segmented-regression-wagner-2002

JournalofClinicalPharmacyandTherapeutics(2002)27,299–309 RESEARCHNOTE Segmentedregressionanalysisofinterruptedtimeseriesstudiesinmedicationuseresearch A.K.Wagner* PharmD,MPH,S.B.Soumerai*ScD,F.Zhang*MSandD.Ross-Degnan*ScD *DepartmentofAmbulatoryCareandPrevention,HarvardMedicalSchoolandHarvardPilgrimHealthCareand DepartmentofEpidemiology,HarvardSchoolofPublicHealth,Boston,MA,USA SUMMARY Interruptedtimeseriesdesignisthestrongest,quasi-experimentalapproachforevaluatinglon-gitudinaleffectsofinterventions.Segmentedregressionanalysisisapowerfulstatisticalmethodforestimatinginterventioneffectsininterruptedtimeseriesstudies.Inthispaper,weshowhowsegmentedregressionanalysiscanbeusedtoevaluatepolicyandeducationalinter-ventionsintendedtoimprovethequalityofmedicationuseand⁄orcontaincosts. Keywords:healthpolicyevaluation,interruptedtimeseriesdesign,longitudinalanalysis,medi-cationuseresearch,quasi-experimentaldesign,segmentedregressionanalysis INTRODUCTION Increasingly,educational,administrativeandpolicyinterventionsarebeingcarriedouttoimprovethequalityofmedicationuseand⁄orcontaincosts.Suchinterventionscanbeimplementedattheinstitu-tional,regionalornationallevel.Forexample,ahealthcareorganizationmightconducteducationandfeedbacktoitsphysicianstoencouragetheuseofapreferredhistamine-2-receptorantagonist(H2RA)(1).Astatemightinitiatealimitonthenumberofpaidmedicationsperpatientpermonth(2).AfederalprogrammelikeMedicaidmightceasetoreimburseforcategoriesofdrugsdeemed SeriesEditor:ParamjitGill,UniversityofBirmingham.Received5April2002,Accepted9May2002 Correspondence:AnitaK.WagnerPharmD,MPH,DepartmentofAmbulatoryCareandPrevention,HarvardMedicalSchoolandHarvardPilgrimHealthCare,133BrooklineAvenue,Boston,MA02215,USA.Tel.:6175099956;fax:6178598112;e-mail:awagner@hsph.harvard.eduÓ2002BlackwellScienceLtd ineffective(3)oradrugwithunfavourableside-effectscanbewithdrawnfromthemarket(4). Interruptedtimeseries(5,6)isthestrongest,quasi-experimentaldesigntoevaluatelongitudinaleffectsofsuchtime-delimitedinterventions.Seg-mentedregressionanalysisofinterruptedtimeseriesdataallowsustoassess,instatisticalterms,howmuchaninterventionchangedanoutcomeofinterest,immediatelyandovertime;instantlyorwithdelay;transientlyorlong-term;andwhetherfactorsotherthantheinterventioncouldexplainthechange. Segmentedregressionanalysisisappropriateforstudyingeffectsofinterventionsconductedaspartofarandomizedtrialaswellasinterventionsthatconstituteanaturalexperiment.Itrequiresdataoncontinuousorcountedoutcomemeasures,sum-marizedatregular,evenlyspacedintervals. TheobjectiveofthisResearchNoteistodescribesegmentedregressionanalysisinthecontextofmedicationuseresearch.Whereasprospectivedatacollectionforinterruptedtimeseriesstudiesisfeasible(7),thepresentdiscussionexcludesaspectsofdatacollectionandfocusesonanalysisandinterpretationoftimeseriesdata.Ourexamplesconsistofstudiesusingexisting,computerizeddatabases. Definitionsandparametersofinterest Atimeseriesisasequenceofvaluesofaparticularmeasuretakenatregularlyspacedintervalsovertime.Segmentsinatimeseriesaredefinedwhenthesequenceofmeasuresisdividedintotwoormoreportionsatchangepoints.Changepointsarespecificpointsintimewherethevaluesofthetimeseriesmayexhibitachangefromtheprevi-ouslyestablishedpatternbecauseofanidentifiable 299 300A.K.Wagneretal. real-worldevent,apolicychange,oranexperi-mentalintervention.Thechoiceofthebeginningandendofeachsegmentdependsonthebeginningandendoftheintervention,withthepossibleadditionofsomepre-specifiedlagtimetoallowtheinterventiontotakeeffect.Segmentedregressionanalysisisamethodforstatisticallymodellingtheinterruptedtimeseriesdatatodrawmoreformalconclusionsabouttheimpactofaninterventionoreventonthemeasureofinterest. Twoparametersdefineeachsegmentofatimeseries:levelandtrend.Thelevelisthevalueoftheseriesatthebeginningofagiventimeinterval(i.e.they-interceptforthefirstsegmentandthevalueimmediatelyfollowingeachchangepointatwhichsuccessivesegmentsjoin).Thetrendistherateofchangeofameasure(inotherwords,theslope)duringasegment.Insegmentedregressionanaly-sis,eachsegmentoftheseriesisallowedtoexhibitbothalevelandatrend.Theanalystexaminesthechangesinleveland⁄ortrendthatfollowanintervention.Achangeinlevel,e.g.ajumpordropintheoutcomeaftertheintervention,constitutesanabruptinterventioneffect.Achangeintrendisdefinedbyanincreaseordecreaseintheslopeofthesegmentaftertheinterventionascomparedwiththesegmentprecedingtheintervention.Achangeintrendrepresentsagradualchangeinthevalueoftheoutcomeduringthesegment.Seg-mentedregressionanalysis(seebelow)usessta-tisticalmodelstoestimatelevelandtrendinthepre-interventionsegmentandchangesinlevelandtrendaftertheintervention(orinterventions). Figure1showsthetimeseriesofmeannumbersofdispensedprescriptionsperpatientpermonthinacohortof860NewHampshireMedicaidenroleeswhoreceivedonaveragethreeormoredrugspermonthinthebaselineyear(2).BeginninginSep-tember1981,NewHampshire’sMedicaidpro-grammerestrictedthenumberofprescriptionsreimbursedtoamaximumofthreeperpatientpermonthtodecreasestatemedicationexpenditures.Implementationofthethree-drugpaymentlimit,orcap,interruptsthetimeseriesandcreatesthetwosegmentsofinterest.Anabruptlevelchangeinthemeannumberofprescriptions,fromaboutfiveperpatienttofewerthanthreeperpatient,followedthecap.Therewasverylittlemonth-to-monthchange(ortrend)inthenumberofprescriptionsbeforeaswellasafterthecap. DATASOURCESANDMEASURES Segmentedregressionanalysisrequiresdatacol-lectedregularlyovertime,andorganizedatequallyspacedintervals.Routinelymaintainedmedicationuseandotherhealthcarerecords,aswellascostdata,arecommonlyusedsourcesoftimeseriesdata.Theyincludepharmacies’dis-pensingfiles,medicalrecords,insuranceclaimsdatabasesthatcontainpharmacydispensing,hos-pitaldischargeandoutpatientclinicrecords.Birth,deathandchronicdiseaseregistryrecordsareotherroutinelycollecteddata.Althoughthesedataarenotgatheredforresearchpurposes,theyoftenprovidereliablemeasuresofrelevantdependentvariablesfortimeseriesstudiesofdruguse. Outcomemeasuresfortimeseriesstudiescanincludemedicationuse,utilizationofotherhealthservicesorclinicalmeasures.Outcomescanbeexpressedasaverages,proportionsorrates.Examplesofdruguse-relatedmeasuresaretheaveragenumberofdrugsprescribedperpatient,averageantibioticprescriptioncost,percentofenroleesreceivingaparticulardrugorpercentofpatientstreatedaccordingtoguidelines.Examplesofotherserviceutilizationwouldbeaveragelengthofhospitalstayormonthlyrateofadmissiontonursinghomes,whereasclinicalmeasuresmightincludeaveragediastolicbloodpressureinagroupofpatients,orpercentagediabeticpatientsachiev-ingadequateglucose control. Ó2002BlackwellScienceLtd,JournalofClinicalPharmacyandTherapeutics,27,299–309 Segmentedregressionanalysis301 Asufficientnumberoftimepointsbeforeandaftertheinterventionisneededtoconductsegmen-tedregressionanalysis.Ageneralrecommendationisfor12datapointsbeforeand12datapointsaftertheintervention(8),althoughthisnumberisnotbasedonestimatesofpower.Rather,with24monthlymeasures,theanalystcanadequatelyevaluateseasonalvariation(seebelow).Therealsoneedstobeasufficientnumberofobservations[aminimumof100isdesirable(8)]ateachdatapointofthetimeseriestoachieveanacceptablelevelofvariabilityoftheestimateateachtimepoint. Yt¼b0þb1Ãtimetþb2Ãinterventiont þb3Ãtimeafterinterventiontþet ð1Þ STATISTICALANALYSISANDINTERPRETATION Estimatingchangesinlevelandtrendthroughsegmentedregression