Methods for scalar-on-function regression

A Bayesian semi-parametric scalar-on-function regression with measurement error using instrumental variables

Wearable devices such as the ActiGraph are now commonly used in research to monitor or track physical activity. This trend corresponds with the growing need to assess the relationships between physical activity and health outcomes, such as obesity, accurately. Device-based physical activity measures are best treated as functions when assessing their associations with scalar-valued outcomes such as body mass index. Scalar-on-function regression (SoFR) is a suitable regression model in this setting. Most estimation approaches in SoFR assume that the measurement error in functional covariates is white noise. Violating this assumption can lead to underestimating model parameters. There are limited approaches to correcting measurement errors for frequentist methods and none for Bayesian methods in this area. We present a non-parametric Bayesian measurement error-corrected SoFR model that relaxes all the constraining assumptions often involved with these models. Our estimation relies on an instrumental variable allowing a time-varying biasing factor, a significant departure from the current generalized method of moment (GMM) approach. Our proposed method also permits model-based grouping of the functional covariate following measurement error correction. This grouping of the measurement error-corrected functional covariate allows additional ease of interpretation of how the different groups differ. Our method is easy to implement, and we demonstrate its finite sample properties in extensive simulations. Finally, we applied our method to data from the National Health and Examination Survey to assess the relationship between wearable device-based measures of physical activity and body mass index in adults in the United States.

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.

Functional support vector machine

Linear and generalized linear scalar-on-function modeling have been commonly used to understand the relationship between a scalar response variable (e.g. continuous, binary outcomes) and functional predictors. Such techniques are sensitive to model misspecification when the relationship between the response variable and the functional predictors is complex. On the other hand, support vector machines (SVMs) are among the most robust prediction models but do not take account of the high correlations between repeated measurements and cannot be used for irregular data. In this work, we propose a novel method to integrate functional principal component analysis with SVM techniques for classification and regression to account for the continuous nature of functional data and the nonlinear relationship between the scalar response variable and the functional predictors. We demonstrate the performance of our method through extensive simulation experiments and two real data applications: the classification of alcoholics using electroencephalography signals and the prediction of glucobrassicin concentration using near-infrared reflectance spectroscopy. Our methods especially have more advantages when the measurement errors in functional predictors are relatively large.

A Study of Pupil Response to Light as a Digital Biomarker of Recent Cannabis Use

Introduction

Given the traffic safety and occupational injury prevention implications associated with cannabis impairment, there is a need for objective and validated measures of recent cannabis use. Pupillary light response may offer an approach for detection.

Method

Eighty-four participants (mean age: 32, 42% female) with daily, occasional, and no-use cannabis use histories participated in pupillary light response tests before and after smoking cannabis ad libitum or relaxing for 15 min (no use). The impact of recent cannabis consumption on trajectories of the pupillary light response was modeled using functional data analysis tools. Logistic regression models for detecting recent cannabis use were compared, and average pupil trajectories across cannabis use groups and times since light test administration were estimated.

Results

Models revealed small, significant differences in pupil response to light after cannabis use comparing the occasional use group to the no-use control group, and similar statistically significant differences in pupil response patterns comparing the daily use group to the no-use comparison group. Trajectories of pupillary light response estimated using functional data analysis found that acute cannabis smoking was associated with less initial and sustained pupil constriction compared to no cannabis smoking.

Conclusion

These analyses show the promise of pairing pupillary light response and functional data analysis methods to assess recent cannabis use.

Functional proportional hazards mixture cure model with applications in cancer mortality in NHANES and post ICU recovery

We develop a functional proportional hazards mixture cure model with scalar and functional covariates measured at the baseline. The mixture cure model, useful in studying populations with a cure fraction of a particular event of interest is extended to functional data. We employ the expectation-maximization algorithm and develop a semiparametric penalized spline-based approach to estimate the dynamic functional coefficients of the incidence and the latency part. The proposed method is computationally efficient and simultaneously incorporates smoothness in the estimated functional coefficients via roughness penalty. Simulation studies illustrate a satisfactory performance of the proposed method in accurately estimating the model parameters and the baseline survival function. Finally, the clinical potential of the model is demonstrated in two real data examples that incorporate rich high-dimensional biomedical signals as functional covariates measured at the baseline and constitute novel domains to apply cure survival models in contemporary medical situations. In particular, we analyze (i) minute-by-minute physical activity data from the National Health And Nutrition Examination Survey 2003-2006 to study the association between diurnal patterns of physical activity at baseline and all cancer mortality through 2019 while adjusting for other biological factors; (ii) the impact of daily functional measures of disease severity collected in the intensive care unit on post intensive care unit recovery and mortality event. Our findings provide novel epidemiological insights into the association between daily patterns of physical activity and cancer mortality. Software implementation and illustration of the proposed estimation method are provided in R.

