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Tensor-based morphometry (TBM) aims at showing local differences in brain volumes with respect to a common template. TBM images are smooth but they exhibit (especially in diseased groups) higher values in some brain regions called lateral ventricles. More specifically, our voxelwise analysis shows both a mean-variance relationship in these areas and evidence of spatially dependent skewness. We propose a model for 3-dimensional functional data where mean, variance, and skewness functions vary smoothly across brain locations. We model the voxelwise distributions as skew-normal. The smooth effects of age and sex are estimated on a reference population of cognitively normal subjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset and mapped across the whole brain. The three parameter functions allow to transform each TBM image (in the reference population as well as in a test set) into a Gaussian process. These subject-specific normative maps are used to derive indices of deviation from a healthy condition to assess the individual risk of pathological degeneration.

This article focuses on the study of lactating sows, where the main interest is the influence of temperature, measured throughout the day, on the lower quantiles of the daily feed intake. We outline a model framework and estimation methodology for quantile regression in scenarios with longitudinal data and functional covariates. The quantile regression model uses a time-varying regression coefficient function to quantify the association between covariates and the quantile level of interest, and it includes subject-specific intercepts to incorporate within-subject dependence. Estimation relies on spline representations of the unknown coefficient functions, and can be carried out with existing software. We introduce bootstrap procedures for bias adjustment and computation of standard errors. Analysis of the lactation data indicates, among others, that the influence of temperature increases during the lactation period.

Wearable devices for continuous monitoring of electronic health increased attention due to their richness in information. Often, inference is drawn from features that quantify some summary of the data, leading to a loss of information that could be useful when one utilizes the functional nature of the response. When functional trajectories are observed repeated over time, it is termed longitudinal functional data. This work is motivated by the interest to assess the efficacy of a noninflammatory medication, meloxicam, on the daily activity levels of household cats with a pre-existing condition of osteoarthritis under a crossover design. These activity profiles are recorded at a minute level by accelerometer over the entire study period. To this aspect, we propose an orthogonal projection-based test pseudo generalized F test for significance of the functional treatment effect under a functional additive crossover model after adjusting for the carryover effect and other baseline covariates. Under mild conditions, we derive the asymptotic null distribution of the test statistic when the projection function for the underlying Hilbert space is estimated from the data. In finite sample numerical studies, the proposed test maintains the size, is powerful to detect the significance of the smooth effect of meloxicam, and is very efficient compared to bootstrap-based alternatives., Comment: made with biometrics template guide, to confine it within 5 pages

The manuscript discusses how to incorporate random effects for quantile regression models for clustered data with focus on settings with many but small clusters. The paper has three contributions: (i) documenting that existing methods may lead to severely biased estimators for fixed effects parameters; (ii) proposing a new two-step estimation methodology where predictions of the random effects are first computed {by a pseudo likelihood approach (the LQMM method)} and then used as offsets in standard quantile regression; (iii) proposing a novel bootstrap sampling procedure in order to reduce bias of the two-step estimator and compute confidence intervals. The proposed estimation and associated inference is assessed numerically through rigorous simulation studies and applied to an AIDS Clinical Trial Group (ACTG) study., Comment: Accepted for Econometrics and Statistics

In many modern applications, a dependent functional response is observed for each subject over repeated time, leading to longitudinal functional data. In this paper, we propose a novel statistical procedure to test whether the mean function varies over time. Our approach relies on reducing the dimension of the response using data-driven orthogonal projections and it employs a likelihood-based hypothesis testing. We investigate the methodology theoretically and discuss a computationally efficient implementation. The proposed test maintains the type I error rate, and shows excellent power to detect departures from the null hypothesis in finite sample simulation studies. We apply our method to the longitudinal diffusion tensor imaging study of multiple sclerosis (MS) patients to formally assess whether the brain's health tissue, as summarized by fractional anisotropy (FA) profile, degrades over time during the study period.

