ÿWPCx  ½¿r•_Ÿ’œ${Ÿ{ÇöÅlNñ È2£„jדOÂíª`è†"IÏ Q'°`ƒˆæ#°”ÙCûVŸe#yíìÛÓ'£â<×cj] ³SH×ä럈Mø#’M't¢Y‹µðÜÅx%£ŒŠ]µ²qi[W2Ó§4ƒY<ý[W¨+¾_ “Úö¾€îT¦uòk¹˜+ÍW§Ñ¡‘s B¬¦6‹²Œ26{5ˆ£oxÃR-Võ¥ÕCÈ+u© -}î n!.åÄ›…Å?ƒŽõ^Ap"C|S ¯e¿è6T댱hPÓŸXšU€*–ê|앇‚ï!=-i á_W—ùâT…¥­U5±0^U©Pìb-¾:…—ˆûvŠs!¡cØ ¹N« øµvp;îŠÔÅ1Ì| gËÊ;èÐ](»qY=÷“ô)h¾¢dÈl9v¨'´;½ª¸ä‘Y KWG°-n aJ})¡â…ƒÝô 3Ëý X&)0rnK±ì\¥j†r ¬<ùŒž³1¶á©ë«­õq77)Óæ­CN°–Ð !F*Hªô*¡®ÐÒ/ì5›¡Ný¼”¢6ÏÕ˜·“Æ„¥9ä\}Äü:çv3í¼BœUN¬ %ú 0(w@( 0Dh£¬OOOžní  [{4ƒ 0D— 0 DÛ 0[  z  ¤ ¹ OÎ n O‹ £Ú UN} } } } } } } D3Ë Bþ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ i i: : : : iY 3|x2(ÖÃ9 Z ‹6Times New Roman RegularX($¡¡ )*+D,ƒ-.E/ƒ0ƒ1E2(3¯$§§Ý ƒ!ÝÝ  Ý ™Ý ƒ¯)ÝÝ  Ýà@ºººìàÔ‡ŠÓ›ŠXXÔÔ#†XXŠŠÓ›-#ÔÚ  Ú3Ú  Úà@@ìì*ìà EÝ ƒ¯)ÝÝ  Ýà@ºº*ìàÚ  Ú3Ú  Ú EÝ ƒ¯)ÝÝ  Ýà@ºº*ìàÚ  Ú2Ú  Ú dÝ ƒ¯)ÝÝ  Ýà@ºº*ìàÚ  Ú2Ú  Úà@@ìì*ìàÔ% € ÔhHeader/Footer§qˆ"H($Ó$‘‘òòÚ  Ú0Ú  Úóó(3¯$©©Ý ƒ!ÝÝ  Ý(3¨à$¥¥Ý ƒ!ÝÝ  ÝÚ  Ú0Ú  Ú. Ñ  Ñ Ñ  Ñ Ñ  Ñ Ñ  Ñ EÝ ƒ¯)ÝÝ  Ýà@ºº*ìàÚ  Ú2Ú  Ú dÝ ƒ¯)ÝÝ  Ýà@ºººìàÚ  Ú2Ú  Úà@@ìì*ìàÔ% € Ô EÝ ƒ¯)ÝÝ  Ýà@ºº*ìàÚ  Ú3Ú  Ú ™Ý ƒ¯)ÝÝ  Ýà@ººˆìàÔ‡Šó‹ŠXXÔÔ#†X%JXŠŠó‹-#ÔÚ  Ú3Ú  Úà@@ìì*ìà(ÖÃ9 Z ‹6Times New Roman Regular""""'ÿÿdxd d!"úÿ!ÿÿÝ ƒ!ÝÝ  Ýññ›ññññññ›ñññññññ ñ›ñ ñññÑH°ÑÑH°ÑÓÓÔ‡ V¿ XXÔÓ  ÓÌSURVIVAL€ANALYSIS€OF€FACULTY€RETENTION€DATA:ÌHOW€LONG€DO€THEY€STAY?ÌÔ#†XX  V¿~#ÔÓÓÓÓÌÔ€¼¼XXÔÌÌÌÓ  ÓÔ€ŠÓ›Š¼¼ÔMike€Tamada€and€Claudia€InmanРȀ  ÐOccidental€CollegeÌLos€Angeles,€CAÌÓ!ÓÓVÓÌÓÓÓ  ÓPresented€at€the€Association€for€Institutional€ResearchÌ€37th€Annual€Forum,€Orlando€FLÌÓýÓÓÓÓÓà0  àÓ˜Óà@&ì(#(#àMay,€1997ˆÌÌÌÌÓÓABSTRACT:ÌÌÓ  Ó€€€€€This€is€an€introduction€to€survival€analysis,€applied€to€faculty€retention€data.€€If€a€collegeÏis€concerned€about€how€long€its€faculty€stay,€€especially€women€faculty€(perhaps€for€TitleÏIX€purposes),€then€two€questions€arise:€€how€to€measure€retention,€and€how€to€discernÏwhether€men€and€women€have€different€"survival€times".€€A€set€of€special€statisticalÏtechniques€known€as€"survival€€analysis"€is€useful€for€answering€questions€such€as€these.€€WeÏinformally€describe€the€techniques€used€and€why€they€are€useful.€€Looking€at€all€tenure€trackÏfaculty€from€1960,€we€found€that€women's€retention€was€essentially€the€same€as€men's;€Ïhowever€the€data€do€not€tell€us€the€reason€for€departures.€€ÌÌÌœñ ñACKNOWLEDGEMENTñ ññ ñACKNOWLEDGMENTñ ñññSññ:›ÌÌ€€€€€We€thank€Amber€Reisz€for€invaluable€research€assistance,€and€the€California€ñ ñAssocationñ ññ ñAssociationñ ñÏfor€Institutional€Research€Conference€Presentations€Committee€for€helpful€comments.€€ThisÏwork€was€supported€in€part€by€a€summer€research€fellowship€from€the€National€ScienceÏFoundation€and€California€Alliance€for€Minority€Participation€(NSF„CAMP).Ð  N.+* Ðñ ñÖ€ ÿÿÖñ ññ ñÖ€ÿÿÖñ ñÔ‡¼¼ŠŠÓ›ÔÑ€P¸ÑÑ€P¸ÑÑ€@¸ÑÑ€P¸ÑSECTION€I.€€INTRODUCTIONÔ#†ŠÓ›Š¼¼à#ÔÐ H ÐÌÌÓôÓñ ñÖ€  ìÖñ ññ ñÖ€ ìÖñ ñÓ àÓHow€long€do€faculty€stay€at€a€college,€and€do€male€and€female€faculty€have€differentÏ"survival€times"?€€Questions€such€as€these€involve€data€with€special€characteristic,€andÏspecial€statistical€methods,€known€variously€as€"survival€analysis"€€or€"duration€analysis"€orÏ"analysis€of€failure€time€data"€are€required.€€This€is€an€informal€introduction€to€survivalÏanalysis,€applied€to€faculty€retention€data.€€ÌSurvival€analysis€is€gradually€becoming€more€widespread€in€the€social€sciences.€€See,Ïfor€example,€a€review€of€the€econometric€literature€by€Kiefer€(1988),€econometric€work€byÏHeckman€(1980),€and€introductory€articles€in€the€psychology€literature€by€Morita€et€al€(1989)Ïand€Singer€and€Willet€(1991).€€Institutional€researchers€are€starting€to€use€survival€analysis;Ïa€prominent€example€is€the€recent€article€for€the€AIR€Professional€File€by€Ronco€(1991)Ïdescribing€the€ð ðcompeting€risksðð€model.€Also,€at€the€1995€California€Association€forÏInstitutional€Research€Conference,€Garcia€(1995)€used€life€table€methodology€to€trackÏstudent€retention€and€graduation€rates.€€€Statistical€packages€such€as€SPSS€are€gradually€beingÏgiven€more€and€more€powerful€survival€analysis€capabilities,€enabling€researcher€to€moreÏeasily€carry€out€such€analyses.ÌSurvival€analysis€is€useful€for€answering€questions€involving€some€sort€of€duration;€theÏquestion€could€be€the€survival€time€of€cancer€patients,€the€duration€of€€unemployment€spells,Ïthe€age€at€which€people€first€get€married,€€the€retention€of€faculty€„„€in€short,€any€questionÏinvolving€the€length€of€time€that€passes€until€a€certain€event€occurs€(death,€employment,Ïmarriage,€termination€or€exit€from€the€school,€and€so€on).Ð ¼+t() ÐÑP¸ÑÑ  ÑÑ  ÑBut€most€undergraduate€and€even€graduate€statistics€courses€in€ñ ñÖ€ ìÖñ ññ ñÖ€ ìÖñ ñmost€disciplines€do€notÏcover€survival€analysis.