1
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Gertheiss J, Rügamer D, Liew BXW, Greven S. Functional Data Analysis: An Introduction and Recent Developments. Biom J 2024; 66:e202300363. [PMID: 39330918 DOI: 10.1002/bimj.202300363] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2023] [Revised: 05/17/2024] [Accepted: 05/27/2024] [Indexed: 09/28/2024]
Abstract
Functional data analysis (FDA) is a statistical framework that allows for the analysis of curves, images, or functions on higher dimensional domains. The goals of FDA, such as descriptive analyses, classification, and regression, are generally the same as for statistical analyses of scalar-valued or multivariate data, but FDA brings additional challenges due to the high- and infinite dimensionality of observations and parameters, respectively. This paper provides an introduction to FDA, including a description of the most common statistical analysis techniques, their respective software implementations, and some recent developments in the field. The paper covers fundamental concepts such as descriptives and outliers, smoothing, amplitude and phase variation, and functional principal component analysis. It also discusses functional regression, statistical inference with functional data, functional classification and clustering, and machine learning approaches for functional data analysis. The methods discussed in this paper are widely applicable in fields such as medicine, biophysics, neuroscience, and chemistry and are increasingly relevant due to the widespread use of technologies that allow for the collection of functional data. Sparse functional data methods are also relevant for longitudinal data analysis. All presented methods are demonstrated using available software in R by analyzing a dataset on human motion and motor control. To facilitate the understanding of the methods, their implementation, and hands-on application, the code for these practical examples is made available through a code and data supplement and on GitHub.
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Affiliation(s)
- Jan Gertheiss
- Departmesnt of Mathematics and Statistics, School of Economics and Social Sciences, Helmut Schmidt University, Hamburg, Germany
| | - David Rügamer
- Department of Statistics, LMU Munich, Munich, Germany
- Munich Center for Machine Learning, Munich, Germany
| | - Bernard X W Liew
- School of Sport, Rehabilitation and Exercise Sciences, University of Essex, Essex, UK
| | - Sonja Greven
- Chair of Statistics, School of Business and Economics, Humboldt-Universität zu Berlin, Berlin, Germany
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2
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Zou H, Xiao L, Zeng D, Luo S. Multivariate functional mixed model with MRI data: An application to Alzheimer's disease. Stat Med 2023; 42:1492-1511. [PMID: 36805635 PMCID: PMC10133011 DOI: 10.1002/sim.9683] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2022] [Revised: 11/09/2022] [Accepted: 01/26/2023] [Indexed: 02/22/2023]
Abstract
Alzheimer's Disease (AD) is the leading cause of dementia and impairment in various domains. Recent AD studies, (ie, Alzheimer's Disease Neuroimaging Initiative (ADNI) study), collect multimodal data, including longitudinal neurological assessments and magnetic resonance imaging (MRI) data, to better study the disease progression. Adopting early interventions is essential to slow AD progression for subjects with mild cognitive impairment (MCI). It is of particular interest to develop an AD predictive model that leverages multimodal data and provides accurate personalized predictions. In this article, we propose a multivariate functional mixed model with MRI data (MFMM-MRI) that simultaneously models longitudinal neurological assessments, baseline MRI data, and the survival outcome (ie, dementia onset) for subjects with MCI at baseline. Two functional forms (the random-effects model and instantaneous model) linking the longitudinal and survival process are investigated. We use Markov Chain Monte Carlo (MCMC) method based on No-U-Turn Sampling (NUTS) algorithm to obtain posterior samples. We develop a dynamic prediction framework that provides accurate personalized predictions of longitudinal trajectories and survival probability. We apply MFMM-MRI to the ADNI study and identify significant associations among longitudinal outcomes, MRI data, and the risk of dementia onset. The instantaneous model with voxels from the whole brain has the best prediction performance among all candidate models. The simulation study supports the validity of the estimation and dynamic prediction method.