Quantile Index Biomarkers Based on Single-Cell Expression Data

Current histocytometry methods enable single-cell quantification of biomolecules in tumor tissue sections by multiple detection technologies, including multiplex fluorescence-based immunohistochemistry or in situ hybridization. Quantitative pathology platforms can provide distributions of cellular signal intensity (CSI) levels of biomolecules across the entire cell populations of interest within the sampled tumor tissue. However, the heterogeneity of CSI levels is usually ignored, and the simple mean signal intensity value is considered a cancer biomarker. Here we consider the entire distribution of CSI expression levels of a given biomolecule in the cancer cell population as a predictor of clinical outcome. The proposed quantile index (QI) biomarker is defined as the weighted average of CSI distribution quantiles in individual tumors. The weight for each quantile is determined by fitting a functional regression model for a clinical outcome. That is, the weights are optimized so that the resulting QI has the highest power to predict a relevant clinical outcome. The proposed QI biomarkers were derived for proteins expressed in cancer cells of malignant breast tumors and demonstrated improved prognostic value compared with the standard mean signal intensity predictors. The R package Qindex implementing QI biomarkers has been developed. The proposed approach is not limited to immunohistochemistry data and can be based on any cell-level expressions of proteins or nucleic acids.

Distributional data analysis via quantile functions and its application to modeling digital biomarkers of gait in Alzheimer's Disease

With the advent of continuous health monitoring with wearable devices, users now generate their unique streams of continuous data such as minute-level step counts or heartbeats. Summarizing these streams via scalar summaries often ignores the distributional nature of wearable data and almost unavoidably leads to the loss of critical information. We propose to capture the distributional nature of wearable data via user-specific quantile functions (QF) and use these QFs as predictors in scalar-on-quantile-function-regression (SOQFR). As an alternative approach, we also propose to represent QFs via user-specific L-moments, robust rank-based analogs of traditional moments, and use L-moments as predictors in SOQFR (SOQFR-L). These two approaches provide two mutually consistent interpretations: in terms of quantile levels by SOQFR and in terms of L-moments by SOQFR-L. We also demonstrate how to deal with multi-modal distributional data via Joint and Individual Variation Explained using L-moments. The proposed methods are illustrated in a study of association of digital gait biomarkers with cognitive function in Alzheimers disease. Our analysis shows that the proposed methods demonstrate higher predictive performance and attain much stronger associations with clinical cognitive scales compared to simple distributional summaries.

Robust scalar-on-function partial quantile regression

Compared with the conditional mean regression-based scalar-on-function regression model, the scalar-on-function quantile regression is robust to outliers in the response variable. However, it is susceptible to outliers in the functional predictor (called leverage points). This is because the influence function of the regression quantiles is bounded in the response variable but unbounded in the predictor space. The leverage points may alter the eigenstructure of the predictor matrix, leading to poor estimation and prediction results. This study proposes a robust procedure to estimate the model parameters in the scalar-on-function quantile regression method and produce reliable predictions in the presence of both outliers and leverage points. The proposed method is based on a functional partial quantile regression procedure. We propose a weighted partial quantile covariance to obtain functional partial quantile components of the scalar-on-function quantile regression model. After the decomposition, the model parameters are estimated via a weighted loss function, where the robustness is obtained by iteratively reweighting the partial quantile components. The estimation and prediction performance of the proposed method is evaluated by a series of Monte-Carlo experiments and an empirical data example. The results are compared favorably with several existing methods. The method is implemented in an R package robfpqr.

