Adaptive Regression for Modeling Nonlinear Relationships PDF
Preview Adaptive Regression for Modeling Nonlinear Relationships
Statistics for Biology and Health George J. Knafl Kai Ding Adaptive Regression for Modeling Nonlinear Relationships Statistics for Biology and Health SeriesEditors MitchellGail JonathanM.Samet B.Singer AnastasiosTsiatis More information about this series at http://www.springer.com/series/2848 George J. Knafl • Kai Ding Adaptive Regression for Modeling Nonlinear Relationships GeorgeJ.Knafl KaiDing UniversityofNorthCarolina UniversityofOklahomaHealthSciencesCenter atChapelHill OklahomaCity,OK,USA ChapelHill,NC,USA Additionalmaterialtothisbookcanbedownloadedfrom http://www.unc.edu/~gknafl/AdaptReg.html ISSN1431-8776 ISSN2197-5671 (electronic) StatisticsforBiologyandHealth ISBN978-3-319-33944-3 ISBN978-3-319-33946-7 (eBook) DOI10.1007/978-3-319-33946-7 LibraryofCongressControlNumber:2016940407 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface This book addresses how to incorporate nonlinearity in one or more predictor (or explanatory or independent) variables in regression models for different types of outcome (or response or dependent) variables. Such nonlinear dependence is often not considered in applied research. While relationships can reasonably be treatedaslinearinsomecases,itisnotunusualforthemtobedistinctlynonlinear. A standard linear analysis in the latter cases can produce misleading conclusions, while a nonlinear analysis can provide novel insights into data not otherwise possible. A variety of examples of the benefits to the modeling of nonlinear relationshipsarepresentedthroughoutthebook. Methods are needed for deciding whether relationships are linear or nonlinear and for fitting appropriate models when they are nonlinear. Methods for these purposes are covered in this book using what are called fractional polynomials based on power transformations of primary predictor variables with real valued (and so possibly fractional) powers. An adaptive approach is used to construct fractional polynomial models based on heuristic (or rule-based) searches through powertransformsofprimarypredictorvariables.Thebookcovershowtoformulate andconductsuchadaptivefractionalpolynomialmodelinginavarietyofcontexts includingadaptiveregressionofcontinuousoutcomes,adaptivelogisticregression ofdiscreteoutcomeswithtwoormorevalues, andadaptivePoissonregressionof countoutcomes,possiblyadjustedintorateoutcomeswithoffsets.Powertransfor- mation of positive valued continuous outcomes is covered as well as modeling of variances/dispersionswithfractionalpolynomials.Thebookalsocoversalternative approaches for modeling nonlinear relationships including standard polynomials, generalized additive models computed using local regression (loess) and spline smoothingapproaches,andmultivariateadaptiveregressionsplines. PartIcoversmodelingofnonlinearrelationshipsforcontinuousoutcomesusing adaptive regression modeling. Adaptive models of this type are linear in the parameters for modeling the means, as are commonly used regression models based on untransformed primary predictor variables. However, adaptive models can depend on nonlinear transformations ofavailable primary predictor variables. v vi Preface Chapters2and3addressnonlinearmodelingofmeansandvariancesofunivariate continuousoutcomesusingfractionalpolynomials.Chapters4and5extendthisto modeling of means and variances of multivariate continuous outcomes, including marginal models based on either maximum likelihood estimation or generalized estimating equations (GEE) and conditional models with current outcome values depending on either prior outcome values (that is, transition models) or all other outcome values.Chapters6and 7cover transformation of positive valuedcontin- uousoutcomesaswellastheirpredictors. Part II extends fractional polynomial modeling to discrete outcomes, either dichotomous with two values or polytomous with more than two values, using adaptive logistic regression. Chapters 8 and 9 address modeling of means and dispersionsofunivariatedichotomousandpolytomousoutcomes.Polytomousout- comes are modeled both with adaptive ordinal regression using cumulative logits undertheproportionaloddsassumptionandwith adaptivemultinomialregression using generalized logits as needed