ÿWPCI )·)Û]1ß Y¡ñžÙöj-»B§9±&‚žmÈ”PRE†•lTjBÃ\lz{mߌ·…'kcEÎ~`ÿ âÂ%Ô,~4f1Ie)Œë;¦¼&þJÒïktìÙGd_'jûz¦eî£H(› *¿ý,¡ìTÓØþBÕVÑ7xÀ,†Ó(]Ô ¹ràéýÞûÑãTbõnÊ:d•u ê}PÆ×ÈÂÚÆ©,S\d*€Äœ%¡°¨¦*“øÎ„Å ïí˜Ä@#üÄÉ1 H.S»Î…ì1 ·À3ÝÁbÇêà/àCBY=Œ­,òPܵô˜pzZBR §=k3§œzhÇ"Ø`†SÝ+'ã7盬jûrU}üØOÌe¯u1wL™dôëþ·ÜqAçßå&‰­òXÒSúðiCÿ€ÙùªQÀºx$üš&(D¡èîfÊá ´ŠªÓ»ÅÄŽú»ÔQ^W@•â?ÝÎ&(ºÂ9~ÚÑî®ÌS³¹®º„J‡´Ÿš¤¿Ž¤ á¿9[1¦t}ªÕ&ó;ÓÆ£]ÊœæÁHTç9%ËY,“’6ÁùÌ(cB¾Hï!TµŸ”UÂiì–Í×µ¬)ûuê†uËEëÙ-P U:~ %¸¾n­Î 04{UN¯4ýU8<þ6X9`("Courier 10cpiX#|x«<þ6X9`("Courier 10cpiXÂxþ6X@É“8åX@( ¤T$¡¡Ó  Ó(ÈhH  Z ‹6Times New Roman Regular8o¬"ÔôÔÿÿÿÿ@ Helv RegularÝ ƒ¤T!ÝÓ  ÓÝ  ÝØØÔ€Xº¯XXXÔÔ€%¢ê%XXº¯ÔÑ  ÑÑ7€fö%%dØdÈ7ÑÑ  ÑÌÌÌÌò òà@Š Š ìàTRACKING€STUDENT€EXIT,€TRANSFER€AND€GRADUATION:ˆÐ h ÐÌà@W W ìàAN€EVENT€HISTORY€ANALYSIS€OF€COMPETING€RISKSˆÌó óÌÌà@&ìàABSTRACTˆÌÌÌEvent€history€models€have€had€limited€application€by€Institutional€Researchers€in€the€study€of€studentÏcareers.€Generally€the€research€that€has€been€conducted€has€focused€on€the€occurrence€or€non„occurrenceÏof€a€singular€event€such€as€graduation,€stop„out€or€change„of„major.€And€there€are€no€studies€in€theÏhigher€education€literature€that€utilize€event€history€methods€to€study€community€college€student€careers.ÏThis€paper€addresses€both€issues.€A€cohort€of€1288€first„time€full„time€community€college€students€isÏtracked€for€9€semesters.€The€resulting€4600€person€record€file€is€used€to€estimate€the€risk€of€stop„out,Ïgraduation,€transfer€to€a€senior€institution,€or€to€a€different€community€college.€The€analysis€of€a€sixÏcovariate€model€finds€comparable€factors€affecting€student€stop„out€and€transfer€to€a€two€year€college,Ïwhile€graduation€and€transfer€to€a€senior€college€share€similar€patterns.€Advantages€of€competing€riskÏanalysis€are€discussed,€and€special€note€is€made€of€the€difficult€issues€associated€with€this€form€ofÏmodeling.ÌÌÌÌÌà@A A ìàPaper€Presented€at€the€Thirty„Seventh€Annual€ForumˆÌà@Ï Ï ìàThe€Association€for€Institutional€ResearchˆÌà@AA"ìàMay€18€„€21,€1997ˆÌà@%%%ìàOrlando,€FLˆÌÌÌÌà@ìì*ìàˆÌÌò òà@ÞÞ$ìàKeith€GuerinˆÐ X#¨$ Ðó óÌà@f f ìàDirector,€Institutional€Research€and€PlanningˆÌà@00ìàCounty€College€of€MorrisˆÌò òÌÐ  š'ê")V Ðà@Š Š ìàTRACKING€STUDENT€EXIT,€TRANSFER€AND€GRADUATION:ˆÌÌà@W W ìàAN€EVENT€HISTORY€ANALYSIS€OF€COMPETING€RISKSˆÌó óÌÌÌÓÓà  àTheoretical€developments€in€retention€research€occupy€a€significant€segment€of€the€existingÏliterature.€Methodological€advances,€though€numerous€they€may€be,€command€much€less€notoriety.€InÏmany€studies,€different€statistical€methods€are€compared€to€demonstrate€the€similarity€of€results.€ArticlesÏfrequently€suggest€the€reader€select€the€method€most€easily€understood,€with€due€regard€given€to€mattersÏof€access€to€computer€hardware€and€software.Ìà  àBut€these€suggestions€are€often€confused€as€methodological€options,€when€in€reality€they€are€aÏmenu€of€statistical€procedures.€They€do€not€offer€the€opportunity€to€address€different€questions.€TheyÏsimply€provide€different€means€to€the€same€end,€answering€the€question€of€what€caused€a€particular€eventÏto€occur€by€the€end€of€a€specified€time€period.€There€is€no€accounting€for€intervening€events€that€occurÏbetween€the€origin€point€and€the€end€of€the€time€period€in€question.€And€the€models€cannot€accommodateÏpredictors€that€vary€over€time.€Event€history€models€address€this€particular€issue.Ìà  àIn€this€study,€event€history€analysis€is€applied€to€the€study€of€student€progress.€A€first„time€full„¼time€cohort€of€1288€community€college€students€is€tracked€for€nine€semesters€to€the€start€of€the€fifth€yearÏfollowing€their€entry.€It€represents€an€extension€of€previous€event€history€applications€in€retentionÏresearch€to€include€estimates€of€competing€risks:€undefined€exits,€graduates,€and€transfers,€to€either€aÏsenior€institution€or€community€college.€The€paper€also€addresses€the€issue€of€time€dependency,€theÏutilization€of€time„varying€covariates,€suggests€a€method€for€identifying€unobserved€heterogeneity,€andÏprovides€a€method€for€examining€violations€of€the€proportionality€assumption.Ìò òDYNAMIC€MODELSÐ t(Ä#' Ðó óà  àEvent€history€models€fall€into€a€general€class€of€statistical€methods€known€as€survival€analysisÐ (*x%( Ð(Allison,€1995).€They€have€also€been€known€as€hazard€rate€models.€This€family€of€applications€has€aÐ Ü+,'* Ðlong€history€in€a€variety€of€fields.