Oneofthegreateststrengthsofinterruptedtimeseriesstudiesistheintuitivegraphicalpresentationofresults,andavisualinspectionoftheseriesovertimeisthefirststepwhenanalysingtimeseriesdata.Visually,wecomparethetimeseriespatternbeforetheinterventionwiththepatternaftertheinterven-tionandassessif,aftertheintervention,thetimeseriespatternhaschangednoticeablyinrelationtothepre-interventionpattern.LookingatthedatapointsinFig.1,wewouldhaveexpectedthepre-interventionseriestocontinueatanaverageofaboutfiveprescriptionsperpatientpermonthhadtheprescriptioncapnotoccurred.Clearly,afterthecap,themeannumberofdispensedprescriptionswasabouthalfofwhatwouldhavebeenexpected. Althoughwecanoftendetectchangesinleveland⁄ortrendofthemeasureofinterestbylookingatatimeseries,wecannoteasilyseewhetherchangesinlevelandtrendcouldbetheresultofchancealoneorthefactorsotherthanintervention.Toassesschanceandcontrolforothereffects,segmentedregressionanalysisisused. Commonsegmentedregressionmodelsfitaleastsquaresregressionlinetoeachsegmentoftheindependentvariable,time,andthusassumealinearrelationshipbetweentimeandtheoutcomewithineachsegment.Wecanspecifythefollowinglinearregressionmodeltoestimatethelevelandtrendinmeannumbersofprescriptionsperpatientbeforethethree-drugcapandthechangesinlevelandtrendfollowingthecapinNewHampshire: Here,Ytisthemeannumberofprescriptionsperpatientinmontht;timeisacontinuousvariableindicatingtimeinmonthsattimetfromthestartoftheobservationperiod;interventionisanindicatorfortimetoccurringbefore(intervention¼0)orafter(intervention¼1)thecap,whichwasimple-mentedatmonth21intheseries;andtimeafterinterventionisacontinuousvariablecountingthenumberofmonthsaftertheinterventionattimet,coded0beforethecapand(time—20)afterthecap.Inthismodel,b0estimatesthebaselineleveloftheoutcome,meannumberofprescriptionsperpatientpermonth,attimezero;b1estimatesthechangeinthemeannumberofprescriptionsperpatientthatoccurswitheachmonthbeforetheintervention(i.e.thebaselinetrend);b2estimatesthelevelchangeinthemeanmonthlynumberofprescriptionsperpatientimmediatelyaftertheintervention,thatis,fromtheendoftheprecedingsegment;andb3estimatesthechangeinthetrendinthemeanmonthlynumberofprescriptionsperpatientafterthecap,comparedwiththemonthlytrendbeforethecap.Thesumofb1andb3isthepost-inter-ventionslope.UsingModel1toestimatelevelandtrendchangesassociatedwiththeintervention,wecontrolforbaselinelevelandtrend,amajorstrengthofsegmentedregressionanalysis.Theer-rortermetattimetrepresentstherandomvari-abilitynotexplainedbythemodel.Itconsistsofanormallydistributedrandomerrorandanerrortermattimetthatmaybecorrelatedtoerrorsatprecedingorsubsequenttimepoints(seebelow).Table1containstheparameterestimatesfromthelinearsegmentedregressionmodel(Model1)ofeffectsoftheNewHampshirethree-drugcaponmeanmonthlynumberofprescriptionsperpatient.Itshouldbenotedthattherewasabriefincreaseinthemeannumberofprescriptionsperpatientinthemonthbeforethecap,inanticipationofthecap.Weexcludedthisvaluefromthemodel(seebelow).Theseresultsindicatethatjustbeforethebeginningoftheobservationperiod,patientsreceivedonaveragefiveprescriptionspermonth.Beforethecap,therewasnosignificantmonth-to-monthchangeinthemeannumberofprescrip-tions(P-valueforbaselinetrend¼0Æ6128).Rightafterthecap,theestimatedmeannumberof Ó2002BlackwellScienceLtd,JournalofClinicalPharmacyandTherapeutics,27,299–309 302A.K.Wagneretal. Table1.Parameterestimates,standarderrorsandP-valuesfromthefullandmostparsimonioussegmentedregressionmodelspredictingmeanmonthlynumbersofprescriptionsperpatientinNewHampshireovertime Coefficient a.FullsegmentedregressionmodelInterceptb05Æ1389Baselinetrendb10Æ003481Levelchangeaftercapb2)2Æ5931Trendchangeaftercapb30Æ0263b.MostparsimonioussegmentedregressionmodelInterceptb05Æ1677Levelchangeaftercapb2)2Æ3736 Standarderror t-statistic P-value 0Æ0748 0Æ0067910Æ15720Æ01930Æ03110Æ0563 68Æ690Æ51)16Æ491Æ36166Æ38)42Æ14 <0Æ00010Æ6128<0Æ00010Æ1849<0Æ0001<0Æ0001 prescriptionsdroppedabruptlyby2Æ6prescrip-tionspermonth.Therewasnosignificantchangeinthemonth-to-monthtrendinthemeannumberofprescriptionsafterthecap(P-valuefortrendchange¼0Æ1849).Afterstepwiseeliminationofnon-significantterms,themostparsimoniousmodelcontainedonlytheinterceptandthesigni-ficantlevelchangeinthemeannumberofprescriptions(Table1b).Expressinginterventioneffects