Circulating microvesicles correlate with radiation proctitis complication after radiotherapy

In a large retrospective study, we assessed the putative use of circulating microvesicles (MVs), as innovative biomarkers of radiation toxicity in a cohort of 208 patients with prostate adenocarcinoma overexposed to radiation. The level of platelet (P)-, monocyte (M)- and endothelial (E)-derived MVs were assessed by flow cytometry. Rectal bleeding toxicity scores were collected at the time of blood sampling and during the routine follow-up and were tested for association with MVs using a multivariate logistic regression. MVs dosimetric correlation was investigated using dose volume histograms information available for a subset of 36 patients. The number of PMVs was significantly increased in patients with highest toxicity grades compared to lower grades. Risk prediction analysis revealed that increased numbers of PMVs, and an increased amount of MMVs relative to EMVs, were associated with worst rectal bleeding grade compared to the time of blood sampling. Moreover, a significant correlation was found between PMV and MMV numbers, with the range of doses up to the median exposure (40 Gy) of bladder/rectum and anterior rectal wall, respectively. MVs could be considered as new biomarkers to improve the identification of patients with high toxicity grade and may be instrumental for the prognosis of radiation therapy complications.

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.

A function-based approach to model the measurement error in wearable devices

Physical activity (PA) is an important risk factor for many health outcomes. Wearable-devices such as accelerometers are increasingly used in biomedical studies to understand the associations between PA and health outcomes. Statistical analyses involving accelerometer data are challenging due to the following three characteristics: (i) high-dimensionality, (ii) temporal dependence, and (iii) measurement error. To address these challenges we treat accelerometer-based measures of PA as a single function-valued covariate prone to measurement error. Specifically, in order to determine the relationship between PA and a health outcome of interest, we propose a regression model with a functional covariate that accounts for measurement error. Using regression calibration, we develop a two-step estimation method for the model parameters and establish their consistency. A test is also proposed to test the significance of the estimated model parameters. Simulation studies are conducted to compare the proposed methods with existing alternative approaches under varying scenarios. Finally, the developed methods are used to assess the relationship between PA intensity and BMI obtained from the National Health and Nutrition Examination Survey data.

Representation Theorem and Functional CLT for RKHS-Based Function-on-Function Regressions

We investigate a nonparametric, varying coefficient regression approach for modeling and estimating the regression effects caused by two functionally correlated datasets. Due to modern biomedical technology to measure multiple patient features during a time interval or intermittently at several discrete time points to review underlying biological mechanisms, statistical models that do not properly incorporate interventions and their dynamic responses may lead to biased estimates of the intervention effects. We propose a shared parameter change point function-on-function regression model to evaluate the pre- and post-intervention time trends and develop a likelihood-based method for estimating the intervention effects and other parameters. We also propose new methods for estimating and hypothesis testing regression parameters for functional data via reproducing kernel Hilbert space. The estimators of regression parameters are closed-form without computation of the inverse of a large matrix, and hence are less computationally demanding and more applicable. By establishing a representation theorem and a functional central limit theorem, the asymptotic properties of the proposed estimators are obtained, and the corresponding hypothesis tests are proposed. Application and the statistical properties of our method are demonstrated through an immunotherapy clinical trial of advanced myeloma and simulation studies.

Scalar on time-by-distribution regression and its application for modelling associations between daily-living physical activity and cognitive functions in Alzheimer's Disease

Wearable data is a rich source of information that can provide a deeper understanding of links between human behaviors and human health. Existing modelling approaches use wearable data summarized at subject level via scalar summaries in regression, temporal (time-of-day) curves in functional data analysis (FDA), and distributions in distributional data analysis (DDA). We propose to capture temporally local distributional information in wearable data using subject-specific time-by-distribution (TD) data objects. Specifically, we develop scalar on time-by-distribution regression (SOTDR) to model associations between scalar response of interest such as health outcomes or disease status and TD predictors. Additionally, we show that TD data objects can be parsimoniously represented via a collection of time-varying L-moments that capture distributional changes over the time-of-day. The proposed method is applied to the accelerometry study of mild Alzheimer's disease (AD). We found that mild AD is significantly associated with reduced upper quantile levels of physical activity, particularly during morning hours. In-sample cross validation demonstrated that TD predictors attain much stronger associations with clinical cognitive scales of attention, verbal memory, and executive function when compared to predictors summarized via scalar total activity counts, temporal functional curves, and quantile functions. Taken together, the present results suggest that SOTDR analysis provides novel insights into cognitive function and AD.