Observational longitudinal studies are a common means to study treatment efficacy and safety in chronic mental illness. In many such studies, treatment changes may be initiated by either the patient or by their clinician and can thus vary widely across patients in their timing, number, and type. Indeed, in the observational longitudinal pathway of the STEP-BD study of bipolar depression, one of the motivations for this work, no two patients have the same treatment history even after coarsening clinic visits to a weekly time-scale. Estimation of an optimal treatment regime using such data is challenging as one cannot naively pool together patients with the same treatment history, as is required by methods based on inverse probability weighting, nor is it possible to apply backwards induction over the decision points, as is done in Q-learning and its variants. Thus, additional structure is needed to effectively pool information across patients and within a patient over time. Current scientific theory for many chronic mental illnesses maintains that a patient's disease status can be conceptualized as transitioning among a small number of discrete states. We use this theory to inform the construction of a partially observable Markov decision process model of patient health trajectories wherein observed health outcomes are dictated by a patient's latent health state. Using this model, we derive and evaluate estimators of an optimal treatment regime under two common paradigms for quantifying long-term patient health. The finite sample performance of the proposed estimator is demonstrated through a series of simulation experiments and application to the observational pathway of the STEP-BD study. We find that the proposed method provides high-quality estimates of an optimal treatment strategy in settings where existing approaches cannot be applied without ad hoc modifications.

Scalar-on-function linear models are commonly used to regress functional predictors on a scalar response. However, functional models are more difficult to estimate and interpret than traditional linear models, and may be unnecessarily complex for a data application. Hypothesis testing can be used to guide model selection by determining if a functional predictor is necessary. Using a mixed effects representation with penalized splines and variance component tests, we propose a framework for testing functional linear models with responses from exponential family distributions. The proposed method can accommodate dense and sparse functional data, and be used to test functional predictors for no effect and form of the effect. We show via simulation study that the proposed method achieves the nominal level and has high power, and we demonstrate its utility with two data applications., Comment: 30 pages, 4 figures

Generalized linear mixed models (GLMMs) are used to model responses from exponential families with a combination of fixed and random effects. For variance components in GLMMs, we propose an approximate restricted likelihood ratio test that conducts testing on the working responses used in penalized quasi-likelihood estimation. This presents the hypothesis test in terms of normalized responses, allowing for application of existing testing methods for linear mixed models. Our test is flexible, computationally efficient, and outperforms several competitors. We illustrate the utility of the proposed method with an extensive simulation study and two data applications. An R package is provided., Comment: 19 pages, 1 figure, Supplementary Materials (additional simulation results) excluded for brevity

To study trends in extreme precipitation across US over the years 1951-2017, we consider 10 climate indexes that represent extreme precipitation, such as annual maximum of daily precipitation, annual maximum of consecutive 5-day average precipitation, which exhibit spatial correlation as well as mutual dependence. We consider the gridded data, produced by the CLIMDEX project (http://www.climdex.org/gewocs.html), constructed using daily precipitation data. In this paper, we propose a multivariate spatial skew-t process for joint modeling of extreme precipitation indexes and discuss its theoretical properties. The model framework allows Bayesian inference while maintaining a computational time that is competitive with common multivariate geostatistical approaches. In a numerical study, we find that the proposed model outperforms multivariate spatial Gaussian processes, multivariate spatial t-processes including their univariate alternatives in terms of various model selection criteria. We apply the proposed model to estimate the average decadal change in the extreme precipitation indexes throughout the United States and find several significant local changes., Comment: 23 pages, 6 Figures