Whenexpressingtheresultsofsegmentedre-gressionmodelling,wecaneitherreportlevelandtrendchangeslikethoseinTable1aor1b,orwecancompareestimatedpost-interventionvaluesoftheoutcometovaluesestimatedatthattimebutbasedonbaselinelevelandtrendonly,asiftheinterventionhadnotoccurred(thecounterfactualvalue).Wecanexpresstheinter-ventioneffectastheabsolutedifferencebetweenthepredictedoutcomebasedontheinterventionandthecounterfactualvalue,orastheratioofthepredictedtothecounterfactualvalue(usuallyexpressedasapercentageincreaseordecrease).ToestimatethecapeffectinNewHampshire,letusexpresstheexpectedresultsfromregressionequation1atmonth26,whichissixmonthsafterthethree-drugcapwasimple-mented:^26ðwithY policyÞ ^26ðwithoutY policyÞ ^þb^Â26¼b01 ð3Þ Thedifferencebetweenequations2and3, ^þb^Â6isthe^26ðwithoutpolicyÞ¼b^26ðwithpolicyÞÀYY23 estimateoftheabsolutepolicyeffect.Therelat-ivechangeinoutcomeassociatedwiththepolicy ^26ðwithpolicyÞÀY^26ðwithoutpolicyÞÞ=Y^26ðwithoutpolicyÞ,whichisðY canbeexpressedasapercentagechangebymultiplyingby100. UsingresultsinTable1b,weestimatedthatinmonth26,patientsreceivedonaverage2Æ8pre-scriptionspermonth.Hadthecapnotbeenintro-duced,themeannumberofprescriptionsperpatientpermonthwouldhavebeen5Æ2.Thus,theaveragenumberofprescriptionsperpatientpermonthdecreasedby2Æ4,or46%(95%confidenceinterval)44%,)48%)afterthecapwasimplemented,com-paredwithwhatitwouldhavebeenwithoutthecap.Estimatingchangesintimeserieswithmorethanonechangepoint Segmentedregressionmodelscanspecifymorethanonechangepoint.Forexample,onemaybeinterestedintheeffectsofdifferentcomponentsofaninterventionintroducedatdifferenttimepoints,orintheeffectsofaninterventionthatwasimplementedandlaterwithdrawn.Onemayalsowanttocontrolforchangesinlevelandslopeoftheseriesthatarecausedbyreasonsotherthanthepolicy.Amodelwithtwochangepointswouldbe: Yt¼b0þb1Âtimetþb2Âintervention1t þb3Âtimeafterintervention1t þb4Âintervention2t þb5Âtimeafterintervention2tþet ^þb^Â26þb^Â1þb^Â6¼b0123 ð2Þ Now,letusconsidertheregressionequation1at month26,hadthepolicynotbeenimplemented(i.e.withoutanypost-interventioneffectsinthemodel): ð4Þ Ó2002BlackwellScienceLtd,JournalofClinicalPharmacyandTherapeutics,27,299–309 Segmentedregressionanalysis303 Figure2containsanexampleofatimeserieswithtwochangepointsofsubstantiveinterest,whichcanbemodelledusingModel4.Elevenmonthsafteritsimplementation,NewHampshire’sMedicaidofficereplacedthethree-drugcapwitha$1copay,with-outrestrictionofthenumberofdrugsperpatientpermonth.Figure2showsthetimeseriesofthemeannumberofprescriptionsperpatientpermonth,interruptedbythetwopolicychanges.Afterthe$1copayhadreplacedthecap,averagepre-scriptionratesroseagain,bothinlevelandslope(toapproximately4Æ7prescriptionsperpatientpermonthbytheendoftheobservationperiod)(2).Additionalchangepointsmayalsoservetocontrolforchangesintheseriesthatarenotofprimaryinterestbutmayinfluencetheresults.Examplesincludediscontinuitiesintheseriesattimepointsotherthanthatoftheinterventionbecauseofadministrativechangesandlaggedormulti-periodinterventions. Modellinglaggedeffectsandinterventionsthatoccuroverseveralperiods Effectsofinterventionsmaytaketimetomanifest.Soumeraietal.studiedtheeffectsofcessationofreimbursementbyMedicaidof141drugsdeemedineffectiveonuseofsubstitutemedications(3).Notallpatientsfilledaprescriptioneachmonth.Afterwithdrawaloftheineffectivedrugs,therewasa brieftransitionperiodoftwotothreemonthsuntilpatientswereplacedonsubstituteregimens(Fig.3).Thatis,therewasatwotothreemonthslagintheeffectoftheintervention.Similarly,in-terventioneffectsmayoccuroverseveralperiods,asandwhenaneducationalprogrammeisrolledoutinahealthsystem.Itisimportanttoaccountforlagsandmulti-periodinterventionsintheanalysistoavoidincorrectspecificationofinterventionef-fects.Tomodeltheseeffects,onecanexcludefromtheanalysisoutcomevaluesthatoccurduringthelagorÔduringinterventionÕperiod.Alternatively,withenoughdatapoints,onecanmodeltheperiodasaseparatesegment.Dataorganizationandlayout