Bayesian flexible beta regression model with functional covariate

Standard parametric regression models are unsuitable when the aim is to predict a bounded continuous response, such as a proportion/percentage or a rate. A possible solution is the flexible beta regression model which is based on a special mixture of betas designed to cope with (though not limited to) bimodality, heavy tails, and outlying observations. This work introduces such a model in the case of a functional covariate, motivated by a spectrometric analysis on milk specimens. Estimation issues are dealt with through a combination of standard basis expansion and Markov chains Monte Carlo techniques. Specifically, the selection of the most significant coefficients of the expansion is done through Bayesian variable selection methods that take advantage of shrinkage priors. The effectiveness of the proposal is illustrated with simulation studies and the application on spectrometric data.

Estimation and model selection for nonparametric function-on-function regression

Regression models with a functional response and functional covariate have received significant attention recently. While various nonparametric and semiparametric models have been developed, there is an urgent need for model selection and diagnostic methods. In this article, we develop a unified framework for estimation and model selection in nonparametric function-on-function regression. We propose a general nonparametric functional regression model with the model space constructed through smoothing spline analysis of variance (SS ANOVA). The proposed model reduces to some of the existing models when selected components in the SS ANOVA decomposition are eliminated. We propose new estimation procedures under either L ₁ or L ₂ penalty and show that the combination of the SS ANOVA decomposition and L ₁ penalty provides powerful tools for model selection and diagnostics. We establish consistency and convergence rates for estimates of the regression function and each component in its decomposition under both the L ₁ and L ₂ penalties. Simulation studies and real examples show that the proposed methods perform well. Technical details and additional simulation results are available in online supplementary materials.

Bayesian inference and dynamic prediction of multivariate joint model with functional data: An application to Alzheimer's disease

Alzheimer's disease (AD) is a severe neurodegenerative disorder impairing multiple domains, for example, cognition and behavior. Assessing the risk of AD progression and initiating timely interventions at early stages are critical to improve the quality of life for AD patients. Due to the heterogeneous nature and complex mechanisms of AD, one single longitudinal outcome is insufficient to assess AD severity and disease progression. Therefore, AD studies collect multiple longitudinal outcomes, including cognitive and behavioral measurements, as well as structural brain images such as magnetic resonance imaging (MRI). How to utilize the multivariate longitudinal outcomes and MRI data to make efficient statistical inference and prediction is an open question. In this article, we propose a multivariate joint model with functional data (MJM-FD) framework that relates multiple correlated longitudinal outcomes to a survival outcome, and use the scalar-on-function regression method to include voxel-based whole-brain MRI data as functional predictors in both longitudinal and survival models. We adopt a Bayesian paradigm to make statistical inference and develop a dynamic prediction framework to predict an individual's future longitudinal outcomes and risk of a survival event. We validate the MJM-FD framework through extensive simulation studies and apply it to the motivating Alzheimer's Disease Neuroimaging Initiative (ADNI) study.

[Trend, spatialization and circumstances associated with violence against vulnerable populations in Brazil, between 2009 and 2017]

The scope of this study was to analyze the trend, spatialization and circumstances associated with violence against vulnerable populations in Brazil between 2009 and 2017. Descriptive analysis, spatialization and polynomial regression were used. A total of 1,116,331 cases of violence were notified, with an increase of 667.5% over the period. In all populations under research there was a predominance of physical violence against women with low levels of schooling, perpetrated by men in residences. Sexual and psychological violence was highlighted in female and LGBT populations. Negligence, however, was significant among people with disabilities, children/adolescents, and the elderly. At least 65% of the notifications of violence in Brazil were characterized as indiscriminate regarding the motivation behind the act. Tendency analysis showed a rise of interpersonal violence against women (+8.5%), disabled people (+7.1%), elderly individuals (+55.4%), natives (+99.4%), LGBT people (+3.4%) and racial violence (+30.2%), except for violence against underaged individuals, which showed a decrease (-11.4%). In North and Northwest regions violence against women and underaged individuals prevailed, while in Midwest, South and Southeast regions the LGBT, disabled and elderly populations were most subject to abuse.