€In€the€social€and€behavioral€sciences,€they€evolved€as€an€extension€ofÏfirst„order€Markov€processes,€and€have€been€applied€to€studying€the€occurrence€and€timing€of€events.ÏThey€are€all€part€of€the€same€family€of€methods€and€generally€derive€their€nomenclature€and€points€ofÏemphasis€from€the€disciplines€in€which€they€were€developed.€Event€history€analysis€represents€aÏsociological€origin€(Tuma,€Hannan,€and€Groeneveld,€1979).€It€is€more€generally€associated€with€multipleÏtypes€of€events€(Clogg,€1991),€whereas€survival€models€are€more€focused€on€the€occurrence€of€a€singleÏevent€(Morita,€Lee€and€Mowday,€1989;€Singer€and€Willett,€1991).€In€the€higher€education€literature,€eachÏof€the€terms€has€been€used.Ìà  àThe€introduction€of€these€models€to€Institutional€Research€has€occurred€only€recentlyÏ(DesJardins,€1993,€1994;€Chizmar€and€Cummins,€1994;€Heffernan,€et€al,€1995).€And€the€extension€of€thisÏanalysis€to€include€assessment€of€competing€risks€has€been€limited€(Ronco,€1996).€Dynamic€models€ofÏcommunity€college€student€progress€are€especially€lacking.€Ìà  àIn€more€traditional€studies,€the€research€question€for€assessing€student€progress€is€phrased€isÏterms€of€the€factors€that€explain€a€singular€event.€The€dependent€variable€is€binary,€and€is€chosen€toÏrepresent€behavior€at€a€point€in€time;€for€example,€(a.)€persistence€to€the€start€of€the€second€academicÏyear,€or€(b.)€persistence€to€graduation€by€the€end€of€the€third€year.€The€researcher€sets€the€dependentÏvariable€for€the€occurrence€of€an€event€or€non„event,€and€estimates€the€effects€of€covariates€that€areÏfixed.€In€example€(a.),€no€accounting€is€made€for€differences€in€the€probability€of€returning€for€theÏsecond€fall€semester€based€on€whether€the€student€attended€in€the€previous€spring.€The€assumption€isÏmade,€as€is€the€case€with€a€first„order€Markov€process€(Markus,€1979),€that€the€transition€process€is€aÏconstant.€€Event€history€analysis€corrects€this€omission.Ìà  àIf€the€analysis€is€directed€toward€estimating€factors€that€contribute€to€student€stop„out,€and€thereÏis€no€recognition€that€the€student€did€not€return€to€campus€because€they€transferred€to€a€senior€institution,Ïthen€the€coefficient€estimate€is€biased.€Stop„out€is€confounded€by€transfer.€In€the€context€of€student€right„Ð Ü+,'+ ÐññÐ Ü+,'+ Ðññto„know,€a€student€success€is€not€recognized.€This€is€a€special€concern€for€research€on€communityÏcollege€cohorts,€since€students€begin€to€transfer€as€early€as€the€end€of€their€first€semester€at€a€two„yearÏinstitution.€This€paper€addresses€the€issue€of€competing€risks.Ìà  àThe€emphasis€for€this€avenue€of€research€is€placed€on€examining€change€as€a€process€that€occursÏover€time.€When€applied€to€retention€studies,€the€risk€of€attrition€from€semester€to€semester€is€directlyÏexamined.€The€probability€of€student€exit€for€each€semester€in€attendance€can€be€modeled€as€a€functionÏof€covariates€that€are€time„varying,€such€as€GPA,€or€invariant€to€time,€such€as€high€school€rank€and€basicÏskills€deficiencies.€With€competing€risk€analysis,€multiple€events€are€modeled,€and€the€effect€ofÏpredictors€on€the€alternative€outcomes€can€be€compared.Ìà@II'ìàò òMETHODó óˆÐ T Ðò òTHE€DATAÐ ¸ Ðó óà  àA€New€Jersey€community€college€provides€the€1991€cohort€used€in€the€analysis.€Data€was€takenÐ l¼ Ðfrom€institutional€files€and€a€state€tracking€system€database.€The€latter€provided€information€on€studentÏtransfers€to€New€Jersey€public€institutions,€both€two€and€four€year.€The€1288€student€cohort€was€selectedÏby€using€criteria€from€the€definition€of€first„time,€full„time€students€contained€in€"student€right„to„know"Ïlegislation.€They€were€tracked€for€nine€semesters€and€converted€to€a€4600€person€record€file,€one€recordÏfor€each€consecutive€semester€in€attendance€prior€to€leaving€the€institution.€Ìà  àThe€creation€of€a€person€period€file€structure€is€a€necessary€first„step€if€transitions€from€state€toÏstate€are€assumed€to€occur€at€discrete€time€intervals€(Sorensen,€1990).€And€while€concerns€have€beenÏraised€about€autocorrelation€across€observations€for€the€same€individual,€Allison€(1995,€p.223)€states€thatÏ"...€the€creation€of€multiple€observations€is€not€an€ad€hoc€method;€rather,€it€follows€directly€fromÏfactoring€the€likelihood€function€for€the€data."€The€method€is€frequently€used€in€event€history€analysisÏâ âwith€discrete€state€data.