Table2illustratesthedatastructurefortheanaly-sisoftheeffectsoftheNewHampshireMedicaidprogrammeinterventions.Theaggregatedoutcomemeasure,meannumberofprescriptions,iscalcu-latedateachmonthlytimepoint.Wespecifiedapre-interventiontrendvariableandtwolevelandtrendchangevariables,oneeachforthethree-drugcapandthe$1copay. Differentdefinitionsofthetimevariableinseg-mentedregressionanalysisarepossible(9).Forexample,timecouldberescaledsothatthestartingpointoftheinterventioniscodedasmonth1,withtimebeingmeasuredbackwardandforwardfromthatpoint.Alternatively,timeatthepointofinterest,forexamplesixmonthsaftertheinter-vention,couldbecodedas1,withtimecountedbackwardandforwardfromthere.Recodingtimeinthesewaysonlychangestheinterpretationof the Ó2002BlackwellScienceLtd,JournalofClinicalPharmacyandTherapeutics,27,299–309 304A.K.Wagneretal. Table2.Structureofdataforanalysisoftheimpactoftwopolicychangesonmeannumberofprescriptionsperpatient(Fig.2)(2) Meannumberof prescriptionsperpatient5Æ175Æ155Æ24Æ85Æ325Æ055Æ254Æ855Æ155Æ424Æ95Æ685Æ45Æ555Æ154Æ855Æ15Æ156Æ22Æ62Æ82Æ752Æ752Æ952Æ82Æ92Æ952Æ852Æ92Æ753Æ853Æ653Æ553Æ753Æ853Æ93Æ554Æ13Æ84Æ14Æ354Æ154Æ654Æ94Æ84Æ454Æ65 Time(month)123456789101112131415161718192021222324252627282930313233343536373839404142434445464748 Intervention1(three-drugcap)000000000000000000001111111111111111111111111111 TimeafterIntervention2Timeafterintervention1($1Copay)intervention20000000000000000000012345678910111213141516171819202122232425262728 000000000000000000000000000000011111111111111111 00000000000000000000000000000001234567891011121314151617 Observation123456789101112131415161718192021222324252627282930313233343536373839404142434445464748 Ó2002BlackwellScienceLtd,JournalofClinicalPharmacyandTherapeutics,27,299–309 Segmentedregressionanalysis305 intercept.Itdoesnotchangetheabsoluteorrelativemeasuresofeffect. Correctingforcorrelationbetweenvaluesoftheoutcomemeasureovertime Ordinaryleastsquaresregressionanalysisassumesthaterrortermsassociatedwitheachobservation(e.g.thedifferencesbetweentheactualoutcomevalueandthosepredictedfromtheregressionmodel)areuncorrelated.Astimeisapredictorinsegmentedregressionanalysis,errortermsofcon-secutiveobservationsareoftencorrelated.Prescri-bingpatternsandotherhealthoutcomesattwotimepointsthatareclosetoeachothermaybemoresimilarthanoutcomesattwotimepointsfurtherapart,resultinginserialautocorrelationoftheerrorterms.Correlationbetweenadjacentdatapointsistermedfirst-orderautocorrelation;corre-lationbetweenthecurrentpointandtwomonthsbeforeorafterwouldbesecond-orderautocorre-lationandsoforth.Theremayalsobeseasonalpatternsinmonthlytimeseries,whereprescribinginJanuaryofoneyearismoresimilartoprescri-binginJanuaryayearagothantoprescribinginothermonths.Thisisanexampleofhigher-orderautocorrelation. Failingtocorrectforautocorrelationmayleadtounderestimatedstandarderrorsandoverestimatedsignificanceoftheeffectsofanintervention.For-tunately,onecandetectautocorrelationintimeseriesdataandavailablestatisticalsoftwarecancontrolforit.FordetailsontheuseofprocautoreginSAS,pleasesee(10). Todetectautocorrelation,onecanvisuallyinspectaplotofresidualsagainsttimeandconductstatisticaltests.Randomlyscatteredresiduals,withoutapattern,indicatethatthereisnoauto-correlation(11).Positiveautocorrelationexistswhenconsecutiveresidualstendtolieonthesamesideoftheregressionline;negativeautocorrelationexistswhenconsecutiveresidualstendtolieondifferentsidesoftheregressionline(11). TheDurbin–Watsonstatistic,reportedbymostleastsquaresregressionprograms,testsforserialautocorrelationoftheerrortermsintheregressionmodel(11,12).Valuescloseto2Æ00indicatenoseriousautocorrelation.Adjustmentforautocorre-lationinvolvesestimatingtheautocorrelationparameterandincludingitinthesegmented