Clinical assessment of W-band spectroscopy for non-invasive detection and monitoring of sustained hyperglycemia

HbA1c is the gold standard test for monitoring medium/long term glycemia conditions in diabetes care, which is a critical factor in reducing the risk of chronic diabetes complications. Current technologies for measuring HbA1c concentration are invasive and adequate assays are still limited to laboratory-based methods that are not widely available worldwide. The development of a non-invasive diagnostic tool for HbA1c concentration can lead to the decrease of the rate of undiagnosed cases and facilitate early detection in diabetes care. We present a preliminary validation diagnostic study of W-band spectroscopy for detection and monitoring of sustained hyperglycemia, using the HbA1c concentration as reference. A group of 20 patients with type 1 diabetes mellitus and 10 healthy subjects were non-invasively assessed at three different visits over a period of 7 months by a millimeter-wave spectrometer (transmission mode) operating across the full W-band. The relationship between the W-band spectral profile and the HbA1c concentration is studied using longitudinal and non-longitudinal functional data analysis methods. A potential blind discrimination between patients with or without diabetes is obtained, and more importantly, an excellent relation (R-squared = 0.97) between the non-invasive assessment and the HbA1c measure is achieved. Such results support that W-band spectroscopy has great potential for developing a non-invasive diagnostic tool for in-vivo HbA1c concentration monitoring in humans.

A functional mixed model for scalar on function regression with application to a functional MRI study

Motivated by a functional magnetic resonance imaging (fMRI) study, we propose a new functional mixed model for scalar on function regression. The model extends the standard scalar on function regression for repeated outcomes by incorporating subject-specific random functional effects. Using functional principal component analysis, the new model can be reformulated as a mixed effects model and thus easily fit. A test is also proposed to assess the existence of the subject-specific random functional effects. We evaluate the performance of the model and test via a simulation study, as well as on data from the motivating fMRI study of thermal pain. The data application indicates significant subject-specific effects of the human brain hemodynamics related to pain and provides insights on how the effects might differ across subjects.

Associations of actigraphic sleep and circadian rest/activity rhythms with cognition in the early phase of Alzheimer's disease

Study Objectives

To compare sleep and circadian rest/activity rhythms (RARs), quantified by standard and novel actigraphic metrics, between controls and participants with mild cognitive impairment (MCI), and to examine the cross-sectional relationships between these measures and cognition.

Methods

Actigraphy data were collected in 179 older individuals (mean age = 72.6 years) with normal cognition (n = 153) and MCI (n = 26). Sleep parameters (e.g. sleep efficiency), and standard nonparametric RARs (e.g. interdaily stability) were generated. Functional principal component analysis (fPCA) was used to generate three novel RAR metrics (fPC1, fPC2, and fPC3). Cognitive composite scores reflecting episodic memory and executive function were derived using factor analysis. Regression models compared sleep and RAR parameters between diagnostic groups and their association with cognitive performance.

Results

Compared to controls, the MCI group exhibited lower levels of the standard RAR parameter: relative amplitude and fPC3—a novel RAR whereby lower scores reflected a lower rhythm peak, as well as greater nighttime activity and less activity in the morning. Across groups, several standard RAR parameters (e.g. interdaily stability) and fPC3 were associated with better episodic memory and executive function performance. Additionally, several standard RAR measures (e.g. relative amplitude) and the novel RAR measure fPC1 (reflecting the total volume of activity and rhythm strength) were associated with better executive function performance.

Conclusions

Individuals with MCI have altered circadian RARs compared to controls, including the novel RAR metric fPC3, reflecting greater nighttime activity and less activity in the morning compared to mean values. Additionally, these measures are significantly associated with cognitive performance.

Time series extrinsic regression: Predicting numeric values from time series data

This paper studies time series extrinsic regression (TSER): a regression task of which the aim is to learn the relationship between a time series and a continuous scalar variable; a task closely related to time series classification (TSC), which aims to learn the relationship between a time series and a categorical class label. This task generalizes time series forecasting, relaxing the requirement that the value predicted be a future value of the input series or primarily depend on more recent values. In this paper, we motivate and study this task, and benchmark existing solutions and adaptations of TSC algorithms on a novel archive of 19 TSER datasets which we have assembled. Our results show that the state-of-the-art TSC algorithm Rocket, when adapted for regression, achieves the highest overall accuracy compared to adaptations of other TSC algorithms and state-of-the-art machine learning (ML) algorithms such as XGBoost, Random Forest and Support Vector Regression. More importantly, we show that much research is needed in this field to improve the accuracy of ML models. We also find evidence that further research has excellent prospects of improving upon these straightforward baselines.