€€Ð (*x%' Ðà  àIt€is€a€traditional€approach€in€conducting€event€history€analysis€(DesJardins,€1993;€Chizmar€andÏâ âCummins,€1994;€Heffernan,€et€al,€1995;€Ronco,€1996):€events€are€measured€at€the€end€of€a€semester,€aÏdiscrete€period€in€time.€A€record€is€created€for€every€term€a€student€is€enrolled,€and€discrete€time€modelsÏare€employed€for€the€analysis.€But€Yamaguchi€(1991)€has€conducted€an€analysis€of€student€dropoutsÏfrom€four€year€colleges€that€measure€transition€states€at€monthly€intervals€throughout€the€course€of€theÏacademic€year,€suggesting€that€events€occur€in€continuous€time.€Continuous€time€methods€are€utilized.ÏThis€is€a€significant€point.Ìà  àDiscrete€state€and€continuous€time€models€employ€different€estimation€procedures.€However,Ïdiscrete€state€models€can€be€used€to€approximate€continuous€time€processes.€And€discrete€state€modelsÏcan€be€estimated€with€continuous€time€techniques€if€the€underlying€dimensions€of€time€are€continuous,Ïeven€though€time€is€measured€in€discrete€intervals.€In€research€on€student€careers,€this€issue€is€generallyÏignored.€It€will€be€addressed€in€this€study.Ìò òDEPENDENT€VARIABLESó óÐ  p Ðà  àIn€this€study€of€competing€risks,€students€could€experience€any€one€of€four€events€over€theÏcourse€of€the€nine€semester€study€period.€Transitions€are€assumed€to€occur€in€the€interval€betweenÏsemesters.€Coding€occurs€in€descending€order€based€on€the€priorities€of€the€institutional€mission.€TheÏfirst€code€assigned€is€for€a€retained€student,€one€experiencing€no€event.€If€the€student€is€not€enrolled€in€aÏsubsequent€semester,€then€event€codes€are€assigned€in€the€following€order.Ìà  àThe€dependent€variable€was€coded€as€graduate€if€they€were€graduated€immediately€after€their€lastÏattendance,€coded€as€a€four„year€transfer€if€the€student€was€reported€enrolled€at€a€senior€institutionÏimmediately€after€their€last€attendance,€and€coded€as€a€two„year€transfer€if€the€student€was€reportedÏenrolled€at€a€different€community€college.€Students€were€coded€as€leavers€if€they€left€prior€to€the€start€ofÏthe€ninth€semester€and€had€not€experienced€one€of€the€above€transitions.€At€the€end€of€the€research€periodÏ14€of€the€original€1288€students€were€right€censored;€no€event€had€occurred.€The€statistics€for€decrementsÐ Ü+,'* ÐtoÏthe€risk€set€and€the€classification€of€event€transitions,Ïempirical€hazards,€and€survival€rates,€are€contained€in€ò òTable€1.ó óÌà..àThe€first€five€columns€of€the€table€are€statistics€typicallyÏreported€for€single€event€transitions.€The€first€columnÏidentifies€the€semester€in€attendance.€The€second€column€includesÏthe€number€of€students€enrolled€at€the€start€of€the€semester,€andÏis€decremented€sequentially€by€the€number€of€studentsÏexperiencing€an€undifferentiated€event,€contained€in€column€3,€atÏthe€end€of€the€semester.€Column€2€totals€4600,€the€number€ofÏperson€period€records€included€in€the€analysis.€Column€3€totalsÏ1274,€plus€14€persisting€students,€to€yield€1288,€the€number€ofÏindividuals€at€the€point€of€first€enrollment.Ìà..àColumn€4€represents€the€empirical€hazard€of€an€event,€and€isÏcalculated€by€dividing€column€3€by€the€corresponding€entry€inÏcolumn€2.€Column€5€represents€the€empirical€survival€rate,Ïcalculated€by€dividing€the€total€at€the€start€of€the€firstÏsemester€(1288)€into€that€number€decreased€by€the€numberÏexperiencing€events€in€the€semester(s)€of€interest.€The€survivalÏrate€at€the€end€of€the€second€semester€is€(1288„204„248)/1288€=Ï.649,€a€number€equivalent€to€the€second€year€retention€rateÏwithout€considering€transfers.Ìà..àColumns€6,€8,€10,€and€12€contain€the€number€of€studentsÏexperiencing€alternative€events€for€each€semester.€The€sum€of€theÏentries€in€the€first€and€each€subsequent€row€for€those€fourÏcolumns€equals€the€corresponding€total€in€column€3.€For€the€firstÏsemester,€181+0+2+21€=€204.€The€empirical€hazard€for€each€eventÏis€calculated€in€the€same€way€the€overall€hazard€was€calculated.ÏAnd€across€each€row,€the€four€event€hazards€total€to€column€4Ï(.141+.000+.002.+.016€=€.158€for€the€first€semester).Ìà..àThe€statistics€from€ò òTable€1.ó ó€are€used€to€generate€the€graphsÏin€ò òFigures€1,€2,ó ó€and€ò ò3ó ó.€For€comparative€purposes,€the€graphedÏhazards€from€Ronco€(1996)€and€DesJardins€(1993),€bothÏrepresenting€universities,€are€displayed€in€ò òFigure€1b.ó ó€and€ò òFigureÏ1c.ó ó€respectively€with€the€baseline€hazard€from€this€study,€ò òFigureÏ1a.ó ó€What€do€these€three€graphs€suggest€about€the€relationshipÏbetween€time€and€the€hazard€rate?