regressionmodelifnecessary.Aftercontrollingforautocorrelation,theDurbin–WatsonstatisticforthefinalregressionmodeloftheNewHampshirecapeffectswas2Æ0822(P-valueforhypothesisofpositiveautocorrelation¼0Æ5149,P-valueforhypothesisofnegativeautocorrelation¼0Æ4851),indicatingnoautocorrelation. ControllingforseasonalchangesinseriesTimeseriessometimesexhibitseasonalfluctua-tions.Useofmanydrugsvariesseasonallybecauseofcyclicvariationsintheillnessesforwhichtheyareprescribed.Detectingseasonalityrequiresbaselineseriesthatspanenoughperiodstodetectthesecyclicpatterns.Ifseasonalityexists,itisimportanttocontrolforitwhenestimatinginterventioneffects,sothatestimatedeffectsaremorelikelytorepresenttrueinterventioneffects.Includingtermstoindicateeachseasoninaregressionmodeldecreasesconfoundingbysea-sonality. Whenseasonalvariationexists,errorsforaparticularmonthmaybemorecorrelatedwitherrorsatthesamemonthoneyearearlier,thanwitherrorsinothermonths.Toestimatethissea-sonalautocorrelation,theauto-regressionmodelneedstoevaluatecorrelationsbetweenerrortermsseparatedbymultiplesof12months.Accountingforseasonallycorrelatederrorsusuallyrequiresatleast24monthlydatapoints.Controllingforwilddatapoints Extremevaluesthatdonotseemtofitintheseries,referredtoaswilddatapoints,mayoccurintimeseries.Sometimesthesepointscanbeexplained.Anexamplemightbeasuddenpeakindruguseamonthbeforearestrictionpolicyisimplemented(e.g.ÔanticipatorydemandÕ),asoccurredinNewHampshirejustpriortoimplementationofthethree-drugcap(Figs1and2).Atothertimes,theseoutliersmightbebecauseofmeasurementerror.Iftheanalystknowstheunderlyingexplanationforawilddatapoint,orisfairlyconfidentthatitresultsfrommeasurementerror,thepointcanbecon-trolledforinthemodelbyenteringanindicatorvariablethathasvalue1inthatperiodand0inallothers.However,ifwilddatapointsarelikelycausedbyrandomvariation,theyshouldbetreated Ó2002BlackwellScienceLtd,JournalofClinicalPharmacyandTherapeutics,27,299–309 306A.K.Wagneretal. asregulardatapoints.Alternatively,onecouldcarryouttheanalysiswithandwithoutthewilddatapointtoevaluateitsimpact.Controllingforpossiblebiases Interruptedtimeseriesdesignsareimmunetomanyofthethreatstovalidityofweakerquasi-experimentalandotherobservationaldesigns.Inparticular,whensegmentsareproperlyspecified,potentialconfoundingintimeseriesstudiesislimitedtofactorsthatarerelatedtotheoutcomeofinterestandthatchangedatthetimeoftheinter-vention.Thoseincludeeffectsofsimultaneouslyoccurringinterventions(knownasÔcointerven-tionsÕ),seasonalchangesintheoutcomethathap-penduringthetimeoftheintervention,changesinthecompositionofthestudypopulation,orchan-gesinthemeasurementoftheoutcomeoccurringatthetimeoftheintervention.Separatinginter-ventioneffectsfromeffectsthatoccuratthesametimerequiresuseofacontrolgroupthatisnotexposedtotheintervention.Wediscussbelowthreechoicesofcontrolgroupsforinterruptedtimeseriesstudies. Overthecourseofalongitudinalstudy,thecom-positionofthestudypopulationmaychangewithrespecttocharacteristicsthatpredicttheoutcomeandcouldberelatedtotheintervention.Forexam-ple,achangeintheaverageproportionofelectiveCesareansectionscouldconfoundtheresultsofastudyoftheeffectsoninfectionratesofcontinuousqualityimprovementeffortstopreventsurgicalsiteinfectionsafterCesareansection(7).Tocontrolforpossibleconfounding,aggregatedmonthlyvaluesofthenumberofdeliveriespersite,thepercentageofdeliveriesbyCesareansection,andthepercentageofnon-electiveCesareansectionswereincludedinthesegmentedregressionmodels(7). Changesinthedefinitionsand⁄ormeasurementofvariablesmayintroducebiasinatimeseriesstudy.Scrutinizingtheconsistencyofrecordingisparticularlyimportantwhenusingroutinelycol-lected,automateddatathatfrequentlyformthebasisforinterruptedtimeseriesstudies.Controlgroupsintimeseriesstudies Atleastthreetypesofcontrolsarepossibleininterruptedtimeseriesstudies.Thecontrolcanbea differentgroupofsubjects,orberepresentedbyarelatedbutdifferentoutcomenotexpectedtochangefollowingtheintervention,inthesamegroupofsubjects.Ideally,acontrolgroupthatisidenticaltothestudygroupbutdoesnotexperiencetheinterventionisfollowedoverthesametimeperiodastheinterventiongroup.Comparingtheeffectintheinterventiongroupwiththatinthecontrolgroupthenallowsseparatingtheinterven-tioneffectfromothersthatmayhaveoccurredatthesametime.Figure4showstheeffectofNewHampshire’scost-containmentpoliciesonamajorhealthoutcome,nursinghomeadmissions(13).Becauseresidentstendtostayinnursinghomeslong-term,thepercentageofchronicallyillenroleesresidinginnursinghomesincreasedovertimeinthecontrolstate,NewJersey,whichdidnotimplementeithercost-containmentstrategy.How-ever,thepercentageofnursinghomeresidentsin-creasedmoresteeplyinNewHampshirethaninthecontrolstate.Becauseofeitherdeterioratinghealthwhendrugswerediscontinuedoradesiretoshifttoanenvironmentexemptfromthecap,limitingac-cesstomedicationswasassociatedwithincreasedadmissiontonursinghomesamonglow-income,elderlyMedicaidpatients;andresultingnursinghomestayswerelong-term. Whenaseparatecontrolgroupisnotavailable,evaluatingarelatedbutdifferentoutcomewithintheinterventiongroupthatshouldnotbeinflu-encedbytheinterventioncanidentifychangesthatwouldhaveoccurredindependentlyoftheintervention.Forexample,inthestudyofreim-bursementchangesforineffectivedrugs (3), Ó2002BlackwellScienceLtd,JournalofClinicalPharmacyandTherapeutics,27,299–309 Segmentedregressionanalysis307 investigatorscompareduseofineffectivedrugsovertimewithuseofdrugsthatwereneitherineffectivedrugsnorplausiblesubstitutesforthewithdrawndrugs(insulin,digoxinandeyeoint-ments)(3).Otherpolicyormeasurementchangesthatmayhaveconfoundedthepolicy’seffectontheuseofineffectivedrugsandtheirsubstituteswouldhaveappearedaschangesintheuseofthesecontrolmedications.Useofthecontroldrugswasstableandsuggestedthattheobservedchangesinthestudydrugswereinfacttheresultofthecessationofreimbursement,ratherthansomeotherfactorthataffecteddruguseinMedicaid(3). Ross-Degnanetal.(4)usedasomewhatdifferentapproachtodefiningacontrolgroupinthestudyoftheeffectsonprescribingofotheranalgesicsofmarketentryandwithdrawalofzomepirac,anon-steroidalanalgesic(NSAID).Evenwithoutzomepirac’smarketentryandwithdrawal,theoutcome,prescribingofotheranalgesics,couldhavechangedovertime.Marketentryandexitofzomepirac,however,occurrednation-wide;theor-eticallyleavingnocontrolgroupthatwasnotexposedtotheinterventions.Whenitenteredthemarket,zomepiracprescribingrosewithintwomonthsto11%oftotalanalgesicprescribingamongcertainphysiciansandremainedremark-ablyconstantatthatleveluntilitwaswithdrawn28monthslater.Otherphysiciansneverprescribedzomepiracatall.Thesephysicianswhoneverpre-scribedzomepiraccouldbeconsideredunaffectedbythemarketchangesandtheauthorsusedthemasacontrolgrouptoexaminechangesinprescri-bingofnon-zomepiracanalgesics. Duringthetimezomepiracwasonthemarket,useofotherNSAIDsincreasedamongphysicianswhoneverprescribedzomepirac,likelybecauseofincreasedmarketingefforts(Fig.5).ControllingforthischangeinNSAIDprescribing,theauthorsestimatedthatentryofzomepiracdecreasedthemarketshareofotherNSAIDsamongphysicianswhoprescribeditby8Æ1%andthatitswithdrawalincreasedotherNSAIDuseby6Æ8%(4). Analysisoftheoutcomeofinterestinthestudygrouponlyislessdesirablebecauseitdoesnotallowcontrolforothereventsthatmayhaveinfluencedtheoutcomeandthatmayhaveoccur-redatthesametimeastheintervention.However,becauselevelandtrendofthepre-intervention segmentserveascontrolforthepost-interventionsegment,singlegrouptimeseriesstilladdressimportantthreatstointernalvalidityandrepresentamethodologicallyacceptabledesignforstudyinginterventioneffects(5,14). Stratificationintimeseriesstudies Anintervention’seffectsmayvaryacrossdifferentgroups.Whenthisisthecase,interventioneffectscanbestudiedseparatelyineachgroup.Forexample,Brufskyetal.(1)expectedthateffectsofinterventionstoencourageprescribingoflesscostlycimetidineratherthanotherH2RAinalargehealthmaintenanceorganization(HMO)woulddifferaccordingtophysicians’incentivestochangeprescribingpatterns,whichdependedontheircontractualarrangementswiththeHMO.Theauthorsstratifiedtheanalysisandexaminedinter-ventioneffectsamongÔstaffmodelÕandÔgroupmodelÕHMOphysiciansseparately.Theycon-firmedthateffectsoftheinterventionsdifferedbycontractualarrangementbetweentheHMOanditsphysicians(1). SpecifyingthefinalsegmentedregressionmodelSeveraldifferentapproachesexistfordecidingwhichvariablestoincludeinafinalmodel.Fre-quently,analystsbeginbyspecifyingthefullregressionmodel(Table1a),includingthe