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Affiliation(s)
- Haotian Zou
- Department of Biostatistics, University of North Carolina, Chapel Hill, North Carolina, United States
| | - Luo Xiao
- Department of Statistics, North Carolina State University, North Carolina, United States
| | - Donglin Zeng
- Department of Biostatistics, University of North Carolina, Chapel Hill, North Carolina, United States
| | - Sheng Luo
- Department of Biostatistics and Bioinformatics, Duke University, North Carolina, United States
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3
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Zhang Y, Wu Y. Robust hypothesis testing in functional linear models. J STAT COMPUT SIM 2023. [DOI: 10.1080/00949655.2023.2195657] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/08/2023]
Affiliation(s)
- Yan Zhang
- Department of Mathematics and Statistics, York University, Toronto, Canada
| | - Yuehua Wu
- Department of Mathematics and Statistics, York University, Toronto, Canada
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4
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Wang W, Sun Y, Wang HJ. Latent group detection in functional partially linear regression models. Biometrics 2023; 79:280-291. [PMID: 34482542 DOI: 10.1111/biom.13557] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2020] [Revised: 08/06/2021] [Accepted: 08/19/2021] [Indexed: 11/28/2022]
Abstract
In this paper, we propose a functional partially linear regression model with latent group structures to accommodate the heterogeneous relationship between a scalar response and functional covariates. The proposed model is motivated by a salinity tolerance study of barley families, whose main objective is to detect salinity tolerant barley plants. Our model is flexible, allowing for heterogeneous functional coefficients while being efficient by pooling information within a group for estimation. We develop an algorithm in the spirit of the K-means clustering to identify latent groups of the subjects under study. We establish the consistency of the proposed estimator, derive the convergence rate and the asymptotic distribution, and develop inference procedures. We show by simulation studies that the proposed method has higher accuracy for recovering latent groups and for estimating the functional coefficients than existing methods. The analysis of the barley data shows that the proposed method can help identify groups of barley families with different salinity tolerant abilities.
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Affiliation(s)
- Wu Wang
- Center for Applied Statistics and School of Statistics, Renmin University of China, Beijing, China
| | - Ying Sun
- Statistics Program, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
| | - Huixia Judy Wang
- Department of Statistics, The George Washington University, Washington, DC, USA
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5
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Ghosal R, Varma VR, Volfson D, Urbanek J, Hausdorff JM, Watts A, Zipunnikov V. Scalar on time-by-distribution regression and its application for modelling associations between daily-living physical activity and cognitive functions in Alzheimer's Disease. Sci Rep 2022; 12:11558. [PMID: 35798763 PMCID: PMC9263176 DOI: 10.1038/s41598-022-15528-5] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2022] [Accepted: 06/24/2022] [Indexed: 11/26/2022] Open
Abstract
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.
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Affiliation(s)
- Rahul Ghosal
- Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD, USA.
| | - Vijay R Varma
- National Institute on Aging (NIA), National Institutes of Health (NIH), Baltimore, MD, USA
| | - Dmitri Volfson
- Neuroscience Analytics, Computational Biology, Takeda, Cambridge, MA, USA
| | - Jacek Urbanek
- Department of Medicine, Johns Hopkins University School of Medicine, Baltimore, MD, USA
| | - Jeffrey M Hausdorff
- Center for the Study of Movement, Cognition and Mobility, Neurological Institute, Tel Aviv Sourasky Medical Center, Tel Aviv, Israel
- Department of Physical Therapy, Sackler Faculty of Medicine, and Sagol School of Neuroscience, Tel Aviv University, Tel Aviv, Israel
- Rush Alzheimer's Disease Center and Department of Orthopedic Surgery, Rush University Medical Center, Chicago, IL, USA
| | - Amber Watts
- Department of Psychology, University of Kansas, Lawrence, KS, USA
| | - Vadim Zipunnikov
- Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD, USA
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6
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Li M, Wang K, Maity A, Staicu AM. Inference in Functional Linear Quantile Regression. J MULTIVARIATE ANAL 2022; 190:104985. [PMID: 35370319 PMCID: PMC8975129 DOI: 10.1016/j.jmva.2022.104985] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
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 R code is published online at https://github.com/xylimeng/fQR-testing.
