Penalized functional regression analysis of white-matter tract profiles in multiple sclerosis

Predicting health outcomes with intensive longitudinal data collected by mobile health devices: a functional principal component regression approach

BACKGROUND

Intensive longitudinal data (ILD) collected in near real time by mobile health devices provide a new opportunity for monitoring chronic diseases, early disease risk prediction, and disease prevention in health research. Functional data analysis, specifically functional principal component analysis, has great potential to abstract trends in ILD but has not been used extensively in mobile health research.

OBJECTIVE

To introduce functional principal component analysis (fPCA) and demonstrate its potential applicability in estimating trends in ILD collected by mobile heath devices, assessing longitudinal association between ILD and health outcomes, and predicting health outcomes.

METHODS

fPCA and scalar-to-function regression models were reviewed. A case study was used to illustrate the process of abstracting trends in intensively self-measured blood glucose using functional principal component analysis and then predicting future HbA1c values in patients with type 2 diabetes using a scalar-to-function regression model.

RESULTS

Based on the scalar-to-function regression model results, there was a slightly increasing trend between daily blood glucose measures and HbA1c. 61% of variation in HbA1c could be predicted by the three preceding months' blood glucose values measured before breakfast (P < 0.0001, [Formula: see text]).

CONCLUSIONS

Functional data analysis, specifically fPCA, offers a unique tool to capture patterns in ILD collected by mobile health devices. It is particularly useful in assessing longitudinal dynamic association between repeated measures and outcomes, and can be easily integrated in prediction models to improve prediction precision.

Subtle microstructural alterations in white matter tracts involved in socio-emotional processing after very preterm birth

Children born very preterm (VPT, < 32 weeks of gestation) have an increased risk of developing socio-emotional difficulties. Possible neural substrates for these socio-emotional difficulties are alterations in the structural connectivity of the social brain due to premature birth. The objective of the current study was to study microstructural white matter integrity in VPT versus full-term (FT) born school-aged children along twelve white matter tracts involved in socio-emotional processing. Diffusion MRI scans were obtained from a sample of 35 VPT and 38 FT 8-to-12-year-old children. Tractography was performed using TractSeg, a state-of-the-art neural network-based approach, which offers investigation of detailed tract profiles of fractional anisotropy (FA). Group differences in FA along the tracts were investigated using both a traditional and complementary functional data analysis approach. Exploratory correlations were performed between the Social Responsiveness Scale (SRS-2), a parent-report questionnaire assessing difficulties in social functioning, and FA along the tract. Both analyses showed significant reductions in FA for the VPT group along the middle portion of the right SLF I and an anterior portion of the left SLF II. These group differences possibly indicate altered white matter maturation due to premature birth and may contribute to altered functional connectivity in the Theory of Mind network which has been documented in earlier work with VPT samples. Apart from reduced social motivation in the VPT group, there were no significant group differences in reported social functioning, as assessed by SRS-2. We found that in the VPT group higher FA values in segments of the left SLF I and right SLF II were associated with better social functioning. Surprisingly, the opposite was found for segments in the right IFO, where higher FA values were associated with worse reported social functioning. Since no significant correlations were found for the FT group, this relationship may be specific for VPT children. The current study overcomes methodological limitations of previous studies by more accurately segmenting white matter tracts using constrained spherical deconvolution based tractography, by applying complementary tractometry analysis approaches to estimate changes in FA more accurately, and by investigating the FA profile along the three components of the SLF.