€Are€there€any€clear€patternsÏthat€might€suggest€a€choice€between€discrete€or€continuous€timeÏmodels?€Is€there€a€functional€form€that€could€be€used€to€specifyÏthe€time€dimension?€These€questions€will€be€addressed€later€in€the€paper.Ìò òCOVARIATESó óÌà..àFour€background€variables,€all€fixed€prior€to€initialÏenrollment,€are€included€in€the€analysis.€They€are€(1)€highÏschool€rank€(HSRANK),€(2)€financial€aid€recipient€(GRANT),€(3)Ïnew€high€school€graduate€(NEWHS),€and€(4)€remedial€coursesÏrequired€(REMREQ).€High€school€rank€is€coded€in€percentiles.ÏFinancial€aid€receipt€is€binary€coded;€1€represents€students€whoÏwere€awarded€a€grant€or€scholarship€not€requiring€repayment,€andÏ0€otherwise.€New€high€school€graduate€is€binary€coded;€a€1Ïrepresents€students€who€graduated€in€the€spring€immediately€priorÏto€their€first€fall€community€college€enrollment,€and€0Ïotherwise.€Remedial€course€requirements€is€ordinal,€and€rangesÏfrom€no€courses€to€a€maximum€of€six.Ìà..àTwo€time„varying€covariates€are€included€in€the€analysis;Ïcumulative€GPA€(AVERS)€at€the€end€of€each€semester€in€attendance,Ïand€the€number€of€courses€dropped€(DROPYR)€in€each€semesterÏattended.€The€analysis€also€includes€measures€of€these€variablesÏas€fixed€effects€determined€in€the€student's€first€semester€ofÏenrollment.€A€comparison€of€the€resulting€parameter€estimates,Ïbased€on€these€two€different€specifications,€allows€forÏdiscussion€of€the€importance€of€including€time„varying€covariatesÏin€analyzing€student€careers.Ìò òAPPROACHES€TO€MODELING€TIMEó óÌà..àA€semester€has€a€start€date€and€ending€date.€It€is€aÏdiscrete€interval€of€time.€But€does€that€mean€that€all€stateÏtransitions€occur€at€the€end€of€that€interval?€Such€is€frequently€the€case€when€Institutional€Researchers€conduct€event€historyÏstudies,€and€the€methods€employed€for€analysis€reflect€thisÏnotion€of€time€as€a€discrete€interval€(DesJardins,€1993;€ChizmarÏand€Cummins,€1994;€Ronco,€1996).€The€many€citations€for€SingerÏand€Willett€or€Willett€and€Singer€reference€discrete€time€methodsÏas€the€appropriate€type€for€studying€student€careers.€Ìà..àBut€Yamaguchi€(1991)€also€studied€student€careers,€examiningÏdropout€behavior€of€four„year€college€students,€and€he€usedÏcontinuous€time€methods.€In€his€study,€dropouts€were€recorded€onÏa€monthly€basis,€suggesting€the€process€occurs€continuously,€andÏnot€just€at€the€end€of€the€semester.€And€students€can€changeÏtheir€major€at€any€point€in€the€term.€Heffernan,€et€al€(1995)Ïmeasured€attrition€as€a€discrete€state€process,€but€specifiedÏtime€as€a€linear€function,€and€included€a€quadratic€term€toÏcapture€a€curvilinear€effect.€Must€time,€in€a€measured€discreteÏstate€model,€that€may€represent€an€underlying€continuous€timeÏdistribution,€necessarily€be€represented€by€a€nonparametricÏapproach?€Allison€(1984,€p.22)€suggests€otherwise.€"The€choiceÏbetween€discrete„€and€continuous„time€methods€should€generally€beÏmade€on€the€basis€of€computational€cost€and€convenience."€If€theÏmodel€fits,€use€it.Ìà..àBut€discrete„time€models€do€have€a€rationale.€With€aÏnonparametric€specification,€they€make€virtually€no€assumptionsÏabout€the€distribution€of€event€times€(Allison,€1984).€They€mayÏbe€represented€in€estimation€procedures€as€dummy€variablesÏindexing€a€time„varying€intercept.€And€they€result€in€less€biasedÏparameter€estimates€when€unobserved€heterogeneity€is€suspected€(Galler€and€Poetter,€1990).€In€the€absence€of€theoreticalÏguidance,€they€appear€a€good€place€to€start.Ìà..àOther€alternatives€are€available€however.€Exponential€modelsÏspecify€no€dependence€on€time€for€event€transitions.€Time€is€aÏconstant,€and€no€measure€of€it€is€included€in€model€estimation.€ÏParametric€methods€are€also€available,€requiring€theÏspecification€of€a€functional€form€for€time.€If€the€effects€ofÏtime€are€presumed€to€be€monotonic,€a€Gompertz€or€Weibull€approachÏmay€be€employed.€In€the€former,€time€has€a€linear€functionalÏform;€in€the€latter,€time€has€a€logged„linear€specification.€InÏeither€case,€a€single€variable€can€be€employed€in€the€analysis€toÏrepresent€time€„€a€much€more€"convenient"€approach€than€a€longÏseries€of€dummy€variables.Ìà..àIn€this€paper,€all€four€approaches€will€be€utilized€toÏestimate€causal€models€of€student€progress.€They€will€be€assessedÏindirectly€with€a€likelihood€ratio€test€to€identify€the€modelÏwith€the€best€fit.