Large wildfires pose a major environmental concern, and precise maps of fire risk can improve disaster relief planning. Fosberg Fire Weather Index (FFWI) is often used to measure wildfire risk; FFWI exhibits non-Gaussian marginal distributions as well as strong spatiotemporal extremal dependence and thus, modeling FFWI using geostatistical models like Gaussian processes is questionable. Extreme value theory (EVT)-driven models like max-stable processes are theoretically appealing but are computationally demanding and applicable only for threshold exceedances or block maxima. Disaster management policies often consider moderate-to-extreme quantiles of climate parameters and hence, joint modeling of the bulk and the tail of the data is required. In this paper, we consider a Dirichlet process mixture of spatial skew-t processes that can flexibly model the bulk as well as the tail. The proposed model has nonstationary mean and covariance structure, and also nonzero spatiotemporal extremal dependence. A simulation study demonstrates that the proposed model has better spatial prediction performance compared to some competing models. We develop spatial maps of FFWI medians and extremes, and discuss the wildfire risk throughout the Santa Ana region of California., Comment: 69 pages, 16 Figures

Wearable devices including accelerometers are increasingly being used to collect high-frequency human activity data in situ. There is tremendous potential to use such data to inform medical decision making and public health policies. However, modeling such data is challenging as they are high-dimensional, heterogeneous, and subject to informative missingness, e.g., zero readings when the device is removed by the participant. We propose a flexible and extensible continuous-time hidden Markov model to extract meaningful activity patterns from human accelerometer data. To facilitate estimation with massive data we derive an efficient learning algorithm that exploits the hierarchical structure of the parameters indexing the proposed model. We also propose a bootstrap procedure for interval estimation. The proposed methods are illustrated using data from the 2003 - 2004 and 2005 - 2006 National Health and Nutrition Examination Survey., Comment: 18 pages, 4 figures

The advent of high-throughput sequencing technologies has made data from DNA material readily available, leading to a surge of microbiome-related research establishing links between markers of microbiome health and specific outcomes. However, to harness the power of microbial communities we must understand not only how they affect us, but also how they can be influenced to improve outcomes. This area has been dominated by methods that reduce community composition to summary metrics, which can fail to fully exploit the complexity of community data. Recently, methods have been developed to model the abundance of taxa in a community, but they can be computationally intensive and do not account for spatial effects underlying microbial settlement. These spatial effects are particularly relevant in the microbiome setting because we expect communities that are close together to be more similar than those that are far apart. In this paper, we propose a flexible Bayesian spike-and-slab variable selection model for presence-absence indicators that accounts for spatial dependence and cross-dependence between taxa while reducing dimensionality in both directions. We show by simulation that in the presence of spatial dependence, popular distance-based hypothesis testing methods fail to preserve their advertised size, and the proposed method improves variable selection. Finally, we present an application of our method to an indoor fungal community found with homes across the contiguous United States., Comment: 26 pages, 4 figures

State-of-the-art robotic hand prosthetics generate finger and wrist movement through pattern recognition (PR) algorithms using features of forearm electromyogram (EMG) signals, but re- quires extensive training and is prone to poor predictions for conditions outside the training data (Peerdeman et al., 2011; Scheme et al., 2010). We propose a novel approach to develop a dynamic robotic limb by utilizing the recent history of EMG signals in a model that accounts for physiological features of hand movement which are ignored by PR algorithms. We do this by viewing EMG signals as functional covariates and develop a functional linear model that quantifies the effect of the EMG signals on finger/wrist velocity through a bivariate coefficient function that is allowed to vary with current finger/wrist position. The model is made par- simonious and interpretable through a two-step variable selection procedure, called Sequential Adaptive Functional Empirical group LASSO (SAFE-gLASSO). Numerical studies show excel- lent selection and prediction properties of SAFE-gLASSO compared to popular alternatives. For our motivating dataset, the method correctly identifies the few EMG signals that are known to be important for an able-bodied subject with negligible false positives and the model can be directly implemented in a robotic prosthetic.

