Functional Additive Mixed Models

<div><p>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, for example, 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 datasets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach.</p></div

Distribution of therapy failures during the maintenance phase of VIT with respect to baseline parameters.

<p>Associations are shown between clinical, demographic and therapeutic parameters and the need for an emergency intervention during an in-hospital sting challenge or, after a field sting by the culprit insect, the development of any type of generalized symptom.</p

Smooth function and pointwise 95% confidence bands (dashed lines) for the effect of baseline tryptase concentration on the risk to need an emergency intervention during sting challenge or to develop generalized symptoms after a field sting (final multivariate generalized additive model).

<p>Odds ratios are referred to those of the median of tryptase concentration. The odds ratio of the latter has been set at 1.</p

Results of the final generalized additive model for the risk to need an emergency intervention during an in-hospital sting challenge or, after a field sting by the culprit insect, to develop any type of generalized symptom.

<p>Those variables are shown, which were selected according to the modeling procedure. P-values and widths of confidence intervals are biased downwards due to the effect of subset selection.</p

Receiver operating characteristic (ROC) curve for the final multiple logistic regression model predicting the risk to need an emergency intervention during sting challenge or to develop generalized symptoms after a field sting.

<p>Models were tested without including the effect of BTC, or with including a smoothed or a linear effect. Corresponding areas under the curve were 0.7449, 0.7084 or 0.7240.</p