Longitudinal Functional Models with Structured Penalties

A functional model for studying common trends across trial time in eye tracking experiments

Eye tracking (ET) experiments commonly record the continuous trajectory of a subject's gaze on a two-dimensional screen throughout repeated presentations of stimuli (referred to as trials). Even though the continuous path of gaze is recorded during each trial, commonly derived outcomes for analysis collapse the data into simple summaries, such as looking times in regions of interest, latency to looking at stimuli, number of stimuli viewed, number of fixations or fixation length. In order to retain information in trial time, we utilize functional data analysis (FDA) for the first time in literature in the analysis of ET data. More specifically, novel functional outcomes for ET data, referred to as viewing profiles, are introduced that capture the common gazing trends across trial time which are lost in traditional data summaries. Mean and variation of the proposed functional outcomes across subjects are then modeled using functional principal components analysis. Applications to data from a visual exploration paradigm conducted by the Autism Biomarkers Consortium for Clinical Trials showcase the novel insights gained from the proposed FDA approach, including significant group differences between children diagnosed with autism and their typically developing peers in their consistency of looking at faces early on in trial time.

Covariate-adjusted hybrid principal components analysis for region-referenced functional EEG data

Electroencephalography (EEG) studies produce region-referenced functional data via EEG signals recorded across scalp electrodes. The high-dimensional data can be used to contrast neurodevelopmental trajectories between diagnostic groups, for example between typically developing (TD) children and children with autism spectrum disorder (ASD). Valid inference requires characterization of the complex EEG dependency structure as well as covariate-dependent heteroscedasticity, such as changes in variation over developmental age. In our motivating study, EEG data is collected on TD and ASD children aged two to twelve years old. The peak alpha frequency, a prominent peak in the alpha spectrum, is a biomarker linked to neurodevelopment that shifts as children age. To retain information, we model patterns of alpha spectral variation, rather than just the peak location, regionally across the scalp and chronologically across development. We propose a covariate-adjusted hybrid principal components analysis (CA-HPCA) for EEG data, which utilizes both vector and functional principal components analysis while simultaneously adjusting for covariate-dependent heteroscedasticity. CA-HPCA assumes the covariance process is weakly separable conditional on observed covariates, allowing for covariate-adjustments to be made on the marginal covariances rather than the full covariance leading to stable and computationally efficient estimation. The proposed methodology provides novel insights into neurodevelopmental differences between TD and ASD children.

Multilevel modeling of spatially nested functional data: Spatiotemporal patterns of hospitalization rates in the US dialysis population

End-stage renal disease patients on dialysis experience frequent hospitalizations. In addition to known temporal patterns of hospitalizations over the life span on dialysis, where poor outcomes are typically exacerbated during the first year on dialysis, variations in hospitalizations among dialysis facilities across the US contribute to spatial variation. Utilizing national data from the United States Renal Data System (USRDS), we propose a novel multilevel spatiotemporal functional model to study spatiotemporal patterns of hospitalization rates among dialysis facilities. Hospitalization rates of dialysis facilities are considered as spatially nested functional data (FD) with longitudinal hospitalizations nested in dialysis facilities and dialysis facilities nested in geographic regions. A multilevel Karhunen-Loéve expansion is utilized to model the two-level (facility and region) FD, where spatial correlations are induced among region-specific principal component scores accounting for regional variation. A new efficient algorithm based on functional principal component analysis and Markov Chain Monte Carlo is proposed for estimation and inference. We report a novel application using USRDS data to characterize spatiotemporal patterns of hospitalization rates for over 400 health service areas across the US and over the posttransition time on dialysis. Finite sample performance of the proposed method is studied through simulations.

