Generalized Functional Linear Models With Semiparametric Single-Index Interactions

Handle URI:
http://hdl.handle.net/10754/598402
Title:
Generalized Functional Linear Models With Semiparametric Single-Index Interactions
Authors:
Li, Yehua; Wang, Naisyin; Carroll, Raymond J.
Abstract:
We introduce a new class of functional generalized linear models, where the response is a scalar and some of the covariates are functional. We assume that the response depends on multiple covariates, a finite number of latent features in the functional predictor, and interaction between the two. To achieve parsimony, the interaction between the multiple covariates and the functional predictor is modeled semiparametrically with a single-index structure. We propose a two step estimation procedure based on local estimating equations, and investigate two situations: (a) when the basis functions are pre-determined, e.g., Fourier or wavelet basis functions and the functional features of interest are known; and (b) when the basis functions are data driven, such as with functional principal components. Asymptotic properties are developed. Notably, we show that when the functional features are data driven, the parameter estimates have an increased asymptotic variance, due to the estimation error of the basis functions. Our methods are illustrated with a simulation study and applied to an empirical data set, where a previously unknown interaction is detected. Technical proofs of our theoretical results are provided in the online supplemental materials.
Citation:
Li Y, Wang N, Carroll RJ (2010) Generalized Functional Linear Models With Semiparametric Single-Index Interactions. Journal of the American Statistical Association 105: 621–633. Available: http://dx.doi.org/10.1198/jasa.2010.tm09313.
Publisher:
Informa UK Limited
Journal:
Journal of the American Statistical Association
KAUST Grant Number:
KUS-CI-016-04
Issue Date:
Jun-2010
DOI:
10.1198/jasa.2010.tm09313
PubMed ID:
20689644
PubMed Central ID:
PMC2915777
Type:
Article
ISSN:
0162-1459; 1537-274X
Sponsors:
Yehua Li is Assistant Professor, Department of Statistics, University of Georgia, Athens, GA 30602 (E-mail: yehuali@uga.edu). Naisyin Wang is Professor, Department of Statistics, University of Michigan, Ann Arbor, MI 48109-1107 (E-mail: nwangaa@umich.edu). Raymond J. Carroll is Distinguished Professor of Statistics, Nutrition and Toxicology, Department of Statistics, Texas A&M University, TAMU 3143, College Station, TX 77843-3143 (E-mail: carroll@stat.tamu.edu). Li's research was supported by the National Science Foundation (DMS-0806131). Wang's research was supported by a grant from the National Cancer Institute (CA74552). Carroll's research was supported by a grant from the National Cancer Institute (CA57030) and by award number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST).
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Full metadata record

DC FieldValue Language
dc.contributor.authorLi, Yehuaen
dc.contributor.authorWang, Naisyinen
dc.contributor.authorCarroll, Raymond J.en
dc.date.accessioned2016-02-25T13:20:06Zen
dc.date.available2016-02-25T13:20:06Zen
dc.date.issued2010-06en
dc.identifier.citationLi Y, Wang N, Carroll RJ (2010) Generalized Functional Linear Models With Semiparametric Single-Index Interactions. Journal of the American Statistical Association 105: 621–633. Available: http://dx.doi.org/10.1198/jasa.2010.tm09313.en
dc.identifier.issn0162-1459en
dc.identifier.issn1537-274Xen
dc.identifier.pmid20689644en
dc.identifier.doi10.1198/jasa.2010.tm09313en
dc.identifier.urihttp://hdl.handle.net/10754/598402en
dc.description.abstractWe introduce a new class of functional generalized linear models, where the response is a scalar and some of the covariates are functional. We assume that the response depends on multiple covariates, a finite number of latent features in the functional predictor, and interaction between the two. To achieve parsimony, the interaction between the multiple covariates and the functional predictor is modeled semiparametrically with a single-index structure. We propose a two step estimation procedure based on local estimating equations, and investigate two situations: (a) when the basis functions are pre-determined, e.g., Fourier or wavelet basis functions and the functional features of interest are known; and (b) when the basis functions are data driven, such as with functional principal components. Asymptotic properties are developed. Notably, we show that when the functional features are data driven, the parameter estimates have an increased asymptotic variance, due to the estimation error of the basis functions. Our methods are illustrated with a simulation study and applied to an empirical data set, where a previously unknown interaction is detected. Technical proofs of our theoretical results are provided in the online supplemental materials.en
dc.description.sponsorshipYehua Li is Assistant Professor, Department of Statistics, University of Georgia, Athens, GA 30602 (E-mail: yehuali@uga.edu). Naisyin Wang is Professor, Department of Statistics, University of Michigan, Ann Arbor, MI 48109-1107 (E-mail: nwangaa@umich.edu). Raymond J. Carroll is Distinguished Professor of Statistics, Nutrition and Toxicology, Department of Statistics, Texas A&M University, TAMU 3143, College Station, TX 77843-3143 (E-mail: carroll@stat.tamu.edu). Li's research was supported by the National Science Foundation (DMS-0806131). Wang's research was supported by a grant from the National Cancer Institute (CA74552). Carroll's research was supported by a grant from the National Cancer Institute (CA57030) and by award number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherInforma UK Limiteden
dc.subjectFunctional data analysisen
dc.subjectGeneralized linear modelsen
dc.subjectInteractionsen
dc.subjectLatent variablesen
dc.subjectLocal estimating equationen
dc.subjectPrincipal componentsen
dc.subjectSingle-index modelsen
dc.titleGeneralized Functional Linear Models With Semiparametric Single-Index Interactionsen
dc.typeArticleen
dc.identifier.journalJournal of the American Statistical Associationen
dc.identifier.pmcidPMC2915777en
dc.contributor.institutionDepartment of Statistics, University of Georgia, Athens, GA 30602, yehuali@uga.edu.en
kaust.grant.numberKUS-CI-016-04en
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