Diurnal Physical Activity Patterns across Ages in a Large UK Based Cohort: The UK Biobank Study

The ability of individuals to engage in physical activity is a critical component of overall health and quality of life. However, there is a natural decline in physical activity associated with the aging process. Establishing normative trends of physical activity in aging populations is essential to developing public health guidelines and informing clinical perspectives regarding individuals' levels of physical activity. Beyond overall quantity of physical activity, patterns regarding the timing of activity provide additional insights into latent health status. Wearable accelerometers, paired with statistical methods from functional data analysis, provide the means to estimate diurnal patterns in physical activity. To date, these methods have been only applied to study aging trends in populations based in the United States. Here, we apply curve registration and functional regression to 24 h activity profiles for 88,793 men (N = 39,255) and women (N = 49,538) ages 42-78 from the UK Biobank accelerometer study to understand how physical activity patterns vary across ages and by gender. Our analysis finds that daily patterns in both the volume of physical activity and probability of being active change with age, and that there are marked gender differences in these trends. This work represents the largest-ever population analyzed using tools of this kind, and suggest that aging trends in physical activity are reproducible in different populations across countries.

Separating the impact of gradual lockdown measures on air pollutants from seasonal variability

Analysis of near-surface measurements at several measuring points in Graz, Austria, reveals the impact of restrictive measures during the COVID-19 pandemic on the emission of atmospheric pollutants. We quantify the effects at traffic hotspots, industrial and residential areas. Using historical data collected over several years, we are able to account for meteorological and seasonal confounders. Our analysis is based on daily means as well as intraday pollution level curves. Nitrogen dioxide (NO₂) has decreased drastically while the levels of particulate matter PM₁₀ and carbon monoxide (CO) mostly exhibit little change. Traffic data shows that the decrease in traffic frequency is parallel to the decline in the levels of NO₂ and NO.

Extracting scalar measures from functional data with applications to placebo response

In controlled and observational studies, outcome measures are often observed longitudinally. Such data are difficult to compare among units directly because there is no natural ordering of curves. This is relevant not only in clinical trials, where typically the goal is to evaluate the relative efficacy of treatments on average, but also in the growing and increasingly important area of personalized medicine, where treatment decisions are optimized with respect to a relevant patient outcome. In personalized medicine, there are no methods for optimizing treatment decision rules using longitudinal outcomes, e.g., symptom trajectories, because of the lack of a natural ordering of curves. A typical practice is to summarize the longitudinal response by a scalar outcome that can then be compared across patients, treatments, etc. We describe some of the summaries that are in common use, especially in clinical trials. We consider a general summary measure (weighted average tangent slope) with weights that can be chosen to optimize specific inference depending on the application. We illustrate the methodology on a study of depression treatment, in which it is difficult to separate placebo effects from the specific effects of the antidepressant. We argue that this approach provides a better summary for estimating the benefits of an active treatment than traditional non-weighted averages.

On the Number of Independent Pieces of Information in a Functional Linear Model with a Scalar Response

In a functional linear model (FLM) with scalar response, the parameter curve quantifies the relationship between a functional explanatory variable and a scalar response. While these models can be ill-posed, a penalized regression spline approach may be used to obtain an estimate of the parameter curve. The penalized regression spline estimate will be dependent on the value of a smoothing parameter. However, the ability to obtain a reasonable parameter curve estimate is reliant on how much information is present in the covariate functions for estimating the parameter curve. We propose to quantify the information present in the covariate functions to estimate the parameter curve. In addition, we examine the influence of this information on the stability of the parameter curve estimator and on the performance of smoothing parameter selection methods in a FLM with a scalar response.

Quantifying uncertainty in brain-predicted age using scalar-on-image quantile regression

Prediction of subject age from brain anatomical MRI has the potential to provide a sensitive summary of brain changes, indicative of different neurodegenerative diseases. However, existing studies typically neglect the uncertainty of these predictions. In this work we take into account this uncertainty by applying methods of functional data analysis. We propose a penalised functional quantile regression model of age on brain structure with cognitively normal (CN) subjects in the Alzheimer's Disease Neuroimaging Initiative (ADNI), and use it to predict brain age in Mild Cognitive Impairment (MCI) and Alzheimer's Disease (AD) subjects. Unlike the machine learning approaches available in the literature of brain age prediction, which provide only point predictions, the outcome of our model is a prediction interval for each subject.