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Affiliation(s)
- Meng Li
- Department of Statistics, Rice University, Houston, TX
| | | | - Arnab Maity
- Department of Statistics, North Carolina State University, Raleigh, NC
| | - Ana-Maria Staicu
- Department of Statistics, North Carolina State University, Raleigh, NC
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7
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Xu W, Lin H, Zhang R, Liang H. Two-sample functional linear models with functional responses. J Stat Plan Inference 2022. [DOI: 10.1016/j.jspi.2021.10.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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8
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Xu J, Cui W. A new RKHS-based global testing for functional linear model. Stat Probab Lett 2022. [DOI: 10.1016/j.spl.2021.109277] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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9
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ZHU H, LI Y, LIU B, YAO W, ZHANG R. Extreme quantile estimation for partial functional linear regression models with heavy-tailed distributions. CAN J STAT 2022; 50:267-286. [PMID: 38239624 PMCID: PMC10795494 DOI: 10.1002/cjs.11653] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2020] [Accepted: 03/15/2021] [Indexed: 11/10/2022]
Abstract
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.
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Affiliation(s)
- Hanbing ZHU
- School of Statistics, Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, East China Normal University, Shanghai, China
| | - Yehua LI
- Department of Statistics, University of California, Riverside, California, USA
| | - Baisen LIU
- School of Statistics, Dongbei University of Finance and Economics, Dalian, China
| | - Weixin YAO
- Department of Statistics, University of California, Riverside, California, USA
| | - Riquan ZHANG
- School of Statistics, Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, East China Normal University, Shanghai, China
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10
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Kutta T, Dierickx G, Dette H. Statistical inference for the slope parameter in functional linear regression. Electron J Stat 2022. [DOI: 10.1214/22-ejs2078] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
Affiliation(s)
- Tim Kutta
- Ruhr-Universität Bochum, D-44780 Bochum
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11
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Yi M, Li Z, Tang Y. F‐type testing in functional linear models. Stat (Int Stat Inst) 2021. [DOI: 10.1002/sta4.420] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Affiliation(s)
- Menghan Yi
- School of Statistics East China Normal University Shanghai China
| | - Zaixing Li
- School of Science, China University of Mining and Technology (Beijing) Beijing China
| | - Yanlin Tang
- School of Statistics East China Normal University Shanghai China
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12
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Ma W, Xiao L, Liu B. A functional mixed model for scalar on function regression with application to a functional MRI study. Biostatistics 2021; 22:439-454. [PMID: 31631222 PMCID: PMC8286587 DOI: 10.1093/biostatistics/kxz046] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2018] [Revised: 09/23/2019] [Accepted: 09/25/2019] [Indexed: 11/12/2022] Open
Abstract
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.
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Affiliation(s)
- Wanying Ma
- Department of Statistics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC 27606, USA
| | - Luo Xiao
- Department of Statistics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC 27606, USA
| | - Bowen Liu
- Department of Statistics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC 27606, USA
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13
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Jadhav S, Ma S. An association test for functional data based on Kendall’s Tau. J MULTIVARIATE ANAL 2021. [DOI: 10.1016/j.jmva.2021.104740] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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14
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15
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Li T, Li T, Zhu Z, Zhu H. Regression Analysis of Asynchronous Longitudinal Functional and Scalar Data. J Am Stat Assoc 2020. [DOI: 10.1080/01621459.2020.1844211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Affiliation(s)
- Ting Li
- School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China
| | - Tengfei Li
- Department of Radiology and Biomedical Research Imaging Center (BRIC), University of North Carolina at Chapel Hill, Chapel Hill, NC
| | - Zhongyi Zhu
- Department of Statistics, Fudan University, Shanghai, China
| | - Hongtu Zhu
- Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC
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16
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Yang SJ, Shin H, Lee SH, Lee S. Functional linear regression model with randomly censored data: Predicting conversion time to Alzheimer ’s disease. Comput Stat Data Anal 2020. [DOI: 10.1016/j.csda.2020.107009] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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17