€€This€paper€will€ñ ñÖ€ìÖñ ññ ñÖ€ìÖñ ñ€introduce€a€few€of€the€concepts€of€survival€analysis,Ïstarting€with€the€basic€definitions€and€moving€up€to€regression€analysis€of€survival€data.€ÏThese€techniques€will€be€applied€to€faculty€retention€data,€in€particular€to€test€the€nullÏhypothesis€that€male€and€female€faculty€at€a€private€liberal€arts€college€have€equal€survivalÏtimes.ÌÌà0  àà ° àPRIOR€RESEARCH€ON€FACULTY€RETENTIONÐ (#(# ÐAlthough€many€studies€of€gender€differences€among€faculty€exist€(see€Dwyer€et€alÏ(1991)€€for€a€survey),€longitudinal€studies€of€faculty€retention€are€much€rarer.€€Most€earlierÏstudies€seem€to€have€found€higher€mobility€rates€(i.e.€lower€retention€rates)€for€female€facultyÏthan€for€male€faculty.€€However,€these€either€were€comparative€rather€than€longitudinal,€orÏdealt€with€only€a€specific€subset€of€faculty,€such€as€psychology€faculty€(Rosenfeld€and€JonesÏ1986)€or€part„time€faculty€(Tuckman€and€Tuckman€1981).€€This€study,€though€covering€onlyÏone€school,€covers€all€tenure„track€faculty€in€all€fields€and€follows€them€longitudinally.€€ByÏcovering€the€faculty€at€one€school€only,€this€study€does€lose€generality,€but€at€the€same€time€Ïavoids€the€complications€involved€in€comparing€faculty€at€research€institutions€with€thoseÏat€teaching€institutions,€and€comparing€faculty€of€widely€divergent€backgrounds€and€qualityÏlevels.€€Moreover,€this€study€illustrates€how€a€wider€ranging€study€could€be€performed,€ifÏlongitudinal€data€on€a€variety€of€institutions€were€gathered.ÌAshenfelter€and€Card€(1996)€are€working€with€TIAA/CREF€and€the€PrincetonÏRetirement€Survey€to€create€a€database€with€which€they€can€study€faculty€retirement€usingÐ  ,X)* Ðsurvival€analysis€techniques.€€However€this€databaseððs€usefulness€in€studying€the€retentionÏof€€junior€faculty€will€be€somewhat€limited€because€it€€will€not€include€professors€who€leftÏtheir€schools€prior€to€1986.ÌÌIn€Section€II€we€will€introduce€some€of€the€fundamental€concepts€used€in€survivalÏanalysis:€€survivor€functions,€hazard€functions,€and€censoring.€€€In€Section€III€we€willÏdescribe€the€issue€being€researched,€namely€faculty€retention€by€gender,€and€describe€theÏdata€set.€€In€section€IV€we€will€describe€simple€techniques€for€analyzing€the€data,€namelyÏusing€life€tables€to€look€at€the€survivor€functions€and€performing€log„rank€tests€forÏdifferences€between€the€genders.€€In€section€V€we€will€discuss€more€complex€techniques,€Ïsuch€as€Coxððs€proportional€hazards€regression€model.ÌÌÓ àÓÔ‡¼¼ŠŠÓ›ÔSECTION€II.€€FUNDAMENTAL€CONCEPTSÔ#†ŠÓ›Š¼¼‚#ÔÐ è  ÐÌLIFE€TABLES€AND€SURVIVOR€FUNCTIONSÌÓ àÓSome€of€the€most€fundamental€concepts€of€survival€analysis€can€be€illustrated€with€aÏlife€table,€similar€to€the€ones€used€by€actuaries€and€demographers.€€Suppose€that€in€the€yearÏ1900,€100€children€were€born€in€Costa€Mesa,€California.€€By€1901,€let€us€suppose€that€90€ofÏthem€were€still€alive;€by€1902,€80€of€them€were€still€alive;€and€by€1903,€70€survived.€€WeÏcould€begin€constructing€a€life€table€that€would€look€like€the€one€in€Exhibit€œ1.ññ€[exhibit1.wpññññ:€Ïâ âWordPerfectWin€ver6.1€or€exhibt1b.wp:€WordPerf€ver5.1/5.2]›Ð Ø*'( ÐThe€first€five€columns€are€fairly€self-explanatory.€€The€"observed€survivor€function"Ïâ âsimply€tells€us,€for€any€given€year,€what€percent€of€the€population€is€still€surviving.€€ThisÏfunction€will€always€be€non-increasing,€as€long€as€we€are€dealing€with€standard€single-eventÏsurvival€models.€€ÌÌÓ àÓHAZARD€FUNCTIONSÌÓ àÓHowever,€researchers€will€often€choose€not€to€focus€on€the€survivor€function,€butÏinstead€will€focus€on€the€"hazard€function"€--€the€percent€of€REMAINING€SURVIVORS€(notÏthe€percent€of€the€total)€who€die€in€a€given€year.€€Notice€that€in€Exhibit€1,€even€though€aÏconstant€number€of€ten€people€are€dying€each€year,€and€thus€a€constant€10%€of€theÏpopulation€is€dying€each€year,€the€hazard€rate€is€INCREASING.€€In€the€second€year,€the€tenÏdeaths€represent€one-NINTH€of€the€survivors,€and€in€the€third€year€one-eighth.€€(1)€ÌWhen€dealing€with€human€mortality,€the€"mortality€rate"€is€in€fact€simply€another€nameÏfor€the€"hazard€rate,"€i.e.€the€value€of€€the€hazard€function.€ÌMathematically,€the€hazard€function€can€be€derived€from€the€survivor€function€andÏvice-versa€(2).€€But€€in€practical€terms€for€researchers,€hazard€functions€are€often€moreÏconvenient€to€study.€€One€reason€is€that€survivor€functions,€when€graphed,€all€look€prettyÏmuch€alike€--€they€are€all€downward-sloping.€€It€is€often€difficult€to€distinguish€betweenÏdifferent€survivor€functions€graphically,€and€to€deduce€what€the€graph€is€telling€us.€€ÌHazard€functions€in€contrast€will€typically€have€very€different€appearances€fromÏpopulation€to€population,€or€from€model€to€model.