baseline Ó2002BlackwellScienceLtd,JournalofClinicalPharmacyandTherapeutics,27,299–309 308A.K.Wagneretal. trend,andalllevelandtrendchanges.Thisfullmodelcontainsthelargestnumberofcovariatesandmayhavetheleastpowertodetectsignificantpredictorsoftheoutcome.Therefore,non-signifi-cantvariablesareoftenremoved.Throughstep-wisebackwardelimination(15),forexample,onemayselectthemostparsimoniousmodel,thatis,theonethatonlyincludesstatisticallysignificantpredictors(atapredeterminedsignificancelevel)(Table1b).Thefullandmostparsimoniousmodelswillnotcorrectlyestimatetheeffectoftheinter-ventionifconfoundersexist.Therefore,importantmeasuredconfoundersshouldbeaddedtothemodel,regardlessofstatisticalsignificance.Fortheoreticalreasons,onemayalsoincludeimportantnon-significantpredictorsthatmaycontrolforthreatstovalidity,suchashistoryormaturation(5).Forexample,thebaselinetrendisoftenaddedtothemodel,assumingthatitisgenerallyanimportantcontrolvariableforseculartrends,regardlessofstatisticalsignificance. Toassessthefitofthefinalmodel,weexamineresidualsaroundthepredictedregressionlines.Residualsthatarenormallydistributedandthatfollownoobservablepatternovertimeindicatethattheassumptionsunderlyingthelinearmodelaremet(15). STRENGTHSOFSEGMENTEDREGRESSIONANALYSIS approachesmayeasilyleadtoinvalidresultsbecauseofthefailuretocontrolforpre-existingtrends. Interruptedtimeseriesalsovisuallydisplaythedynamicsofresponseofapopulationtoaninter-ventionbyshowingwhetheraneffectisimmediateordelayed,abruptorgradual,andwhetherornotaneffectpersistsoristemporary.Segmentedregressionanalysiscanestimatethesizeoftheeffectatdifferenttimepoints,aswellaschangesinthetrendoftheeffectovertime. LIMITATIONSOFSEGMENTEDREGRESSIONANALYSIS Randomizedcontrolledtrialsarerarelypossibletoassesstheimpactofpolicychanges.Timeseriesdesignsarethestrongest,quasi-experimentaldesignstoestimateinterventioneffectsinnon-randomizedsettings.Incontrasttocross-sectionalobservationalstudies,segmentedregressionana-lysisofinterruptedtimeseriesdataallowsanalyststocontrolforpriortrendsintheoutcomeandtostudythedynamicsofchangeinresponsetoanintervention. Evenwithoutacontrolgroup,segmentedregressionanalysisaddressesimportantthreatstointernalvalidity(suchashistoryandmaturation)bymakingmultipleassessmentsoftheoutcomevariablebothbeforeandaftertheintervention.Incontrast,cross-sectionalanalysescomparetheout-comebetweengroupsatonepointaftertheinter-ventionandpre–poststudiescomparetwopoints,onebeforeandoneaftertheintervention.Both Aswithotheranalyses,therearelimitations.Themodelswediscussedassumealineartrendintheoutcomewithineachsegment.Theassumptionoflinearityoftenmayholdonlyovershortintervals.Changesmayfollownon-linearpatterns.Forexample,interventioneffectsmaydiffuseacrossapopulationwithanincreasingordecreasingcur-vilineartrend.Someeffectscanbeaccommodatedinlinearmodelsbyuseofgeometricallyincreasingordecreasingtrendterms,butsomenon-lineartrendsmayrequireothertypesofmathematicalmodelssuchasBox-Jenkinsmodels(16).Althoughthesemodelsarewidelyusedforpredictingfuturetrends,theyareoflessuseinexaminingchangesintrendthatoccuratdefinedtimepoints.Inaddition,medicationuseresearchfrequentlylackstherequiredminimumof50timepointsforemployingthesemodels. Segmentedregressiontypicallyaggregatesin-dividual-leveldatabytimepoint.Forexample,inthestudyoftheNewHampshirecostcontain-mentinterventions,theinvestigatorscalculatedmeanmonthlynumbersofprescriptionsperpa-tient.Theunitofanalysisinthesegmentedre-gressionmodelwasthemonthlymeanprescriptionnumber,ratherthaneachindivid-ual’snumberofprescriptionspermonth.Con-trarytocross-sectionalanalysismethods,suchaslogisticregression,segmentedregressionanalysisoftimeseriesdatadoesnotallowcontrolforindividual-levelcovariates.Individual-levelchar-acteristics,however,wouldonlyconfoundthetimeseriesresultsiftheypredictedtheoutcomeandchangedinrelationshiptothetimeoftheintervention. Ó2002BlackwellScienceLtd,JournalofClinicalPharmacyandTherapeutics,27,299–309