Differences in functional connectivity distribution after transcranial direct-current stimulation: A connectivity density point of view

In this manuscript, we consider the problem of relating functional connectivity measurements viewed as statistical distributions to outcomes. We demonstrate the utility of using the distribution of connectivity on a study of resting-state functional magnetic resonance imaging association with an intervention. The method uses the estimated density of connectivity between nodes of interest as a functional covariate. Moreover, we demonstrate the utility of the procedure in an instance where connectivity is naturally considered an outcome by reversing the predictor/response relationship using case/control methodology. The method utilizes the density quantile, the density evaluated at empirical quantiles, instead of the empirical density directly. This improved the performance of the method by highlighting tail behavior, though we emphasize that by being flexible and non-parametric, the technique can detect effects related to the central portion of the density. To demonstrate the method in an application, we consider 47 primary progressive aphasia patients with various levels of language abilities. These patients were randomly assigned to two treatment arms, transcranial direct-current stimulation and language therapy versus sham (language therapy only), in a clinical trial. We use the method to analyze the effect of direct stimulation on functional connectivity. As such, we estimate the density of correlations among the regions of interest and study the difference in the density post-intervention between treatment arms. We discover that it is the tail of the density, rather than the mean or lower order moments of the distribution, that demonstrates a significant impact in the classification. The new approach has several benefits. Among them, it drastically reduces the number of multiple comparisons compared with edge-wise analysis. In addition, it allows for the investigation of the impact of functional connectivity on the outcomes where the connectivity is not geometrically localized.

ESTIMATION FOR EXTREME CONDITIONAL QUANTILES OF FUNCTIONAL QUANTILE REGRESSION

Quantile regression as an alternative to modeling the conditional mean function provides a comprehensive picture of the relationship between a response and covariates. It is particularly attractive in applications focused on the upper or lower conditional quantiles of the response. However, conventional quantile regression estimators are often unstable at the extreme tails, owing to data sparsity, especially for heavy-tailed distributions. Assuming that the functional predictor has a linear effect on the upper quantiles of the response, we develop a novel estimator for extreme conditional quantiles using a functional composite quantile regression based on a functional principal component analysis and an extrapolation technique from extreme value theory. We establish the asymptotic normality of the proposed estimator under some regularity conditions, and compare it with other estimation methods using Monte Carlo simulations. Finally, we demonstrate the proposed method by empirically analyzing two real data sets.

Extreme quantile estimation for partial functional linear regression models with heavy-tailed distributions

In this article, we propose a novel estimator of extreme conditional quantiles in partial functional linear regression models with heavy-tailed distributions. The conventional quantile regression estimators are often unstable at the extreme tails due to data sparsity, especially for heavy-tailed distributions. We first estimate the slope function and the partially linear coefficient using a functional quantile regression based on functional principal component analysis, which is a robust alternative to the ordinary least squares regression. The extreme conditional quantiles are then estimated by using a new extrapolation technique from extreme value theory. We establish the asymptotic normality of the proposed estimator and illustrate its finite sample performance by simulation studies and an empirical analysis of diffusion tensor imaging data from a cognitive disorder study.

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.

Scalar-on-function regression for predicting distal outcomes from intensively gathered longitudinal data: Interpretability for applied scientists

Researchers are sometimes interested in predicting a distal or external outcome (such as smoking cessation at follow-up) from the trajectory of an intensively recorded longitudinal variable (such as urge to smoke). This can be done in a semiparametric way via scalar-on-function regression. However, the resulting fitted coefficient regression function requires special care for correct interpretation, as it represents the joint relationship of time points to the outcome, rather than a marginal or cross-sectional relationship. We provide practical guidelines, based on experience with scientific applications, for helping practitioners interpret their results and illustrate these ideas using data from a smoking cessation study.

A smoothing-based goodness-of-fit test of covariance for functional data

Functional data methods are often applied to longitudinal data as they provide a more flexible way to capture dependence across repeated observations. However, there is no formal testing procedure to determine if functional methods are actually necessary. We propose a goodness-of-fit test for comparing parametric covariance functions against general nonparametric alternatives for both irregularly observed longitudinal data and densely observed functional data. We consider a smoothing-based test statistic and approximate its null distribution using a bootstrap procedure. We focus on testing a quadratic polynomial covariance induced by a linear mixed effects model and the method can be used to test any smooth parametric covariance function. Performance and versatility of the proposed test is illustrated through a simulation study and three data applications.