We study the problem of sparse signal detection on a spatial domain. We propose a novel approach to model continuous signals that are sparse and piecewise smooth as product of independent Gaussian processes (PING) with a smooth covariance kernel. The smoothness of the PING process is ensured by the smoothness of the covariance kernels of Gaussian components in the product, and sparsity is controlled by the number of components. The bivariate kurtosis of the PING process shows more components in the product results in thicker tail and sharper peak at zero. The simulation results demonstrate the improvement in estimation using the PING prior over Gaussian process (GP) prior for different image regressions. We apply our method to a longitudinal MRI dataset to detect the regions that are affected by multiple sclerosis (MS) in the greatest magnitude through an image-on-scalar regression model. Due to huge dimensionality of these images, we transform the data into the spectral domain and develop methods to conduct computation in this domain. In our MS imaging study, the estimates from the PING model are more informative than those from the GP model.

A scalar-response functional model describes the association between a scalar response and a set of functional covariates. An important problem in the functional data literature is to test the nullity or linearity of the effect of the functional covariate in the context of scalar-on-function regression. This article provides an overview of the existing methods for testing both the null hypotheses that there is no relationship and that there is a linear relationship between the functional covariate and scalar response, and a comprehensive numerical comparison of their performance. The methods are compared for a variety of realistic scenarios: when the functional covariate is observed at dense or sparse grids and measurements include noise or not. Finally, the methods are illustrated on the Tecator data set., Comment: 38 pages, 14 pages of Supplementary Material, 40 images in total

This article develops flexible methodology to study the association between scalar outcomes and functional predictors observed over time, at many instances, in longitudinal studies. We propose a parsimonious modeling framework to study time-varying regression that leads to superior prediction properties and allows to reconstruct full trajectories of the response. The idea is to model the time-varying functional predictors using orthogonal basis functions and expand the time-varying regression coefficient 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 21st days of their lactation period. R code and an R illustration are provided at http://www4.stat.ncsu.edu/~staicu/software, Comment: 27 pages, 4 Figures

We propose simple inferential approaches for the fixed effects in complex functional mixed effects models. We estimate the fixed effects under the independence of functional residuals assumption and then bootstrap independent units (e.g. subjects) to estimate the variability of and conduct inference in the form of hypothesis testing on the fixed effects parameters. Simulations show excellent coverage probability of the confidence intervals and size of tests. Methods are motivated by and applied to the Baltimore Longitudinal Study of Aging (BLSA), though they are applicable to other studies that collect correlated functional data.

We study additive function-on-function regression where the mean response at a particular time point depends on the time point itself as well as the entire covariate trajectory. We develop a computationally efficient estimation methodology based on a novel combination of spline bases with an eigenbasis to represent the trivariate kernel function. We discuss prediction of a new response trajectory, propose an inference procedure that accounts for total variability in the predicted response curves, and construct pointwise prediction intervals. The estimation/inferential procedure accommodates realistic scenarios such as correlated error structure as well as sparse and/or irregular designs. We investigate our methodology in finite sample size through simulations and two real data applications., Comment: 26 pages, 4 figures

We propose a novel modeling framework to study the effect of covariates of various types on the conditional distribution of the response. The methodology accommodates flexible model structure, allows for joint estimation of the quantiles at all levels, and involves a computationally efficient estimation algorithm. Extensive numerical investigation confirms good performance of the proposed method. The methodology is motivated by and applied to a lactating sow study, where the primary interest is to understand how the dynamic change of minute-by-minute temperature in the farrowing rooms within a day (functional covariate) is associated with low quantiles of feed intake of lactating sows, while accounting for other sow-specific information (vector covariate)., Comment: 30 pages, 4 figures, 3 tables

The focus of this work is on spatial variable selection for scalar-on-image regression. We propose a new class of Bayesian nonparametric models, soft-thresholded Gaussian processes and develop the efficient posterior computation algorithms. Theoretically, soft-thresholded Gaussian processes provide large prior support for the spatially varying coefficients that enjoy piecewise smoothness, sparsity and continuity, characterizing the important features of imaging data. Also, under some mild regularity conditions, the soft-thresholded Gaussian process leads to the posterior consistency for both parameter estimation and variable selection for scalar-on-image regression, even when the number of true predictors is larger than the sample size. The proposed method is illustrated via simulations, compared numerically with existing alternatives and applied to Electroencephalography (EEG) study of alcoholism., Comment: 39 pages, 5 figures