€The€parameter€estimates€will€also€be€comparedÏas€a€test€for€unobserved€heterogeneity.Ìò òà@ì..HàRESULTSˆÌBASELINE€ESTIMATESó óÌà..àStandard€procedure€is€used€to€estimate€the€baseline€hazardÏfunction.€Eight€dummy€variables€are€used€to€index€time.€TheÏintercept€is€suppressed.€The€covariate€vector€is€set€to€zero.€TheÏmaximum€likelihood€estimator€employs€logit€analysis€for€fourÏequations.€When€applied€to€a€procedure€for€competing€risks,€it€isÏmore€commonly€known€as€a€multinomial€logit€model€(MNL).€"TheÏprocedure€is€justified€as€a€form€of€conditional€likelihood.€The€resulting€estimates€are€consistent€and€asymptotically€normal"Ï(Allison,€1995).Ìà..àThe€results€are€supportive€of€the€empirical€hazards€reportedÏearlier.€For€leavers,€the€risk€of€exit€increases€with€time.€TheÏhighest€relative€risks€are€at€the€end€of€the€even€numberedÏsemesters,€between€spring€and€fall.€The€last€semester€coefficientÏis€not€significant.€The€heightened€risk€of€event€occurrence€afterÏthe€spring€term€is€also€evident€for€graduates,€however,€theÏcoefficients€for€the€sixth€and€eighth€semesters€are€notÏsignificant.€Four€year€transfers€also€experience€an€increasingÏrisk€with€time,€with€odd/even€swings€also€noted.€The€risk€of€two„¼year€transfer€is€greatest€in€the€first€two€semesters€and€theÏfourth€semester€before€falling€to€zero.€Time€dependence€of€theÏhazard€rate€is€established.€And€a€reasonable€approximation€of€aÏpattern€for€the€coefficients€is€evident.Ìò òTIME€VARYING€COVARIATESó óÌà..àFor€those€inexperienced€in€the€use€of€event€history€models,Ïthe€meaning€and€importance€of€time„varying€covariates€may€not€beÏclearly€understood.€Two€are€included€in€this€discussion,Ïcumulative€GPA€and€the€number€of€courses€dropped€in€a€semester.ÏThey€are€coded€in€two€different€ways,€and€added€to€the€baselineÏmodel€where€comparisons€can€be€made€for€model€fit€and€coefficientÏinterpretation.Ìà..àCourse€drops€are€coded€in€one€model€as€time„varying.€TheÏvalues€change€for€each€semester,€dependent€upon€changes€inÏstudent€behavior.€Course€drops€are€fixed€in€the€other€model,€andÏrepresent€only€the€number€of€courses€dropped€in€the€first€semester€of€attendance.€This€value€remains€a€part€of€each€studentÏrecord€for€as€long€as€they€are€enrolled.Ìà..àThe€mean€values€for€these€variables€are€plotted€in€ò òFigureÏ4a.ó ó€and€ò òFigure€4b.ó ó€In€4a.€the€plots€are€consistently€divergentÏfor€two„year€transfers.€The€plots€for€four„year€transfers€divergeÏin€the€seventh€and€eight€semester.€In€4b.€the€plots€areÏdrastically€different€for€leavers,€but€follow€a€similar€track€forÏgraduates.€Where€divergence€occurs,€one€expects€dissimilarÏresults€for€the€coefficient€estimates.€ò òTable€3.ó ó€contains€theÏresults.€The€greatest€differences€in€coefficient€estimates€appearÏfor€leavers€and€two„year€transfers,€just€as€the€plots€suggested.ÏFor€leavers,€the€number€of€courses€dropped€in€the€first€semesterÏhas€no€long€term€effect,€but€for€continuing€students€there€is€aÏstrong€positive€effect;€the€more€courses€dropped,€the€greater€theÏrisk€of€exit.€For€two„year€transfers,€it€is€clear€that€droppingÏcourses€in€the€first€semester€is€a€signal€for€departure.€Most€ofÏthese€transfers€occur€early€in€the€student's€career,€explainingÏwhy€there€is€no€continuing€effect.Ìà..àThe€coefficients€for€GPA€are€all€significant€in€the€sameÏdirection,€though€the€magnitudes€are€always€greater€for€the€time„¼varying€specification.€This€suggest€that€in€the€long€term,Ïstudents€can€recover€from€a€poor€start,€but€GPA€will€continue€toÏhave€a€major€influence€in€their€career.€The€GPA€means€plotted€inÏò òFigure€5aó ó€and€ò òFigure€5bó ó€support€the€similarities€for€theÏcoefficient€estimates;€there€is€very€little€divergence.Ìà..àThe€model€comparisons€with€the€baseline€measure€indicatesÏboth€fixed€and€time„varying€specifications€are€an€improved€fit,€but€the€time„varying€covariate€model€is€clearly€superior.Ìò òFULL€MODEL€ESTIMATESó óÌà..àAlternative€approaches€to€specifying€the€functional€form€ofÏtime€were€introduced€earlier.€In€this€section,€we€apply€each€ofÏthose€specifications€noted€previously.€A€nonparametric,Ïexponential,€Gompertz,€and€Weibull€model€will€be€estimatedÏseparately€for€each€event€type.€Multinomial€logit€analysis€(MNL)Ïis€used€to€estimate€the€equations.€I€also€use€an€alternateÏestimation€procedure,€appropriate€for€competing€risk€models,Ïwhich€Allison€(1995)€labels€a€full„information€maximum€likelihoodÏ(FIML)€estimator.