Modeling and Prediction of Multiple Correlated Functional Outcomes

We propose a copula-based approach for analyzing functional data with correlated multiple functional outcomes exhibiting heterogeneous shape characteristics. To accommodate the possibly large number of parameters due to having several functional outcomes, parameter estimation is performed in two steps: first, the parameters for the marginal distributions are estimated using the skew t family, and then the dependence structure both within and across outcomes is estimated using a Gaussian copula. We develop an estimation algorithm for the dependence parameters based on the Karhunen-Loève expansion and an EM algorithm that significantly reduces the dimension of the problem and is computationally efficient. We also demonstrate prediction of an unknown outcome when the other outcomes are known. We apply our methodology to diffusion tensor imaging data for multiple sclerosis (MS) patients with three outcomes and identify differences in both the marginal distributions and the dependence structure between the MS and control groups. Our proposed methodology is quite general and can be applied to other functional data with multiple outcomes in biology and other fields.

Parametrization of white matter manifold-like structures using principal surfaces

In this manuscript, we are concerned with data generated from a diffusion tensor imaging (DTI) experiment. The goal is to parameterize manifold-like white matter tracts, such as the corpus callosum, using principal surfaces. The problem is approached by finding a geometrically motivated surface-based representation of the corpus callosum and visualized fractional anisotropy (FA) values projected onto the surface. The method also applies to any other diffusion summary. An algorithm is proposed that 1) constructs the principal surface of a corpus callosum; 2) flattens the surface into a parametric 2D map; 3) projects associated FA values on the map. The algorithm is applied to a longitudinal study containing 466 diffusion tensor images of 176 multiple sclerosis (MS) patients observed at multiple visits. For each subject and visit the study contains a registered DTI scan of the corpus callosum at roughly 20,000 voxels. Extensive simulation studies demonstrate fast convergence and robust performance of the algorithm under a variety of challenging scenarios.

Longitudinal High-Dimensional Principal Components Analysis with Application to Diffusion Tensor Imaging of Multiple Sclerosis

We develop a flexible framework for modeling high-dimensional imaging data observed longitudinally. The approach decomposes the observed variability of repeatedly measured high-dimensional observations into three additive components: a subject-specific imaging random intercept that quantifies the cross-sectional variability, a subject-specific imaging slope that quantifies the dynamic irreversible deformation over multiple realizations, and a subject-visit specific imaging deviation that quantifies exchangeable effects between visits. The proposed method is very fast, scalable to studies including ultra-high dimensional data, and can easily be adapted to and executed on modest computing infrastructures. The method is applied to the longitudinal analysis of diffusion tensor imaging (DTI) data of the corpus callosum of multiple sclerosis (MS) subjects. The study includes 176 subjects observed at 466 visits. For each subject and visit the study contains a registered DTI scan of the corpus callosum at roughly 30,000 voxels.

Variable-Domain Functional Regression for Modeling ICU Data

We introduce a class of scalar-on-function regression models with subject-specific functional predictor domains. The fundamental idea is to consider a bivariate functional parameter that depends both on the functional argument and on the width of the functional predictor domain. Both parametric and nonparametric models are introduced to fit the functional coefficient. The nonparametric model is theoretically and practically invariant to functional support transformation, or support registration. Methods were motivated by and applied to a study of association between daily measures of the Intensive Care Unit (ICU) Sequential Organ Failure Assessment (SOFA) score and two outcomes: in-hospital mortality, and physical impairment at hospital discharge among survivors. Methods are generally applicable to a large number of new studies that record a continuous variables over unequal domains.

Smooth Scalar-on-Image Regression via Spatial Bayesian Variable Selection

We develop scalar-on-image regression models when images are registered multidimensional manifolds. We propose a fast and scalable Bayes inferential procedure to estimate the image coefficient. The central idea is the combination of an Ising prior distribution, which controls a latent binary indicator map, and an intrinsic Gaussian Markov random field, which controls the smoothness of the nonzero coefficients. The model is fit using a single-site Gibbs sampler, which allows fitting within minutes for hundreds of subjects with predictor images containing thousands of locations. The code is simple and is provided in less than one page in the Appendix. We apply this method to a neuroimaging study where cognitive outcomes are regressed on measures of white matter microstructure at every voxel of the corpus callosum for hundreds of subjects.