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García‐Portugués E, Álvarez‐Liébana J, Álvarez‐Pérez G, González‐Manteiga W. A goodness‐of‐fit test for the functional linear model with functional response. Scand Stat Theory Appl 2020. [DOI: 10.1111/sjos.12486] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Eduardo García‐Portugués
- Department of Statistics Carlos III University of Madrid, Spain
- UC3M‐Santander Big Data Institute Carlos III University of Madrid, Spain
| | - Javier Álvarez‐Liébana
- Department of Statistics and Operations Research and Mathematics Didactics University of Oviedo, Spain
| | | | - Wenceslao González‐Manteiga
- Department of Statistics, Mathematical Analysis and Optimization University of Santiago de Compostela, Spain
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18
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Rank method for partial functional linear regression models. J Korean Stat Soc 2020. [DOI: 10.1007/s42952-020-00075-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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19
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Nonparametric testing of lack of dependence in functional linear models. PLoS One 2020; 15:e0234094. [PMID: 32589640 PMCID: PMC7319281 DOI: 10.1371/journal.pone.0234094] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/31/2019] [Accepted: 05/18/2020] [Indexed: 11/19/2022] Open
Abstract
An important inferential task in functional linear models is to test the dependence between the response and the functional predictor. The traditional testing theory was constructed based on the functional principle component analysis which requires estimating the covariance operator of the functional predictor. Due to the intrinsic high-dimensionality of functional data, the sample is often not large enough to allow accurate estimation of the covariance operator and hence causes the follow-up test underpowered. To avoid the expensive estimation of the covariance operator, we propose a nonparametric method called Functional Linear models with U-statistics TEsting (FLUTE) to test the dependence assumption. We show that the FLUTE test is more powerful than the current benchmark method (Kokoszka P,2008; Patilea V,2016) in the small or moderate sample case. We further prove the asymptotic normality of our test statistic under both the null hypothesis and a local alternative hypothesis. The merit of our method is demonstrated by both simulation studies and real examples.
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20
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Robust estimation with a modified Huber’s loss for partial functional linear models based on splines. J Korean Stat Soc 2020. [DOI: 10.1007/s42952-020-00052-x] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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21
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22
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Dziak JJ, Coffman DL, Reimherr M, Petrovich J, Li R, Shiffman S, Shiyko MP. Scalar-on-function regression for predicting distal outcomes from intensively gathered longitudinal data: Interpretability for applied scientists. STATISTICS SURVEYS 2019; 13:150-180. [PMID: 31745402 PMCID: PMC6863606 DOI: 10.1214/19-ss126] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023]
Abstract
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.
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Affiliation(s)
- John J Dziak
- The Methodology Center, The Pennsylvania State University, University Park, PA
| | - Donna L Coffman
- Department of Epidemiology and Biostatistics, College of Public Health, Temple University, Philadelphia, PA
| | - Matthew Reimherr
- Department of Statistics, The Pennsylvania State University, University Park, PA
| | - Justin Petrovich
- Department of Business Administration, St. Vincent College, Latrobe, PA
| | - Runze Li
- Department of Statistics and The Methodology Center, The Pennsylvania State University, University Park, PA
| | - Saul Shiffman
- Department of Psychology, University of Pennsylvania, Pittsburgh, PA
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23
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Yuan M, Zhang Y. Test for the parametric part in partial functional linear regression based on B-spline. COMMUN STAT-SIMUL C 2019. [DOI: 10.1080/03610918.2019.1667391] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
- Mingao Yuan
- Department of Statistics, North Dakota State University, Fargo, North Dakota, USA
| | - Yue Zhang
- Department of Mathematics, Indiana University-Purdue University, Indianapolis, Indiana, USA
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24
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Li Q, Tan X, Wang L. Testing for error correlation in partially functional linear regression models. COMMUN STAT-THEOR M 2019. [DOI: 10.1080/03610926.2019.1642492] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
- Qian Li
- School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China
| | - Xiangyong Tan
- School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China
| | - Liming Wang
- School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China
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25
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Chen ST, Xiao L, Staicu A. A smoothing-based goodness-of-fit test of covariance for functional data. Biometrics 2019; 75:562-571. [PMID: 30450612 PMCID: PMC6526086 DOI: 10.1111/biom.13005] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/06/2018] [Accepted: 11/06/2018] [Indexed: 11/28/2022]
Abstract
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.