€€€The€researcher€can€more€easily€interpretÏand€tell€a€story€about€a€given€hazard€function.€€For€example,€if€I€asked€you€what€the€hazardÐ  ,X)* Ðfunction€for€human€beings€looked€like€(realistically,€not€using€the€fake€data€in€Exhibit€1),Ïafter€a€little€thought€you€would€probably€realize€that€it€is€U-shaped:€€the€mortality€rate€forÏinfants€is€relatively€high,€then€it€falls€for€children€and€young€adults,€then€€it€rises€continuallyÏfor€older€people.ÌIn€contrast€to€the€hazard€function,€it€would€be€difficult€for€you€to€tell€me€much€aboutÏhuman€beings'€survivor€function,€except€that€it€is€downward-sloping.€€€Ì€€Radioactive€decay€provides€another€example€of€a€hazard€function.€€€How€manyÏCesium€137€atoms€are€left€after€a€period€of€time?€€Radioactive€decay€is€usually€assumed€toÏbe€constant,€and€since€Cesium€137€has€a€half-life€of€30.0€years,€about€2.3%€of€the€atoms€willÏdecay€per€year.€€This€is€a€CONSTANT€hazard€rate,€with€2.3%€decay€per€year.ÌHazard€functions€often€exhibit€"negative€time€dependence,"€that€is,€the€hazard€rateÏdecreases€over€time.€€Unemployment€spells€often€are€an€example:€€quite€a€few€unemploymentÏspells€end€after€€two€or€three€months,€but€by€the€time€an€unemployment€spell€has€lasted,€sayÏsixty€months,€the€job-seeker's€probability€of€finding€a€€job€in€the€next€month€is€quite€smallÏ--€i.e.€his€or€her€hazard€rate€is€low.€€€We€will€see€that€the€hazard€function€for€faculty€typicallyÏexhibits€positive€time€dependence€initially,€but€after€a€few€years€the€exhibits€negative€timeÏdependence.€€After€a€professor€has€been€around€for€15€years,€the€probability€that€she€willÏleave€in€the€16th€year€is€low.ÌÌÓ àÓCENSORINGÌÓ àÓSurvival€data€frequently€are€"censored,"€meaning€that€the€true€value€of€an€subject'sÏsurvival€time€is€unknown,€except€that€it€exceeds€a€certain€value.€€Here€is€an€example€ofÐ  ,X)* Ðcensoring:€€as€of€€1996,€any€professor€who€arrived€in€1994€would€have€completed€two€yearsÏand€the€value€of€their€€duration€variable€would€be€2.€€But€these€faculty€are€very€different€fromÏfaculty€who€arrived€in,€say,€1968€and€left€in€1970.€€Both€have€durations€of€2€years,€€but€theÏ1994€cohort€of€faculty€will€ultimately€have€durations€GREATER€than€2€--€but€we€do€notÏknow€what€their€final,€true,€duration€will€be.€€Thus€returning€faculty€are€all€censored€(3).€€We€Ïonly€know€the€true€durations€of€faculty€who€have€arrived€AND€left€the€school.€ÌHow€do€we€deal€with€censoring?€€Clearly,€it€is€undesirable€to€take€the€durationÏvariables€at€face€value€and€to€consider€the€1994€€faculty€to€have€durations€of€2€years.€€€OneÏpossibility€is€to€drop€the€censored€subjects€from€the€data€€set.€€This€usually€creates€two€majorÏproblems€however.€€First,€the€data€set€may€as€a€result€shrink€to€an€unacceptably€small€numberÏof€€subjects.€€Second,€the€sample€will€probably€be€biased€--€because€professors€with€longÏdurations€are€especially€likely€to€be€censored,€and€these€long-lived€faculty€are€thus€gettingÏdropped€from€the€sample.€€The€sample€will€be€biased€towards€short-lived€faculty.€ÌA€better€way€of€dealing€with€censoring€is€to€use€the€survival€analysis€techniques€whichÏhave€been€developed€over€the€years€to€deal€with€the€problem€of€censoring.€€These€will€beÏexplained€after€we€describe€the€data€set.ÌÌÓ àÓÔ‡¼¼ŠŠÓ›ÔSECTION€III.€€EXAMINING€FACULTY€RETENTIONÔ#†ŠÓ›Š¼¼f0#ÔÐ À$x!" Ѐ€€€€ÌTHE€ISSUESÌÓ àÓThis€study€covers€faculty€at€a€selective€private€liberal€arts€college.€€During€the€lateÏ1980s€and€early€1990s€there€seemed€to€be€an€unusually€large€number€of€junior€€femaleÐ Ð,ˆ)* Ðfaculty€who€left€the€school,€for€various€reasons.€€Also,€in€the€early€1990s€the€schoolÏappointed€a€professor€to€be€its€Title€IX€coordinator.€€Thus€questions€of€faculty€retention,Ïespecially€female€faculty€retention,€arose.€€Although€the€school€has€had€a€good€record€ofÏhiring€female€faculty,€and€although€the€proportion€of€women€in€the€faculty€has€been€rising,Ïthere€is€still€the€question€of€whether€these€newly„hired€women€faculty€were€actually€stayingÏat€the€school.ÌUnlike€the€situation€with€students,€whose€ð ðsurvivalðð€can€be€measured€with€indicatorsÏsuch€as€graduation€and€retention€rates,€there€are€no€widely€used€overall€measures€of€facultyÏretention,€with€the€exception€of€tenure€and€promotions.€€However,€faculty€in€general€haveÏto€stay€about€6€years€before€they€can€get€tenure€„„€and€thus€information€on€tenure€will€€notÏcover€faculty€in€the€first€three€or€four€years€at€the€school.€€Many€professors€were€simply€tooÏnew€to€be€eligible€for€tenure,€others€left€before€they€became€eligible.€€Also,€tenure€doesnððtÏtell€us€how€long€the€professor€actually€stayed€at€the€school;€it€merely€tells€us€that€they€stayedÏlong€enough€to€get€tenure.