€It€yields€more€precise€coefficient€estimatesÏthan€the€MNL.€But€the€procedure€estimates€all€equationsÏsimultaneously,€and€requires€considerably€more€computation€time.ÏAnd€Hanushek€and€Jackson€(1977)€warn€that€FIML€estimators€areÏvery€sensitive€to€specification€errors.€But€this€is€a€qualityÏthat€will€prove€useful€in€examining€the€issue€of€unobservedÏheterogeneity.€There€is€no€consensus€on€how€to€proceed.Ìà..àAllison€(1984,€p.33)€notes€that€biostatisticians€see€theÏeffect€of€unobserved€heterogeneity€as€a€change€in€the€shape€ofÏthe€distribution€of€event€timing€(T),€"and€that€this€can€beÏaccommodated€by€specifying€a€different€distribution€of€T€...€orÏby€using€a€more€general€model...€ò òThis€position€is€reasonable€soÏlong€as€one€is€primarily€concerned€with€estimating€the€effects€ofÏthe€explanatory€variables€and€is€not€particularly€interested€inÏtesting€hypotheses€about€the€effects€of€time.ó ó"€For€Blossfeld€andÏRohwer€(1995,€p.239),€unobserved€heterogeneity€is€often€a€resultÏof€excluding€important€causal€factors.€They€suggest€estimating€"a€series€of€models€with€different€specifications€and€then€check€toÏsee€whether€the€estimation€results€are€stable€or€not."€TheÏprocedure€in€this€study€is€to€use€alternative€specifications€ofÏtime.€Ìà..àThe€results€of€the€analysis€for€the€four€event€types,€usingÏnonparametric,€Gompertz,€and€Weibull€specifications,€andÏestimated€with€both€an€MNL€and€an€FIML€procedure€are€contained€inÏò òTables€4a€„€4dó ó.€The€estimates€for€the€different€functional€formsÏof€time,€since€they€are€not€equivalent,€are€not€included.€ForÏeach€event,€the€parameter€estimates€with€the€extreme€scores€forÏthe€individual€covariates€are€highlighted,€ò òboldó ó€for€the€largest,Ïand€òòitalicóó€for€the€smallest.€This€makes€the€range€of€estimatesÏmore€apparent.€In€virtually€every€case,€the€signs€andÏsignificance€levels€for€the€estimates€are€equal.€And€theÏmagnitude€of€the€coefficients€are€very€similar,€suggesting€aÏsurprising€degree€of€robustness.€The€estimates€are€stable.€TheÏextreme€scores€do€not€appear€to€be€characteristic€of€anyÏparticular€specification€or€estimation€procedure.Ìà..àThe€next€decision€is€to€select€the€model€with€the€best€fitÏfor€the€data.€Model€chi„square€statistics€for€the€FIML€procedureÏare€ignored,€since€the€estimates€were€generated€to€identifyÏpossible€specification€errors.€The€focus€will€be€on€the€modelsÏwith€alternate€specifications€of€time.€Since€a€direct€likelihoodÏratio€test€is€not€appropriate€for€models€with€differentÏfunctional€forms€of€time,€indirect€comparisons€are€made€with€theÏrestricted€model,€the€exponential.€The€„2€Log€Likelihood€scoresÏfor€that€model€are€in€the€lower€left€corner€of€each€table.€For€each€event€estimated,€the€exponential€model€provides€the€worstÏfit€with€one€exception.€It€provides€the€best€fit€for€two„yearÏtransfers.€In€all€other€cases,€the€likelihood€ratio€chi„squareÏstatistics€are€significant€at€better€than€.001,€making€modelÏselection€difficult.€Ìà..àThe€Weibull€specification€appears€to€be€a€better€fit€thanÏthe€Gompertz,€but€the€most€significant€difference€is€for€theÏgraduates.€And€in€that€case,€the€nonparametric€specificationÏappears€superior.€For€four„year€transfers,€the€nonparametricÏspecification€appears€to€have€the€edge.€But€for€the€leavers,Ïthere€is€no€clearly€superior€model.€A€reexamination€of€Figure€3.Ïsuggests€an€increase€in€the€hazard€over€time€that€is€much€moreÏstable€than€other€risk€sets.€Possibly€linear?€For€convenience,Ïand€that€alone,€I€will€use€the€nonparametric€specification€as€theÏbase€to€conclude€this€paper.Ìò òTESTING€THE€PROPORTIONALITY€ASSUMPTIONó óÌà..àWhen€the€proportionality€assumption€is€violated,€and€noÏinteraction€with€time€is€included€in€the€model,€then€theÏcoefficients€represent€an€average€effect€of€time.€If€theÏinteraction€with€time€is€properly€specified,€the€method€shouldÏresult€in€more€efficient€estimates€of€the€other€covariatesÏ(Allison,€1995).€The€interaction€effects€of€time€can€beÏadequately€characterized€by€specifying€a€functional€form€for€timeÏ(Yamaguchi,€1991).€Ìà..àIn€a€test€for€violations€of€the€proportionality€assumption,Ïthe€interaction€of€a€logged€(logòòtóó)€specification€of€time€isÏtested€for€each€covariate€in€the€model.€For€those€adding€significantly€to€the€fit€of€the€model,€multiple€interactions€areÏtested.€This€procedure€is€continued€until€the€best€fitting€modelÏis€obtained€for€each€event€transition.