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Affiliation(s)
- Stephanie T. Chen
- Department of StatisticsNorth Carolina State UniversityRaleighNorth Carolina
| | - Luo Xiao
- Department of StatisticsNorth Carolina State UniversityRaleighNorth Carolina
| | - Ana‐Maria Staicu
- Department of StatisticsNorth Carolina State UniversityRaleighNorth Carolina
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26
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Affiliation(s)
- Ruiyan Luo
- Division of Epidemiology and Biostatistics, School of Public Health, Georgia State University, Atlanta, GA
| | - Xin Qi
- Department of Mathematics and Statistics, Georgia State University, Atlanta, GA
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27
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28
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Tekbudak MY, Alfaro-Córdoba M, Maity A, Staicu AM. A comparison of testing methods in scalar-on-function regression. ASTA ADVANCES IN STATISTICAL ANALYSIS 2018. [DOI: 10.1007/s10182-018-00337-x] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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29
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Liu R, Wang H, Wang S. Functional variable selection via Gram–Schmidt orthogonalization for multiple functional linear regression. J STAT COMPUT SIM 2018. [DOI: 10.1080/00949655.2018.1530776] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
Affiliation(s)
- Ruiping Liu
- School of Economics and Management, Beihang University, Beijing, People's Republic of China
| | - Huiwen Wang
- School of Economics and Management, Beihang University, Beijing, People's Republic of China
- Beijing Key Laboratory of Emergence Support Simulation Technologies for City Operations, Beijing, People's Republic of China
| | - Shanshan Wang
- School of Economics and Management, Beihang University, Beijing, People's Republic of China
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30
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Affiliation(s)
- Łukasz Smaga
- Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland
| | - Jin-Ting Zhang
- Department of Statistics and Applied Probability, National University of Singapore, Singapore
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Kong D, Ibrahim JG, Lee E, Zhu H. FLCRM: Functional linear cox regression model. Biometrics 2017; 74:109-117. [PMID: 28863246 DOI: 10.1111/biom.12748] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2016] [Revised: 04/01/2017] [Accepted: 06/01/2017] [Indexed: 11/27/2022]
Abstract
We consider a functional linear Cox regression model for characterizing the association between time-to-event data and a set of functional and scalar predictors. The functional linear Cox regression model incorporates a functional principal component analysis for modeling the functional predictors and a high-dimensional Cox regression model to characterize the joint effects of both functional and scalar predictors on the time-to-event data. We develop an algorithm to calculate the maximum approximate partial likelihood estimates of unknown finite and infinite dimensional parameters. We also systematically investigate the rate of convergence of the maximum approximate partial likelihood estimates and a score test statistic for testing the nullity of the slope function associated with the functional predictors. We demonstrate our estimation and testing procedures by using simulations and the analysis of the Alzheimer's Disease Neuroimaging Initiative (ADNI) data. Our real data analyses show that high-dimensional hippocampus surface data may be an important marker for predicting time to conversion to Alzheimer's disease. Data used in the preparation of this article were obtained from the ADNI database (adni.loni.usc.edu).
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Affiliation(s)
- Dehan Kong
- Department of Statistical Sciences, University of Toronto, Ontario, Canada
| | - Joseph G Ibrahim
- Department of Biostatistics, University of North Carolina at Chapel Hill, North Carolina, U.S.A
| | - Eunjee Lee
- Department of Biostatistics, University of Michigan, Michigan, U.S.A
| | - Hongtu Zhu
- Department of Biostatistics, The University of Texas MD Anderson Cancer Center, Texas, U.S.A
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Bai J, Ivanescu A, Crainiceanu CM. Discussion of the paper ‘A general framework for functional regression modelling’. STAT MODEL 2017. [DOI: 10.1177/1471082x16681335] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
This discussion provides our reaction to the article by Greven and Scheipl. It contains an overview of their article and a description of the many areas of research that remain open and could benefit from further methodological and computational development.
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Affiliation(s)
- Jiawei Bai
- Department of Biostatistics, Johns Hopkins University, Baltimore, MD, USA
| | - Andrada Ivanescu
- Department of Mathematical Sciences, Montclair State University, Montclair, NJ, USA
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