ÌA€better€measure€of€retention€is€to€literally€count€how€many€years€each€professor€stayedÏat€the€school.ÌÌÓ àÓDATAÌÓ àÓWe€used€the€collegeððs€catalogs€to€identify€339€full-time€tenure-track€faculty€who€hadÏstarted€working€in€1960€up€to€1994,€and€to€determine€their€final€year€at€the€college.€€ManyÏâ âof€them€of€course€are€still€at€the€college.€€Ð ¨*`'( ÐÑP¸ ÑÑ  ÑÑ  ÑThe€catalog€also€supplied€us€with€the€following€variables:€€PhD/ABD€status€whenÏâ âhired,€year€of€Phd,€department,€entry€rank,€year€of€full„time€tenure-track€status€(someÏprofessors€started€as€adjunct€or€visiting€faculty),€and€tenure€status€upon€entry€(a€fewÏprofessors€enter€with€tenure€in€hand).€€We€also€collected€data€on€€years€to€tenure€and€to€fullÏprofessorship,€but€those€variables€are€not€used€in€this€study.€ÌThe€catalog€did€not€directly€supply€us€with€gender€€€information,€but€by€looking€at€theÏnames€and€consulting€with€veteran€employees€we€were€able€to€determine€the€gender€of€allÏbut€two€faculty.€€We€do€not€have€ethnicity€information,€especially€for€faculty€from€the€1960s.ÌWe€do€not€have€information€on€the€REASON€for€exit;€the€€professor€may€have€left€dueÏto€a€better€offer€elsewhere,€or€may€have€been€turned€down€for€tenure€or€contract€renewal.€ÏThus€this€study€only€measures€overall€retention;€it€does€not€measure€retention€of€"desirable"Ïfaculty,€or€the€rate€at€which€"undesirable"€faculty€were€gotten€rid€of.ÌDescriptive€statistics€for€the€data€set€are€in€Exhibit€œ2.€[exhibit2.wp€or€exhibt2b.wp]›ÌIf€there€were€no€censoring€problems,€we€could€simply€find€the€mean€duration€of€maleÏand€female€professors,€do€a€t„test€and€be€done.€€We€could€also€do€linear€regressions€to€seeÏif€other€variables€affect€duration.€€ÌHowever,€our€data€our€heavily€censored€„„€about€140€of€the€339€professors€in€ourÏsample€are€still€at€the€school€and€thus€we€do€not€know€their€ultimate€duration.€€Thus€weÏutilized€survival€analysis.ÌÌÓ àÓÔ‡¼¼ŠŠÓ›ÔSECTION€IV.€€SIMPLE€COMPARISONSÔ#†ŠÓ›Š¼¼ ?#ÔÐ ¨*`'( ÐÐ Ð,ˆ)* ÐSURVIVOR€FUNCTIONSÌÓ àÓInitial€analyses€of€our€data€quickly€revealed€that€faculty€who€arrived€in€the€earlier€yearsÏ„„€the€1960s€and€1970s€„„€had€much€lower€durations€that€faculty€who€arrived€later.€€(We€laterÏperformed€a€log„rank€test€showing€a€large€and€highly€significant€difference.)€€Thus€weÏdecided€to€split€the€data€set,€since€it€seemed€apparent€that€the€faculty€who€arrived€in€€1980„94€Ïhad€survival€and€hazard€rates€which€were€different€from€the€1960„79€faculty.€€Also,€sinceÏmost€of€the€earlier€faculty€were€men,€a€simple€comparison€of€men€vs.€women€would€tend€toÏshow€men€having€a€low€retention€rate€simply€due€to€the€fact€that€so€many€of€them€arrivedÏduring€the€years€when€retention€rates€were€low.ÌExhibit€œ3€[exhibit4.wb1:€quattro€pro€win€5.0]›€shows€life€tables€for€the€faculty€whoÏarrived€from€1960€to€1979,€and€the€faculty€who€arrived€from€1980€to€1994.€€The€estimatedÏsurvivor€functions€are€calculated€using€Kaplan„Meier€(also€known€as€product€limit)Ïestimates.€€Notice€that€the€censored€observations€are€utilized€ð ðfor€as€long€as€they€canðð€„„€thatÏis,€if€a€€professor€has€been€at€the€school€for€three€years€and€is€still€there€right€now,€we€do€notÏknow€her€ultimate€duration.€€But€we€do€know€that€she€did€not€ð ðattritðð€(that€is,€leave€theÏschool)€after€her€first€or€second€years,€and€so€she€does€contribute€to€the€calculation€of€theÏone„€and€two„year€retention€rates.€€Most€full„featured€statistical€packages,€such€as€SPSS,€willÏcalculate€life€tables€and€survivor€and€hazard€functions.ÌExhibit€4€œ[exhibit4.wb1]€graphs›€the€observed€survivor€rates€and€Exhibit€œ5Ï[exhibit4.wb1]€›€the€observed€hazard€rates€for€the€1960„79€and€1980„94€faculty.ÌWe€do€not€know€the€explanation€for€the€very€high€attrition€rates€of€the€1960„79€faculty.€ÏA€non„trivial€proportion€(one€out€of€eight)€only€lasted€one€year.€€One€potential€factor€is€thatÐ  ,X)* Ðthe€school€offered€only€1„year€contracts€to€new€faculty€for€much€of€that€period.€€However,Ïit€seems€unlikely€that€this€is€a€complete€explanation:€the€vast€majority€of€faculty€hired€inÏrecent€years€would€probably€stay€longer€than€a€year€even€if€they€were€limited€to€1„yearÏcontracts.€€Possibly€faculty€quality€became€higher€in€the€1980s€and€1990s,€and€new€facultyÏare€more€likely€to€qualify€for€contract€renewal,€tenure,€etc.€€ÌThe€hazard€functions€in€Exhibit€5€of€course€show€the€very€high€hazard€rates€that€theÏearly€professors€experienced,€especially€in€their€first€few€years.€€Post„1979€professors€inÏcontrast€have€a€very€low€hazard€rate€their€first€two€years€„„€94%€of€stayed€for€at€least€theirÏthird€year€„„€and€even€when€their€hazard€rate€increases€it€still€lower€than€that€of€the€pre„1979Ïprofessors.