for model (9) with FPC-based random intercepts; empirical covariance for ${\widehat{\epsilon }}_{ij}\left(t\right)$ for model (9) with spline-based random intercepts; legend for covariance values. Bottom row: Empirical correlations.