Use of Functional Linear Models to Detect Associations between Characteristics of Walking and Continuous Responses Using Accelerometry Data

Various methods exist to measure physical activity. Subjective methods, such as diaries and surveys, are relatively inexpensive ways of measuring one’s physical activity; however, they are prone to measurement error and bias due to self-reporting. Wearable accelerometers offer a non-invasive and objective measure of one’s physical activity and are now widely used in observational studies. Accelerometers record high frequency data and each produce an unlabeled time series at the sub-second level. An important activity to identify from the data collected is walking, since it is often the only form of activity for certain populations. Currently, most methods use an activity summary which ignores the nuances of walking data. We propose methodology to model specific continuous responses with a functional linear model utilizing spectra obtained from the local fast Fourier transform (FFT) of walking as a predictor. Utilizing prior knowledge of the mechanics of walking, we incorporate this as additional information for the structure of our transformed walking spectra. The methods were applied to the in-the-laboratory data obtained from the Developmental Epidemiologic Cohort Study (DECOS).

Longitudinal dynamic functional regression

The paper develops a parsimonious modelling framework to study the time-varying association between scalar outcomes and functional predictors observed at many instances, in longitudinal studies. The methods enable us to reconstruct the full trajectory of the response and are applicable to Gaussian and non-Gaussian responses. The idea is to model the time-varying functional predictors by using orthogonal basis functions and to expand the time-varying regression coefficient by using the same basis. Numerical investigation through simulation studies and data analysis show excellent performance in terms of accurate prediction and efficient computations, when compared with existing alternatives. The methods are inspired and applied to an animal science application, where of interest is to study the association between the feed intake of lactating sows and the minute-by-minute temperature throughout the 21 days of their lactation period. R code and an R illustration are provided.

Bayesian Semiparametric Functional Mixed Models for Serially Correlated Functional Data, with Application to Glaucoma Data

Glaucoma, a leading cause of blindness, is characterized by optic nerve damage related to intraocular pressure (IOP), but its full etiology is unknown. Researchers at UAB have devised a custom device to measure scleral strain continuously around the eye under fixed levels of IOP, which here is used to assess how strain varies around the posterior pole, with IOP, and across glaucoma risk factors such as age. The hypothesis is that scleral strain decreases with age, which could alter biomechanics of the optic nerve head and cause damage that could eventually lead to glaucoma. To evaluate this hypothesis, we adapted Bayesian Functional Mixed Models to model these complex data consisting of correlated functions on spherical scleral surface, with nonparametric age effects allowed to vary in magnitude and smoothness across the scleral surface, multi-level random effect functions to capture within-subject correlation, and functional growth curve terms to capture serial correlation across IOPs that can vary around the scleral surface. Our method yields fully Bayesian inference on the scleral surface or any aggregation or transformation thereof, and reveals interesting insights into the biomechanical etiology of glaucoma. The general modeling framework described is very flexible and applicable to many complex, high-dimensional functional data.

A multi-dimensional functional principal components analysis of EEG data

The electroencephalography (EEG) data created in event-related potential (ERP) experiments have a complex high-dimensional structure. Each stimulus presentation, or trial, generates an ERP waveform which is an instance of functional data. The experiments are made up of sequences of multiple trials, resulting in longitudinal functional data and moreover, responses are recorded at multiple electrodes on the scalp, adding an electrode dimension. Traditional EEG analyses involve multiple simplifications of this structure to increase the signal-to-noise ratio, effectively collapsing the functional and longitudinal components by identifying key features of the ERPs and averaging them across trials. Motivated by an implicit learning paradigm used in autism research in which the functional, longitudinal, and electrode components all have critical interpretations, we propose a multidimensional functional principal components analysis (MD-FPCA) technique which does not collapse any of the dimensions of the ERP data. The proposed decomposition is based on separation of the total variation into subject and subunit level variation which are further decomposed in a two-stage functional principal components analysis. The proposed methodology is shown to be useful for modeling longitudinal trends in the ERP functions, leading to novel insights into the learning patterns of children with Autism Spectrum Disorder (ASD) and their typically developing peers as well as comparisons between the two groups. Finite sample properties of MD-FPCA are further studied via extensive simulations.