for nominal outcomes. Chapters 10 and 11 extendthismodelingtomultivariatediscreteoutcomes. PartIIIextendsfractionalpolynomialmodelingfurthertocountoutcomesusing adaptive Poisson regression, possibly adjusted to models of rate outcomes using offsets.Chapters12and13addressmodelingofmeansanddispersionsofunivar- iatecount/rateoutcomes.Chapters14and15extendthismodelingtomultivariate count/rateoutcomes. PartIVcoversmodelingofnonlinearrelationshipsforunivariatecontinuousand dichotomous outcomes using generalized additive models (GAMs) and multivari- ateadaptiveregressionsplines(MARS)models.ItalsocomparesGAMsasgener- ated by SAS® PROC GAM and MARS models as generated by PROC ADAPTIVEREG to associated adaptive regression models. Chapters 16 and 17 address modeling of nonlinear relationships using GAMs for means of univariate continuous and dichotomous outcomes. Chapters 18 and 19 address modeling of nonlinear relationships using MARS models for means of these two types of out- comes. Modeling of variances/dispersions, correlated multivariate outcomes, and polytomous discrete outcomes are not covered since PROC GAM and PROC ADAPTIVEREG do notcurrentlysupportsuch modeling. Modeling of count/rate outcomesisalsonotconsideredforbrevity. Chapters2–19presentaseriesofanalysesofselecteddatasets.Theseanalyses demonstrate how to conduct adaptive regression modeling, generalized additive modeling,andMARSmodelingintheregression,logisticregression,andPoisson regressioncontextsaswellastheneedforsuchnonlinearmodeling.Overviewsof analysis results are provided in even-numbered chapters. Statistical formulations forassociatedregressionmodelsarealsoprovidedinsomeofthesechapters.PartV (Chap. 20) provides a summary of these formulations and their extensions to distributions in the exponential family. It also covers the heuristics underlying the adaptive modeling process. Familiarity with vector and matrix notation and calculus is needed to understand the formulations, but not the analyses and the examplecode.Aninformaloverviewoftheadaptivemodelingprocessisprovided inSect.1.3. Preface vii Directsupportforadaptiveregressionmodelingbasedonfractionalpolynomials isnotcurrentlyavailableinstandardstatisticalsoftwaretoolslikeSASversion9.4 (SASInstituteInc.,Cary,NC).Consequently,SASmacroshavebeendevelopedfor thesepurposes.Detaileddescriptionsofhowtousethesemacrosandoftheiroutput areprovidedinodd-numberedchapters(exceptforChap.1).Aworkingknowledge ofSASisassumed,sothebookdoesnotprovideanintroductiontotheuseofSAS. The intended audience includes data analysts, both applied researchers conducting analyses of their own data and statisticians conducting analyses for applied researchers. Readers can choose to focus on a specific type of regression analysis (for example, logistic regression of univariate dichotomous outcomes as coveredinChaps.8and9)butshouldreviewChaps.2and3firstforanintroduction toadaptiveregressionmodelingandSects.4.5.3and4.5.4onmoderationanalyses using geometric combinations. Practice exercises are provided at the end of odd-numbered chapters (except for Chap. 1) for readers to practice conducting analyseslikethoseintherelatedeven-numberedchapters,andsothebookcanbe used as a text for a course or workshop on adaptive regression modeling. The lecturescanpresenttheanalysesinthetextalongwithunderlyingformulationsand studentscanusetheexercisestopracticeconductingadaptiveregressionanalyses. The data sets are primarily taken from the health sciences, but the methods apply generallytoallapplicationareas. Referencesareprovidedattheendofeachchapter.Supplementarymaterialsare available on the Internet (http://www.unc.edu/~gknafl/AdaptReg.html) including InternetsourcesfordatasetsusedinChaps.2–19,theSASmacrosusedinanalyses reported in Chaps. 2–19, detailed descriptions of those macros, and code for conductingtheanalysesreportedinChaps.2–19. ChapelHill,NC,USA GeorgeJ.Knafl OklahomaCity,OK,USA KaiDing Acknowledgments WethankHannahBrackenofSpringerforhersupportandourfamiliesforalltheir encouragement. The development of the genreg macro used in reported analyses was partially supported by grants R01 AI57043 from the National Institute of Allergy and Infectious Diseases and R03 MH086132 from the National Institute ofMentalHealth. ix
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