ANALYSIS OF MULTIPLE SCLEROSIS LESIONS VIA SPATIALLY VARYING COEFFICIENTS

Magnetic resonance imaging (MRI) plays a vital role in the scientific investigation and clinical management of multiple sclerosis. Analyses of binary multiple sclerosis lesion maps are typically "mass univariate" and conducted with standard linear models that are ill suited to the binary nature of the data and ignore the spatial dependence between nearby voxels (volume elements). Smoothing the lesion maps does not entirely eliminate the non-Gaussian nature of the data and requires an arbitrary choice of the smoothing parameter. Here we present a Bayesian spatial model to accurately model binary lesion maps and to determine if there is spatial dependence between lesion location and subject specific covariates such as MS subtype, age, gender, disease duration and disease severity measures. We apply our model to binary lesion maps derived from T2-weighted MRI images from 250 multiple sclerosis patients classified into five clinical subtypes, and demonstrate unique modeling and predictive capabilities over existing methods.

Quantifying the reliability of image replication studies: the image intraclass correlation coefficient (I2C2)

This article proposes the image intraclass correlation (I2C2) coefficient as a global measure of reliability for imaging studies. The I2C2 generalizes the classic intraclass correlation (ICC) coefficient to the case when the data of interest are images, thereby providing a measure that is both intuitive and convenient. Drawing a connection with classical measurement error models for replication experiments, the I2C2 can be computed quickly, even in high-dimensional imaging studies. A nonparametric bootstrap procedure is introduced to quantify the variability of the I2C2 estimator. Furthermore, a Monte Carlo permutation is utilized to test reproducibility versus a zero I2C2, representing complete lack of reproducibility. Methodologies are applied to three replication studies arising from different brain imaging modalities and settings: regional analysis of volumes in normalized space imaging for characterizing brain morphology, seed-voxel brain activation maps based on resting-state functional magnetic resonance imaging (fMRI), and fractional anisotropy in an area surrounding the corpus callosum via diffusion tensor imaging. Notably, resting-state fMRI brain activation maps are found to have low reliability, ranging from .2 to .4. Software and data are available to provide easy access to the proposed methods.

Bayesian scalar-on-image regression with application to association between intracranial DTI and cognitive outcomes

Diffusion tensor imaging (DTI) measures water diffusion within white matter, allowing for in vivo quantification of brain pathways. These pathways often subserve specific functions, and impairment of those functions is often associated with imaging abnormalities. As a method for predicting clinical disability from DTI images, we propose a hierarchical Bayesian "scalar-on-image" regression procedure. Our procedure introduces a latent binary map that estimates the locations of predictive voxels and penalizes the magnitude of effect sizes in these voxels, thereby resolving the ill-posed nature of the problem. By inducing a spatial prior structure, the procedure yields a sparse association map that also maintains spatial continuity of predictive regions. The method is demonstrated on a simulation study and on a study of association between fractional anisotropy and cognitive disability in a cross-sectional sample of 135 multiple sclerosis patients.

VARYING COEFFICIENT MODEL FOR MODELING DIFFUSION TENSORS ALONG WHITE MATTER TRACTS

Diffusion tensor imaging provides important information on tissue structure and orientation of fiber tracts in brain white matter in vivo. It results in diffusion tensors, which are 3×3 symmetric positive definite (SPD) matrices, along fiber bundles. This paper develops a functional data analysis framework to model diffusion tensors along fiber tracts as functional data in a Riemannian manifold with a set of covariates of interest, such as age and gender. We propose a statistical model with varying coefficient functions to characterize the dynamic association between functional SPD matrix-valued responses and covariates. We calculate weighted least squares estimators of the varying coefficient functions for the Log-Euclidean metric in the space of SPD matrices. We also develop a global test statistic to test specific hypotheses about these coefficient functions and construct their simultaneous confidence bands. Simulated data are further used to examine the finite sample performance of the estimated varying co-efficient functions. We apply our model to study potential gender differences and find a statistically significant aspect of the development of diffusion tensors along the right internal capsule tract in a clinical study of neurodevelopment.