€That€determination€is€madeÏwith€a€likelihood€ratio€test€comparing€the€newly€fitted€modelÏwith€the€base€model,€which€contains€the€8€dummy€variables€forÏtime€(they€have€no€inherent€interpretation€and€are€not€reported)Ïand€the€six€covariates.€The€results€are€contained€in€ò òTable€5.ó ó€Ìà..àThe€final€models€selected€are€as€follows:€(a)€ò òLeaversó óÏincludes€an€interaction€term€with€NEWHS,€ò òGraduatesó ó€includes€anÏinteraction€with€REMREQ,€and€(c)€ò ò4„Year€Transfersó ó€includesÏinteractions€with€GRANT€and€DROPYR.€No€improvements€were€foundÏfor€ò ò2„Year€Transfersó ó.€The€parameter€estimates€for€the€finalÏmodels€are€included€in€ò òTable€6ó ó.€Comparisons€with€ò òTable€4.ó ó€entriesÏclearly€demonstrate€the€importance€of€including€measures€forÏtime„dependency.Ìà..àSorensen€(1990)€provides€the€interpretation€for€theÏsignificant€logòòtóó*covariate€interaction€terms.€Those€with€aÏnegative€coefficient€suggest€a€"negative€time€dependency";€theÏeffect€of€the€coefficient€on€the€hazard€decreases€with€time.€AÏpositive€coefficient€suggests€the€opposite;€the€effect€of€theÏcoefficient€on€the€hazard€decreases€with€time.€The€totalÏcovariate€effect€is€calculated€as€the€sum€of€the€main€and€theÏinteraction€effect.Ìà..àThe€ò òLeaveró ó€model€has€one€interaction€term.€It€is€positive,Ïsuggesting€the€effect€of€being€a€new€high€school€studentÏincreases€with€time.€Since€the€main€effect€is€negative,€theÏoverall€interpretation€is€that€students€just€out€of€high€school€have€a€declining€risk€of€exit€as€the€number€of€semesters€enrolledÏincreases.€The€total€effect€of€the€NEWHS€covariate€is€Ì(„0.9416)€+€(0.4281)€=€„0.5135,€a€close€comparison€with€the€baseÏestimate€of€„0.5857.€The€remaining€parameters€are€relativelyÏunchanged€in€comparison€with€the€base€model.€The€interpretationsÏfollow:€(a)€as€GPA€increases,€the€risk€of€leaving€decreases;€(b)Ïhigh€school€rank€has€no€effect;€(c)€receipt€of€financial€aidÏdecreases€the€risk€of€leaving;€(d)€as€the€number€of€remedialÏcourse€requirements€increases,€the€risk€of€leaving€decreases;€(e)Ïan€increase€in€the€number€of€courses€dropped€increases€the€riskÏof€leaving.Ìà..àIn€the€ò òGraduateó ó€model,€there€is€one€interaction€included,Ïand€its€effect€is€striking.€In€the€base€model,€the€coefficientÏfor€remedial€requirements€is€not€significant,€but€the€inclusionÏof€the€interaction€term€makes€both€the€main€effect€and€theÏinteraction€significant.€And€the€improvement€in€the€model€fit€isÏconsiderable.€This€suggests€that€a€failure€to€account€forÏviolations€of€the€proportionality€assumption€has€seriousÏconsequences€leading€to€biased€parameter€estimates.€TheÏcoefficients€suggest€that€increases€in€remedial€courseÏrequirements€decreases€the€odds€of€graduating,€and€that€the€riskÏof€not€graduating€increases€with€time.Ìà..àThe€remaining€coefficients€are€interpreted€as€follows:€(a)Ïincreases€in€GPA€increase€the€odds€of€graduating;€(b)€HSRANK€hasÏno€effect;€(c)€financial€aid€has€no€effect;€(d)€newly€graduatedÏhigh€school€students€have€better€odds€of€graduating;€and€(e)€theÏmore€courses€a€student€drops,€the€greater€the€risk€they€will€not€graduate.Ìà..àThe€ò ò4„Year€Transferó ó€model€has€two€interaction€terms.€BothÏmain€effect€covariates€in€the€base€model€were€nonsignificantÏwithout€the€interactions,€but€in€the€final€model,€both€the€mainÏand€interaction€terms€are€significant.€Again,€by€not€estimatingÏthe€extent€of€nonproportionality,€the€covariates€in€the€baseÏmodel€are€biased.€The€risk€ratios€do€vary€with€time.€ForÏfinancial€aid€recipients,€the€risk€of€transfer€to€a€seniorÏinstitution€is€much€greater€than€nonrecipients,€but€the€riskÏdeclines€with€increases€in€the€number€of€semesters€enrolled.€AndÏthe€increases€in€the€number€of€courses€dropped€reduces€the€oddsÏthat€a€student€will€transfer,€and€that€reduction€in€riskÏincreases€with€time.Ìà..à€The€remaining€coefficients€are€interpreted€as€follows:€(a)Ïstudents€with€higher€GPAs€are€at€greater€risk€for€transfer€thanÏthose€with€lower€grades;€(b)€HSRANK€has€no€effect;€(c)€NEWHS€hasÏno€effect;€(d)€increases€in€the€number€of€remedial€courseÏrequirements€lead€to€decreases€in€the€risk€of€transfer.Ìà..àThe€ò ò2„Year€Transferó ó€model€has€only€one€significantÏcovariate.€This€can€be€attributed,€in€part,€to€the€small€numberÏof€students€experiencing€the€event.€But€improvements€to€theÏestimate€of€this€equation€can€be€made.