Combining information from related regressions

CombiningInformationfromRelatedRegressionsFrancescaDominici;GiovanniParmigiani;KennethH.Reckhow;RobertL.WolpertJournalofAgricultural,Biological,andEnvironmentalStatistics,Vol.2,No.3.(Sep.,1997),pp.313-332.StableURL:=1085-7117%28199709%292%3A3%3C313%3ACIFRR%3E2.0.CO%3B2-BJournalofAgricultural,Biological,andEnvironmentalStatisticsiscurrentlypublishedbyAmericanStatisticalAssociation.YouruseoftheJSTORarchiveindicatesyouracceptanceofJSTOR'sTermsandConditionsofUse,availableat://@jstor.org.:162007CombiningInformationFromRelatedRegressionsFrancescaDOMINICI,GiovanniPARMIGIANI,KennethH.RECKHOW,andRobertL.WOLPERTWeproposeandillustrateanapproachforcombininginformationfromseveralre-gressionstudies,eachconsideringonlyasubsetofthevariablesofinterest.OurapproachusesacombinationofBayesianhierarchicalmodelinganddataaugmentation.Hierarchi-calmodelsareaflexibletoolformodelingstudy-to-studyaswellaswithin-studyvari-ability.Dataaugmentationmethodsaddressfullytheuncertaintyresultingfrommissingdataandprovidevenuesforcombininginformationinawaythatpreservesthemean-ingoftheregressioncoefficientsacrossstudies.Wediscussindetailanonnal-nonnalmodel.wesuggestasimpleandefficientnumericalimplementationbasedonablockGibbssampler.andweprovideexplicitfullconditionaldistributionsforanarbitrarypatternofvariablesmissingbystudy.Wediscussanapplicationofourmodeltoinvestigatingthelevelofchlorophyll-ainwaterqualitymanagement.Chlorophyll-nisoneofthemostimportantindicatorsoflakewaterquality.Scientistshavedevelopedanumberandvarietyofforecastingmodelsrelatingchlorophyll-atonutrientssuchasphosphorusandnitrogen.Thesemodelsoftenhavetorelyonsparseinformationfrommultiplesources-inthiscaselakes.Westudytherelationshipamongchlorophyll-aandphosphorusin12northerntemperatelakesbyusingdatafromtheliterature.Animportantcovariateisnitrogen.whichisreportedonlyinsomeofthestudies.KeyWords:Hierarchicalmodels;Missingcovariates:Waterquality.1.INTRODUCTIONIfthereareseveralstudiesthataddressthesameresearchquestion,onemightbeinterestedincombiningtheinformationfromtheindividualstudiesinordertodrawover-allconclusionsabouttheresearchquestionofinterest.Thecombiningoftheindividualstudiesinorderlearnaboutthewholeisreferredtointheliteratureasmeta-analysis.Inthisarticlewefocusonmeta-analysisofregressionstudies.Inparticular,wediscussFrancescaDominiciisVisitingAssistantProfessor,DepartmentofBiostatistics,JohnsHopkinsUniversity.GiovanniParmigianiisAssistantProfessor,InstituteofStatisticsandDecisionSciencesandCenterforHealthPolicyResearchandEducation;RobertWolpertisAssociateProfessor,InstituteofStatisticsandDecisionSciences;andKennethReckhowisAssociateProfessor,NicholasSchooloftheEnvironmentandInstituteofStatisticsandDecisionSciences;DukeUniversity,Durham,NC27708-0251.ThisworkwascompletedwhileFrancescaDominiciwasavisitingscholarattheInstituteofStatisticsandDecisionSciences.01997AmericanSratisticalAssociariorzarzdthelnrernatiorzalBiometricSociehJortnialofAgricultrtral,Biological,andEnvironmenralStaristics,Volrtme2,Nrtniber3,Pages313-332howtocombineseveralmultivariateregressiondatasets,eachrecordingoverlapping,butpossiblydifferent,setsofvariables.Thisisacommonsituation:frequently,aninitialstudywillidentifyapotentiallyinterestingrelationshipbetweenvariables.Newstudiesarethenlikelytofollow,withmorecomprehensivedesignsandmorevariables,perhapsinanattempttoclarifypotentialconfoundingeffectsorbiasesintheinitialstudy.Interestinsimilarquestionsfromotheragencies,technologicalprogressinmeasuringpotentialexplanatoryvariables,andemergenceofnewandinterestingexplanatoryvariablesarealllikelytoleadtomorestudieswithyetdifferentsetsofvariables.Often,studieshavemultipleendpointsorusedifferentproxiesforresponsesofinterest.Inpractice,mul-tistudyregressionanalysescarriedouttosupportimportantpolicydecisionswillveryoftenrequirecombiningstudieswithdifferentvariables.Thegoalofthisarticleistoprovideaframeworkforhandlingsomeofthemosturgentmodelingproblemsarisinginthesituationjustdescribed.Examplesare1.combiningseveralstudieswithacommonresponsevariableandoverlapping,butdifferentcovariates;2.combiningstudieswiththesamecovariatesbutdifferentendpoints(responses),withtheaidofoneormorefurtherstudiesinvestigatingthedependencebetweentheendpoints;and3.combiningmultivariateanalyseswithdiffering