ÌIn€addition€it€is€interesting€to€note€that€the€hazard€rates€for€the€pre„1979€professorsÏpeaked€in€their€5th€and€7th€years€„„€not€too€surprising€given€the€€timing€of€tenure€decisionsÏand€contract€renewals.€€The€hazard€for€the€post„1979€professors€peaks€in€their€5th€and€8thÏyears,€which€is€somewhat€surprising.€€Possibly€more€tenure€decisions€are€getting€deferred€orÏdelayed€in€recent€years.€€For€all€professors,€the€8th€year€seems€to€be€the€cutoff€point€„„€if€aÏprofessor€has€stayed€for€8€years,€the€chances€are€quite€good€that€he€or€she€will€be€back€forÏthe€9th€and€subsequent€years.€€This€same€phenomenon€can€be€observed€in€the€survival€graphsÏin€Exhibit€4€„„€after€the€8th€year€the€survival€curves€flatten€out.ÌExhibits€6€and€œ7€[exhibit6.wb1:€quattro€pro€win€ver5.0]›€show€the€life€tables€for€maleÏand€female€faculty€who€entered€after€1979.€€Exhibit€8€œ[exhibit6.wb1]€shows›€a€graph€of€theirÏobserved€survivor€rates.€€It€appears€that€men€have€a€slightly€higher€survivor€or€retention€rateÏthan€women,€but€it€partly€depends€on€how€where€one€measures€the€survivor€rate€„„€forÐ  ,X)* Ðexample,€women€have€a€100%€one„year€retention€rate€whereas€men€only€have€a€98%Ïretention€rate.€€But€women€have€only€a€60%€6„year€retention€rate€whereas€men€have€a€72%Ïrate.€€On€the€whole€the€differences€do€not€seem€terribly€large€„„€but€how€can€we€tell€whatÏð ðlargeðð€is?€€To€some€degree€this€is€a€decision€for€policy„makers€to€decide.€€But€we€can€alsoÏmake€an€overall€comparison€of€the€two€survival€functions,€and€measure€the€statisticalÏsignificance€of€the€difference.€€A€simple€way€of€testing€for€the€difference€between€twoÏsurvival€functions€is€to€perform€a€log„rank€test.ÌÌÓ àÓLOG€RANK€TESTS¹Ó àÓLog„rank€tests€are€relatively€simple€to€perform€(statistical€packages€such€as€SPSS€willÏperform€these€tests).€€They€can€be€interpreted€as€a€generalization€of€rank€tests€such€as€theÏWilcoxon€test;€essentially€the€number€of€attritions€in€a€given€period€is€compared€to€theÏnumber€of€attritions€expected€under€the€null€hypotheses.€€See,€for€example,€Kalbfleisch€andÏPrentice€(1980)€for€a€discussion€and€derivation.ÌThe€log„rank€test€yields€a€statistic€which€is€distributed€as€chi„squared,€with€r„1€degreesÏof€freedom,€where€r€is€the€number€of€samples€being€compared.€€In€our€case,€we€have€twoÏpost„1979€samples,€men€and€women.€€The€log„rank€test€was€significant€at€the€p=36%€level,Ïnowhere€close€to€the€standard€significance€levels€and€suggesting€that€the€differences€betweenÏmenððs€and€womenððs€survival€times€could€have€been€caused€by€random€variation.ÌFor€the€1960„79€samples,€men€actually€seemed€to€have€lower€survival€rates€thanÏwomen.€€However€a€log„rank€test€performed€on€these€samples€again€showed€no€significantÏdifferences.Ð  ,X)* ÐA€log„rank€test€comparing€all€post„1979€faculty€to€all€1960„79€faculty€was€highlyÏsignificant€however,€with€p€well€below€1/10€of€1%.ÌThe€log„rank€test€has€an€important€weakness€in€that€it€simply€compares€two€(or€more)Ïentire€samples.€€It€does€not€take€into€account€the€effects€of€other€variables,€such€as€PhD/ABDÏstatus,€entering€rank,€entering€tenure€status,€and€time€trends.€€To€control€for€these€otherÏvariables,€a€€multivariate€approach€is€preferable.ÌÔ€¼¼ŠŠÓ›ÔÓ àÓÌSECTION€V.€€MORE€COMPLEX€TESTSÌÔ€ŠÓ›Š¼¼ÔTHE€COX€PROPORTIONAL€HAZARDS€MODELÐ h  ÐÓ àÓThere€are€several€different€regression€models€that€can€be€applied€to€survival€data.€ÏMany€of€them€are€based€on€parametric€hazard€functions;€that€is,€one€has€to€assume€that€theÏpopulation€has€an€underlying€hazard€function€with€a€specific€functional€form.€€The€simplestÏsuch€functional€form€would€be€the€exponential€model€„„€in€the€exponential€model,€the€hazardÏrate€is€constant€(that€is,€a€constant€proportion,€h,€of€the€population€exits€each€period,€and€theÏsurviving€population€thus€declines€exponentially)€and€a€€regression€might€estimate€the€valueÏof€h,€as€well€as€the€value€of€the€slope€parameters€of€the€righthand€side€variables€used€in€theÏregression.ÌFew€survival€processes€have€such€simple€functional€forms€„„€typically€the€hazard€rateÏwill€vary€with€the€subjectððs€duration.€€€For€such€situations€there€are€many€more€complexÏparametric€models€which€can€be€used.€€€Some€of€them€can€flexibly€fit€data€with€positive€timeÏâ âdependence,€negative€time€dependence,€or€both.Ð +À'( ÐIn€our€case€however,€we€were€unwilling€to€make€prior€assumptions€about€the€shape€ofÏâ âthe€hazard€function,€and€thus€unwilling€to€choose€one€specific€parametric€model.€€ÌThe€"Cox€proportional€hazards€regression€model"€is€a€regression€model€€frequentlyÏused€in€such€situations.