€For€community€collegeÏstudents,€this€transition€is€very€likely€a€function€of€county€ofÏresidence.€The€one€significant€variable,€GPA,€suggests€thatÏstudents€making€the€lateral€transfer€to€another€community€collegeÏhave€lower€grades€than€persisting€students.€There€is€considerableÏsimilarity€with€the€coefficient€in€the€ò òLeaversó ó€model,€suggesting€that€both€events€have€similar€motivations.Ìà@ì..”àò òDISCUSSIONó óˆÌà..àSubstantively,€many€of€the€parameter€estimates€have€logicalÏinterpretations.€Students€with€better€grades€are€more€likely€toÏpersist€to€graduation,€but€they€are€also€likely€to€transfer€to€aÏsenior€institution.€From€a€policy€perspective,€both€eventsÏsatisfy€a€community€college's€(CC)€mission.€From€a€financialÏperspective,€early€transfer€leads€to€a€decline€in€revenue.ÏTransfers€are€also€more€likely€to€be€a€result€of€studentÏfinancial€aid€receipt.€The€common€argument€used€by€CC's€is€thatÏearly€transfer€will€cost€the€student€more€money€in€increasedÏtuition,€But€if€the€student's€tuition€is€covered€by€aid,€there€isÏno€financial€loss.€Ìà..àCourse€dropping€behavior€does€not€lead€to€student€success.ÏIt€increases€the€risk€of€leaving,€and€decreases€the€risk€of€bothÏgraduation€and€transfer.€This€finding€presents€a€challenge€to€theÏnotion€that€students€with€good€grades€drop€courses€to€preserve€aÏhigher€GPA.€More€research€is€needed€in€this€area.€Would€stricterÏcontrols€on€drops€produce€more€successes?Ìà..àStudents€who€delay€college€entry€are€at€greater€risk€ofÏleaving,€and€have€lower€odds€of€graduating.€Should€the€admissionsÏoffice€be€more€aggressive€in€recruiting€high€school€students?€DoÏolder€students€need€more€support€to€graduate?€There€appears€to€beÏno€effect€on€transfer€however.€Perhaps€older€students€are€just€asÏinterested€in€transfer€as€their€younger€counterparts,€and€are€notÏinterested€in€the€Associate's€Degree.Ìà..àAnd€finally,€requiring€a€student€to€take€more€remedial€courses€can€retain€them€longer,€but€they€have€much€lower€odds€ofÏgraduating€or€transferring.€Longer€student€stays€benefit€theÏinstitution€financially,€but€eventually€a€full€schedule€of€creditÏcourses€will€take€its€toll,€reducing€the€odds€of€a€studentÏsuccess.€Should€students€with€severe€deficiencies€be€deniedÏadmissions,€or€is€the€community€college€providing€them€their€bestÏopportunity?Ìà@ì..l àò òSUGGESTIONS€FOR€FURTHER€RESEARCHó óˆÌà..àThis€paper€has€presented€the€benefits€of€using€an€eventÏhistory€analysis€of€competing€risks.€Suggestions€have€also€beenÏmade€for€a€rethinking€of€the€need€to€conduct€research€usingÏnonparametric€models€based€solely€on€discrete€time€procedures.ÏAre€student€careers,€measured€in€discrete€intervals,€really€aÏreflection€of€an€underlying€continuous€time€process.€And€whenÏcausal€models€are€the€primary€research€interest,€can€time€beÏrelegated€to€the€status€of€a€"nuisance€variable"?€These€areÏmethodological€issues€requiring€further€investigation.Ìà..àThis€paper€has€also€identified€the€importance€of€identifyingÏand€controlling€for€violations€of€the€proportionality€assumption.ÏThe€use€of€a€logòòtóó*covariate€interaction€term€has€provided€clearÏevidence€of€the€bias€that€can€result€when€proportionality€isÏincorrectly€assumed.€It€is€a€much€more€convenient€specificationÏthan€a€long€string€of€time€dummy€variable€interactions.€PerhapsÏit€is€simply€an€issue€between€research€on€community€collegeÏstudents€and€their€senior€institution€colleagues.Ìà..àOne€cannot€overlook€the€importance€of€poorly€measuredÏvariables.€This€is€especially€important€in€a€competing€risk€model,€where€the€inability€to€obtain€data€on€studentsÏtransferring€to€private€institutions€or€out„of„state€schoolsÏleads€to€biased€estimates€in€leaver€models.€Many€leavers€areÏreally€transfers.Ìà..àAnd€finally,€the€general€category€of€student€leavers€wasÏleft€undefined.€No€attempt€was€made€to€differentiate€between€aÏ"stopout"€and€a€"dropout".€Future€research€will€need€to€addressÏthis€issue€by€adding€analysis€of€repeat€events.€And€in€the€longÏrun,€perhaps€we€can€all€move€toward€including€greaterÏconsideration€of€structural€factors€in€our€models,€and€focus€onÏthe€issue€of€student€career€"patterns".€Emphasize€sequences€ofÏevent€transitions€and€compare€the€patterns€with€institutions€Ïhaving€very€different€profiles.€If€we€do€not€move€in€thisÏdirection,€we€will€always€seek€answers€to€our€questions€byÏexamining€characteristics€of€individuals.Ì