Evaluation of longitudinal surrogate markers

The use of surrogate markers to examine the effectiveness of a treatment has the potential to decrease study length and identify effective treatments more quickly. Most available methods to investigate the usefulness of a surrogate marker involve restrictive parametric assumptions and tend to focus on settings where the surrogate is measured at a single point in time. However, in many clinical settings, the potential surrogate marker is often measured repeatedly over time, and thus, the surrogate marker information is a trajectory of measurements. In addition, it is often difficult in practice to correctly specify the relationship between a treatment, primary outcome, and surrogate marker trajectory. In this paper, we propose a model-free definition for the proportion of the treatment effect on the primary outcome that is explained by the treatment effect on the longitudinal surrogate markers. We propose three novel flexible methods to estimate this proportion, develop the asymptotic properties of our estimators, and investigate the robustness of the estimators under multiple settings via a simulation study. We apply our proposed procedures to an AIDS clinical trial dataset to examine a trajectory of CD4 counts as a potential surrogate.

Efficient Estimation for Varying-Coefficient Mixed Effects Models with Functional Response Data

In this article, we focus on the estimation of varying-coefficient mixed effects models for longitudinal and sparse functional response data, by using the generalized least squares method coupling a modified local kernel smoothing technique. This approach provides a useful framework that simultaneously takes into account the within-subject covariance and all observation information in the estimation to improve efficiency. We establish both uniform consistency and pointwise asymptotic normality for the proposed estimators of varying-coefficient functions. Numerical studies are carried out to illustrate the finite sample performance of the proposed procedure. An application to the white matter tract dataset obtained from Alzheimer’s Disease Neuroimaging Initiative study is also provided.

Predicting plant disease epidemics from functionally represented weather series

Epidemics are often triggered by specific weather patterns favouring the pathogen on susceptible hosts. For plant diseases, models predicting epidemics have therefore often emphasized the identification of early season weather patterns that are correlated with a disease outcome at some later point. Toward that end, window-pane analysis is an exhaustive search algorithm traditionally used in plant pathology for mining correlations in a weather series with respect to a disease endpoint. Here we show, with reference to Fusarium head blight (FHB) of wheat, that a functional approach is a more principled analytical method for understanding the relationship between disease epidemics and environmental conditions over an extended time series. We used scalar-on-function regression to model a binary outcome (FHB epidemic or non-epidemic) relative to weather time series spanning 140 days relative to flowering (when FHB infection primarily occurs). The functional models overall fit the data better than previously described standard logistic regression (lr) models. Periods much earlier than heretofore realized were associated with FHB epidemics. The findings were used to create novel weather summary variables which, when incorporated into lr models, yielded a new set of models that performed as well as existing lr models for real-time predictions of disease risk. This article is part of the theme issue 'Modelling infectious disease outbreaks in humans, animals and plants: approaches and important themes'. This issue is linked with the subsequent theme issue 'Modelling infectious disease outbreaks in humans, animals and plants: epidemic forecasting and control'.

Bayesian bandwidth estimation and semi-metric selection for a functional partial linear model with unknown error density

This study examines the optimal selections of bandwidth and semi-metric for a functional partial linear model. Our proposed method begins by estimating the unknown error density using a kernel density estimator of residuals, where the regression function, consisting of parametric and nonparametric components, can be estimated by functional principal component and functional Nadayara-Watson estimators. The estimation accuracy of the regression function and error density crucially depends on the optimal estimations of bandwidth and semi-metric. A Bayesian method is utilized to simultaneously estimate the bandwidths in the regression function and kernel error density by minimizing the Kullback-Leibler divergence. For estimating the regression function and error density, a series of simulation studies demonstrate that the functional partial linear model gives improved estimation and forecast accuracies compared with the functional principal component regression and functional nonparametric regression. Using a spectroscopy dataset, the functional partial linear model yields better forecast accuracy than some commonly used functional regression models. As a by-product of the Bayesian method, a pointwise prediction interval can be obtained, and marginal likelihood can be used to select the optimal semi-metric.