€€It€does€not€make€prior€assumptions€about€the€shape€of€the€hazardÏfunction€„„€the€ð ðbaselineðð€hazard€function€is€estimated€from€the€data.€€It€does€howeverÏassume€that€all€the€right€hand€side€variables€affect€the€hazard€function€proportionately.€€ForÏexample,€a€change€in€the€value€of€one€righthand€side€variable€might€double€the€entire€hazardÏfunction;€a€change€in€another€variable€might€reduce€the€entire€hazard€function.€€The€impactÏof€a€righthand€side€variable€is€assumed€to€always€be€a€proportional€change€in€the€entireÏhazard€function€(4).ÌSome€packages€such€as€the€Windows€version€of€SPSS€can€perform€Cox€proportionalÏhazards€regressions,€as€can€many€econometric€packages.€€The€coefficients€cannot€beÏcalculated€directly;€€iterative€maximum€likelihood€techniques€are€necessary,€just€as€with€logit€Ï(also€known€as€logistic)€regressions.ÌWe€ran€the€regression€with€several€different€sets€of€variables;€in€all€of€them€gender€hadÏonly€a€very€small€coefficient€and€was€nowhere€close€to€significance€at€the€5%€or€even€10%Ïlevel.€€We€did€find€however€that€faculty€who€entered€with€a€PhD€had€significantly€higherÏsurvival€rates€than€faculty€who€entered€ABD€and€faculty€who€entered€with€tenure€also€hadÏhigher€survival€rates€(not€surprisingly).€€There€was€some€evidence€that€faculty€who€startedÏas€adjuncts€also€had€higher€survival€rates€(however€remember€that€this€sample€is€of€tenureÏtrack€faculty€only,€and€only€a€small€proportion€of€adjuncts€are€able€to€switch€into€a€tenureÏtrack€position).€€And€of€course€the€post„1979€faculty€had€much€higher€survival€rates.€€TheÐ  ,X)* Ðprofessorsðð€departments€did€not€seem€to€affect€survival€rates.€€The€results€from€an€illustrativeÏregression€are€in€Exhibit€œ9€[exhibit9.wp:€wordperfwin€ver6.1€or€exhibt9b.wp:€wordperfÏvñ)ñer5.1/5.2]ñ)ñ.›€€€Remember€that€the€dependent€variable€is€the€hazard€rate,€so€the€negativeÏcoefficient€on€post„1979€faculty€means€that€they€have€LOWER€hazard€rates,€and€thusÏHIGHER€retention€and€survival€rates.ÌÌÓ àÓSOME€REGRESSION€DIAGNOSTICSÌÓ àeÓThere€are€alternative€ways€of€running€the€regression,€for€example€the€sample€can€beÏsplit€into€subsamples€called€strata.€€Each€stratum€has€its€own€baseline€hazard€function,€whichÏas€before€is€completely€flexible,€without€any€parametric€assumptions.€€However€all€strataÏshare€the€same€righthand€side€variables€and€the€same€slope€coefficient.€€ÌHow€should€we€decide€whether€we€need€to€split€the€sample€into€strata?€€MoreÏgenerally,€what€sorts€of€regression€diagnostics€are€available,€so€that€we€can€evaluate€theÏð ðgoodness€of€fitðð€of€the€regression?ÌFirst,€the€bad€news.€€There€is€no€equivalent€to€the€Ròò2óó€or€the€mean€squared€error€that€canÐ Ø Ðbe€used€to€evaluate€Ordinary€Least€Squared€regressions.€€One€can€perform€a€log„likelihoodÏtest€(which€is€distributed€as€a€chi„squared€statistic)€which€compares€the€overall€fittedÏregression€to€the€null€regression€„„€but€as€with€OLS€regressions,€almost€any€sort€of€reasonable€Ïrighthand€side€variables€will€give€extremely€significant€results,€and€thus€one€doesnððt€get€aÏstrong€sense€of€how€well€the€regression€fit€the€data.€€Some€pseudo„Ròò2óó€formulas€based€on€theÐ °(h%& Ðâ âchange€in€the€log„likelihood€have€been€suggested.Ð ¨*`'( ÐThe€good€news:€there€are€several€graphical€techniques€for€evaluating€the€results€ofÏâ âsurvival€regressions.€€However€they€are€very€heuristic€in€nature;€there€do€not€seem€to€be€anyÏfixed€formulas€for€defining€when€a€fit€is€ð ðgoodðð€or€ð ðbadðð;€rather€one€simply€looks€at€theÏgraph€and€tries€to€decide€if€the€fit€is€good€enough.€€Also,€most€statistical€packages€will€notÏproduce€these€graphs€for€you;€you€have€to€download€the€parameters€and€data€and€produceÏthe€graphs€yourself.ÌHere€is€a€brief€description€of€a€couple€examples€of€these€graphical€regression€diagnosticÏtechniques.€€One€standard€technique€is€the€ð ðlog„minus„logðð€plot:€a€plot€of€the€logarithm€ofÏminus€the€logarithm€of€the€estimated€survival€functions€of€the€possible€strata,€plotted€with€Ïduration€on€the€horizontal€axis.€€In€other€words,€plot€ln(„ln(S(t))€against€t,€where€S(t)€is€theÏestimated€survival€rate€at€time€t.€€(Remember€that€survival€rates€are€always€between€0€andÏ1;€thus€the€logarithm€of€the€survival€function€will€always€be€negative.€€The€log„minus„logÏplot€uses€the€logarithm€of€€MINUS€this€logarithm.)ÌWhen€the€different€strata€are€plotted€on€the€log„minus„log€plot,€their€plotted€curvesÏshould€ideally€stay€roughly€the€same€distance€from€each€other.€€If€they€do€not€have€thisÏconstant€separation,€then€the€proportional€hazards€assumption€may€be€violated,€and€theÏregression€should€be€stratified€(rather€than€using€the€stratum€variable€as€a€righthand€sideÏvariable).