Existing approaches for multivariate functional principal component analysis are restricted to data on the same one-dimensional interval. The presented approach focuses on multivariate functional data on different domains that may differ in dimension, e.g. functions and images. The theoretical basis for multivariate functional principal component analysis is given in terms of a Karhunen-Loève Theorem. For the practically relevant case of a finite Karhunen-Loève representation, a relationship between univariate and multivariate functional principal component analysis is established. This offers an estimation strategy to calculate multivariate functional principal components and scores based on their univariate counterparts. For the resulting estimators, asymptotic results are derived. The approach can be extended to finite univariate expansions in general, not necessarily orthonormal bases. It is also applicable for sparse functional data or data with measurement error. A flexible R implementation is available on CRAN. The new method is shown to be competitive to existing approaches for data observed on a common one-dimensional domain. The motivating application is a neuroimaging study, where the goal is to explore how longitudinal trajectories of a neuropsychological test score covary with FDG-PET brain scans at baseline. Supplementary material, including detailed proofs, additional simulation results and software is available online.

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.

Viewing a stressful soccer match more than doubles the risk of an acute cardiovascular event. In view of this excess risk, particularly in men with known coronary heart disease, preventive measures are urgently needed.

In linear mixed models, model selection frequently includes the selection of random effects. Two versions of the Akaike information criterion, aic, have been used, based either on the marginal or on the conditional distribution. We show that the marginal aic is not an asymptotically unbiased estimator of the Akaike information, and favours smaller models without random effects. For the conditional aic, we show that ignoring estimation uncertainty in the random effects covariance matrix, as is common practice, induces a bias that can lead to the selection of any random effect not predicted to be exactly zero. We derive an analytic representation of a corrected version of the conditional aic, which avoids the high computational cost and imprecision of available numerical approximations. An implementation in an R package (R Development Core Team, 2010) is provided. All theoretical results are illustrated in simulation studies, and their impact in practice is investigated in an analysis of childhood malnutrition in Zambia.

BackgroundNumerous studies have found that ambient air pollution has been associated with cardiovascular disease exacerbation.ObjectivesGiven previous findings, we hypothesized that particulate air pollution might induce systemic inflammation in myocardial infarction (MI) survivors, contributing to an increased vulnerability to elevated concentrations of ambient particles.MethodsA prospective longitudinal study of 1,003 MI survivors was performed in six European cities between May 2003 and July 2004. We compared repeated measurements of interleukin 6 (IL-6), fibrinogen, and C-reactive protein (CRP) with concurrent levels of air pollution. We collected hourly data on particle number concentrations (PNC), mass concentrations of particulate matter (PM) < 10 μm (PM10) and < 2.5 μm (PM2.5), gaseous pollutants, and meteorologic data at central monitoring sites in each city. City-specific confounder models were built for each blood marker separately, adjusting for meteorology and time-varying and time-invariant covariates. Data were analyzed with mixed-effects models.ResultsPooled results show an increase in IL-6 when concentrations of PNC were elevated 12–17 hr before blood withdrawal [percent change of geometric mean, 2.7; 95% confidence interval (CI), 1.0–4.6]. Five day cumulative exposure to PM10 was associated with increased fibrinogen concentrations (percent change of arithmetic mean, 0.6; 95% CI, 0.1–1.1). Results remained stable for smokers, diabetics, and patients with heart failure. No consistent associations were found for CRP.ConclusionsResults indicate an immediate response to PNC on the IL-6 level, possibly leading to the production of acute-phase proteins, as seen in increased fibrinogen levels. This might provide a link between air pollution and adverse cardiac events.

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We introduce models for the analysis of functional data observed at multiple time points. The dynamic behavior of functional data is decomposed into a time-dependent population average, baseline (or static) subject-specific variability, longitudinal (or dynamic) subject-specific variability, subject-visit-specific variability and measurement error. The model can be viewed as the functional analog of the classical longitudinal mixed effects model where random effects are replaced by random processes. Methods have wide applicability and are computationally feasible for moderate and large data sets. Computational feasibility is assured by using principal component bases for the functional processes. The methodology is motivated by and applied to a diffusion tensor imaging (DTI) study designed to analyze differences and changes in brain connectivity in healthy volunteers and multiple sclerosis (MS) patients. An R implementation is provided. 87

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