In this paper, we study statistical inference in functional quantile regression for scalar response and a functional covariate. Specifically, we consider a functional linear quantile regression model where the effect of the covariate on the quantile of the response is modeled through the inner product between the functional covariate and an unknown smooth regression parameter function that varies with the level of quantile. The objective is to test that the regression parameter is constant across several quantile levels of interest. The parameter function is estimated by combining ideas from functional principal component analysis and quantile regression. An adjusted Wald testing procedure is proposed for this hypothesis of interest, and its chi-square asymptotic null distribution is derived. The testing procedure is investigated numerically in simulations involving sparse and noisy functional covariates and in a capital bike share data application. The proposed approach is easy to implement and the {\tt R} code is published online at \url{https://github.com/xylimeng/fQR-testing}.

Although there are established graphics that accompany the most common functional data analyses, generating these graphics for each dataset and analysis can be cumbersome and time consuming. Often, the barriers to visualization inhibit useful exploratory data analyses and prevent the development of intuition for a method and its application to a particular dataset. The refund.shiny package was developed to address these issues for several of the most common functional data analyses. After conducting an analysis, the plot_shiny() function is used to generate an interactive visualization environment that contains several distinct graphics, many of which are updated in response to user input. These visualizations reduce the burden of exploratory analyses and can serve as a useful tool for the communication of results to non-statisticians.

We propose a Bayesian modeling framework for jointly analyzing multiple functional responses of different types (e.g. binary and continuous data). Our approach is based on a multivariate latent Gaussian process and models the dependence among the functional responses through the dependence of the latent process. Our framework easily accommodates additional covariates. We offer a way to estimate the multivariate latent covariance, allowing for implementation of multivariate functional principal components analysis (FPCA) to specify basis expansions and simplify computation. We demonstrate our method through both simulation studies and an application to real data from a periodontal study.

We consider analysis of dependent functional data that are correlated because of a longitudinal-based design: each subject is observed at repeated time visits and for each visit we record a functional variable. We propose a novel parsimonious modeling framework for the repeatedly observed functional variables that allows to extract low dimensional features. The proposed methodology accounts for the longitudinal design, is designed for the study of the dynamic behavior of the underlying process, and is computationally fast. Theoretical properties of this framework are studied and numerical investigation confirms excellent behavior in finite samples. The proposed method is motivated by and applied to a diffusion tensor imaging study of multiple sclerosis. Using Shiny (Chang et al., 2015) we implement interactive plots to help visualize longitudinal functional data as well as the various components and prediction obtained using the proposed method., Comment: 32 pages, 4 figures

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., Comment: 26 pages, 8 figures

Second order approximate ancillaries have evolved as the primary ingredient for recent likelihood development in statistical inference. This uses quantile functions rather than the equivalent distribution functions, and the intrinsic ancillary contour is given explicitly as the plug-in estimate of the vector quantile function. The derivation uses a Taylor expansion of the full quantile function, and the linear term gives a tangent to the observed ancillary contour. For the scalar parameter case, there is a vector field that integrates to give the ancillary contours, but for the vector case, there are multiple vector fields and the Frobenius conditions for mutual consistency may not hold. We demonstrate, however, that the conditions hold in a restricted way and that this verifies the second order ancillary contours in moderate deviations. The methodology can generate an appropriate exact ancillary when such exists or an approximate ancillary for the numerical or Monte Carlo calculation of $p$-values and confidence quantiles. Examples are given, including nonlinear regression and several enigmatic examples from the literature., Comment: Published in at http://dx.doi.org/10.3150/10-BEJ248 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)