€€Exhibit€œ10ñ,ñ€[residmkt.wb1:€quattroprowin€ver5.0]ñ,ñ›€œshoñ*ñwñ*ññ+ñtñ+ñs›€an€example€of€a€log„¼minus„log€plot,€with€the€sample€stratified€by€pre„1979€(actually€1960„79)€and€post„1979Ïstatus.€€ÌThe€two€curves€show€a€certain€amount€of€change€in€their€distance€from€each€other,€andÏthey€even€cross€at€year€8.€€There€may€not€be€any€exact€guidelines€for€deciding€when€toÐ  ,X)* Ðstratify,€but€this€would€seem€to€be€a€situation€where€stratification€is€called€for.€€(TheÏstratified€regressions€gave€results€very€similar€to€the€ones€in€Exhibit€9.)ÌAnother€diagnostic€device€is€the€ð ðgeneralized€residual,ðð€a€concept€suggested€by€CoxÏand€Snell€(1968).€€In€the€context€of€survival€analysis,€generalized€residuals€are€generated€by€Ïcalculating€the€ð ðintegrated€hazardðð€„„€the€sum,€across€time,€of€the€values€of€the€hazardÏfunction€(or€the€integral€with€respect€to€time€if€continuous€time€is€being€used).€€As€KieferÏ(1988)€notes,€ð ðthe€integrated€hazard€does€not€have€a€particularly€convenient€interpretation,ððÏbut€ð ðit€is€the€basic€ingredient€in€a€variety€of€specification€checks.ðð€€For€the€Cox€proportionalÏhazards€model,€a€generalized€residual€for€a€duration€t€can€be€calculated€by€taking€theÏintegrated€hazard€at€time€t€and€multiplying€it€by€the€exponent€of€€the€product€of€righthandÏside€variables€and€their€coefficients€(i.e.€e(t)€=€H(t)exp(xb)€where€e(t)€is€the€generalizedÏresidual€for€time€t,€H(t)€is€the€integrated€hazard€at€time€t,€xb€is€the€vector€product€of€theÏrighthand€side€variables€and€their€coefficients,€and€exp()€is€the€exponential€function).€€TheseÏresiduals€can€be€plotted€with€a€residual€of€size€r€on€the€horizontal€axis€and€the€logarithm€ofÏthe€proportion€of€residuals€greater€than€r€on€the€vertical€axis.€€The€resulting€plot,€if€theÏregression€has€a€good€fit,€should€ideally€follow€the€45„degree€line€from€the€origin.€€See€alsoÏCrowley€and€Hu€(1977)€€for€a€discussion€and€example.ÌExhibit€œ11ñ.ññ-ñ[reñ-ññ.ññ/ñ€[residmkt.wb1]ñ/ñ›€shows€the€plot€of€the€generalized€residuals€from€anÏunstratified€regression.€€Again€there€seem€to€be€no€hard„and„fast€formulas€for€determiningÏwhen€the€residuals€are€sufficiently€close€to€the€ideal.€€However€the€graph€in€Exhibit€11€seemsÏâ âto€exhibit€a€good€fit.Ð ¨*`'( ÐÑ@¸ÑÑ  ÑÑ  Ññ2ñÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  Ññ2ññ1ñÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  Ññ1ññ0ñÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  Ññ0ññ/ñÑ@¸ÑÑ  ÑÑ  Ññ/ññ.ñÑ@¸ÑÑ  ÑÑ  Ññ.ñExhibit€œ12ñ0ñ€[ñ0ññ1ññ0ñexhibñ0ññ1ññ2ñresidmkt.wb1]ñ2ñ›€shows€the€plot€of€the€generalized€residuals€from€the€sameÏâ âregression,€stratified€by€pre„1979€and€post„1979€status.€€If€anything€these€residuals€seem€toÏhave€a€worse€ñ/ñÑ@¸ÑÑ  ÑÑ  Ññ/ññ.ñÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  ÑÑ@¸ÑÑ  ÑÑ  Ññ.ñfit€that€those€from€the€unstratified€regression,€which€seems€counterintuitive.€ÏAgain€it€is€not€clear€if€the€generalized€residuals€in€this€graph€could€be€considered€to€be€ð ðcloseÏenoughðð€to€the€45„degree€line.ÌThus€the€results€from€the€log„minus„log€and€generalized€residual€graphs€are€notÏdefinitive,€but€do€not€seem€to€indicate€a€gross€lack€of€goodness€of€fit€in€the€regression.ÌÔ€¼¼ŠŠÓ›ÔÓ àÓÔ  ÔÔ  ÔÔ  ÔÌñ/ñÑ@¸ÑÑ  ÑÑ  Ññ/ñÔ  ÔCONCLUSIONÌÔ€ŠÓ›Š¼¼ÔÓ àÓÌThere€are€many€other€statistical€techniques€used€in€survival€analysis,€but€this€paper€hasÏprovided€an€introduction.€€It€seems€safe€to€conclude€from€the€survival€graphs,€log€rank€tests,Ïand€regression€results€that€the€survival€rates€of€male€and€female€faculty€did€not€exhibit€largeÏdifferences€in€a€statistical€sense.€€To€decide€whether€the€differences€are€large€enough€toÏworry€about€in€a€non„statistical€sense€is€a€largely€subjective€judgement,€but€the€life€tables€andÏestimated€survival€functions€at€least€provide€numerical€measures€for€comparing€the€retentionÏof€men€and€women.ÌOne€crucial€piece€of€information€that€our€data€set€does€not€provide€is€the€reason€forÏfaculty€attrition.€€The€college€may€have€deliberately€made€some€professors€leave,€while€itÏmay€have€wished€to€retain€other€professors€who€left€the€school.€€And€of€course€the€reasonsÏfor€attrition€are€typically€complex€and€cannot€be€captured€in€a€single€variable€„„€someÏprofessors€may€have€wanted€to€stay€on€the€whole€but€some€aspect€of€the€school€made€the€jobÐ -¸)* ÐÑP¸ÑÑ  ÑÑ  Ññ2ñÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  Ññ2ññ1ñÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  Ññ1ññ0ñÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  Ññ0ñunattractive;€some€professors€may€have€been€deemed€desirable€by€some€members€of€theÏcollege€community€and€undesirable€by€others.