Variable Selection in Generalized Functional Linear Models

Modern research data, where a large number of functional predictors is collected on few subjects are becoming increasingly common. In this paper we propose a variable selection technique, when the predictors are functional and the response is scalar. Our approach is based on adopting a generalized functional linear model framework and using a penalized likelihood method that simultaneously controls the sparsity of the model and the smoothness of the corresponding coefficient functions by adequate penalization. The methodology is characterized by high predictive accuracy, and yields interpretable models, while retaining computational efficiency. The proposed method is investigated numerically in finite samples, and applied to a diffusion tensor imaging tractography data set and a chemometric data set.

Spatial variability and changes of metabolite concentrations in the cortico-spinal tract in multiple sclerosis using coronal CSI

We characterized metabolic changes along the cortico-spinal tract (CST) in multiple sclerosis (MS) patients using a novel application of chemical shift imaging (CSI) and considering the spatial variation of metabolite levels. Thirteen relapsing-remitting (RR) and 13 primary-progressive (PP) MS patients and 16 controls underwent (1)H-MR CSI, which was applied to coronal-oblique scans to sample the entire CST. The concentrations of the main metabolites, i.e., N-acetyl-aspartate, myo-Inositol (Ins), choline containing compounds (Cho) and creatine and phosphocreatine (Cr), were calculated within voxels placed in regions where the CST is located, from cerebral peduncle to corona radiata. Differences in metabolite concentrations between groups and associations between metabolite concentrations and disability were investigated, allowing for the spatial variability of metabolite concentrations in the statistical model. RRMS patients showed higher CST Cho concentration than controls, and higher CST Ins concentration than PPMS, suggesting greater inflammation and glial proliferation in the RR than in the PP course. In RRMS, a significant, albeit modest, association between greater Ins concentration and greater disability suggested that gliosis may be relevant to disability. In PPMS, lower CST Cho and Cr concentrations correlated with greater disability, suggesting that in the progressive stage of the disease, inflammation declines and energy metabolism reduces. Attention to the spatial variation of metabolite concentrations made it possible to detect in patients a greater increase in Cr concentration towards the superior voxels as compared to controls and a stronger association between Cho and disability, suggesting that this step improves our ability to identify clinically relevant metabolic changes.

Corrected confidence bands for functional data using principal components

Functional principal components (FPC) analysis is widely used to decompose and express functional observations. Curve estimates implicitly condition on basis functions and other quantities derived from FPC decompositions; however these objects are unknown in practice. In this article, we propose a method for obtaining correct curve estimates by accounting for uncertainty in FPC decompositions. Additionally, pointwise and simultaneous confidence intervals that account for both model- and decomposition-based variability are constructed. Standard mixed model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition. Iterated expectation and variance formulas combine model-based conditional estimates across the distribution of decompositions. A bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities. Our method compares favorably to competing approaches in simulation studies that include both densely and sparsely observed functions. We apply our method to sparse observations of CD4 cell counts and to dense white-matter tract profiles. Code for the analyses and simulations is publicly available, and our method is implemented in the R package refund on CRAN.

Longitudinal Penalized Functional Regression for Cognitive Outcomes on Neuronal Tract Measurements

We describe and analyze a longitudinal diffusion tensor imaging (DTI) study relating changes in the microstructure of intracranial white matter tracts to cognitive disability in multiple sclerosis patients. In this application the scalar outcome and the functional exposure are measured longitudinally. This data structure is new and raises challenges that cannot be addressed with current methods and software. To analyze the data, we introduce a penalized functional regression model and inferential tools designed specifically for these emerging types of data. Our proposed model extends the Generalized Linear Mixed Model by adding functional predictors; this method is computationally feasible and is applicable when the functional predictors are measured densely, sparsely or with error. An online appendix compares two implementations, one likelihood-based and the other Bayesian, and provides the software used in simulations; the likelihood-based implementation is included as the lpfr() function in the R package refund available on CRAN.