€€The€dataset€does€not€provide€even€a€hint€of€Ïwhat€the€reasons€for€attrition€were;€it€simply€records€who€stayed€and€who€left,€and€when.€€ÌThus€it€is€possible€that€a€school€could€still€have€a€problem€with€retaining€female€facultyÏeven€if€their€retention€rate€ñ ñequalledñ ññ ñequaledñ ñ€that€of€the€male€faculty.€€Possibly€the€males€who€left€wereÏnot€deemed€desirable€by€the€school€while€the€females€were€„„€or€vice„versa.Ìâ âÓ àÓÐ Ø*'( ÐÑP¸ÑÑ  ÑÑ  ÑññÑP¸ÑÑ  ÑÑ  ÑññññÑP¸ÑÑ  ÑÑ  ÑññññÑP¸ÑÑ  ÑÑ  Ññññ ñÑP¸ÑÑ  ÑÑ  ÑÑP¸ÑÑ  ÑÑ  Ññ ññ ñÑP¸ÑÑ  ÑÑ  Ññ ñÔ  Ôâ âWHERE€TO€GO€FROM€HEREÌÓ àÓThis€paper€has€only€discussed€single„spell,€single„outcome€models.€€Some€events€suchÏas€unemployment€or€marriage€can€happen€repeatedly€to€the€same€person€over€time.€€AlsoÏsometimes€there€are€multiple€possible€events€which€we€wish€to€measure:€€a€student€mightÏstay€enrolled€until€he€or€she€eventually€graduates,€transfers€or€drops€out€„„€this€is€the€subjectÏof€Roncoððs€AIR€Professional€File€article€(1996),€and€in€a€life€table€context,€Garciaððs€CAIRÏconference€presentation€(1995).ÌFor€people€who€wish€to€perform€survival€analyses€of€their€own,€we€have€found€SingerÏand€Willettððs€(1991€and€1993)€articles€to€be€clearly€written€and€easy€to€understand.€€MoritaÏet€al€provide€another€good,€slightly€more€technical€introduction.€€For€a€more€mathematicalÏapproach,€Kieferððs€survey€article€(1988)€and€Heckmanððs€work€(1984,€e.g.)€€represent€theÏeconometric€approach.€€For€a€general€statistical€approach,€Kalbfleisch€and€Prenticeððs€bookÏ(1980)€is€cited€extremely€often€and€provides€a€good€but€mathematical€introduction.€€It€isÏgetting€a€little€dated€now,€however.ÌWe€used€an€econometrics€package€called€Limdep€(ver€6.0)€and€SPSS€for€Windows€(verÏ6.1.2)€to€perform€these€calculations.€€Many€lower€cost€statistical€packages€do€not€have€theÏcapability€of€performing€survival€analysis.€€On€the€other€hand,€if€you€have€discrete€time€data,ÏWillett€and€Singer€(1993)€describe€how€some€survival€analysis€can€be€performed€simply€byÏdoing€a€series€of€logit€regressions€(also€known€as€logistic€regressions),€which€many€statisticalÏpackages€can€perform.ÌSurvival€analysis€will€not€replace€the€t„test€and€the€contingency€table€in€terms€of€being€Ïa€ð ðmust€knowðð€statistical€technique.€€But€if€you€have€a€data€situation€where€you€areÐ  ,X)* Ðmeasuring€time€duration,€especially€in€the€presence€of€censoring,€then€survival€analysisÏcomes€in€handy€indeed.ÌÌÓ àÓÔ‡¼.;¼ŠŠs£ÔENDNOTES:Ô#†Šs£Š¼¼.;ÕŠ#ÔÐ 0 è ÐÌÓ àÇŠÓÓ àÓ(1)€€In€calculating€the€observed€hazard€function,€there€are€some€technicalities€associated€withÏthe€question€of€whether€time€is€being€measured€as€a€continuous€or€discrete€variable.€€MostÏstatistical€packages,€including€SPSS,€will€assume€that€time€is€continuous,€and€will€makeÏadjustments€to€the€calculated€hazard€function€instead€of€using€the€simple€calculations€inÏExhibit€1.€€In€this€example,€we€are€measuring€time€in€years.€€But€most€people€do€not€literallyÏlive€exactly€1.00€years€or€2.00€years€and€then€drop€dead.€€Instead€they€may€die€at€any€age,Ïsuch€as€1.032€or€2.964.€€But€life€tables€put€people€into€age€categories,€such€as€0€to€1€years,Ïand€1€to€2€years,€and€do€not€record€the€exact€age€at€death.€€Still,€knowing€that€,€for€example,Ïin€the€first€year€we€started€with€100€people€and€ended€with€90€people,€we€might€assume€thatÏpeople€died€at€an€even€rate€throughout€the€year€and€assume€that€during€that€first€year€theÏaverage€size€of€the€surviving€population€was€95.€€Thus€one€possible€simple€adjustment€to€theÏhazard€rate€would€be€to€calculate€it€as€10/95€instead€of€10/100.€€With€discrete€time,€suchÏadjustments€are€not€necessary€„„€for€example,€faculty€duration€typically€can€be€measured€inÏinteger€years.ÌÌ(2)€€If€we€assume€that€survival€time€is€a€random€variable,€and€denote€the€survivor€functionÏas€S(t),€where€S€denotes€the€proportion€of€the€population€surviving€at€time€t,€then€theÐ Ð,ˆ)* Ðcumulative€distribution€function€of€survival€time€is€F(t)€=€1-S(t).€€€If€time€is€continuousÏ(rather€than€discrete),€then€the€density€function€of€survival€time€is€f(t)€=€F'(t).€€And€the€hazardÏfunction€€is€h(t)€=€f(t)/S(t).€€Conversely,€the€survivor€function€can€be€derived€from€the€hazardÏfunction:€€S(t)€=€exp(-int(h(t)))€where€€"int(h(t))"€denotes€the€integral€of€h(t)€ñ ñfrom€fromñ ññ ñfromñ ñ€0€to€t.€€€ÌÌ(3)€€This€is€known€as€"right€censoring,"€where€the€subject's€date€of€EXIT€is€unknown.€€InÏother€types€of€research,€subjects€can€be€"left€censored,"€with€the€date€of€ENTRY€unknown.€ÏFor€example,€if€one€wishes€to€measure€the€life€expectancy€of€AIDS€patients€from€the€dateÏof€infection€(as€opposed€to€the€date€of€diagnosis),€many€€patients€will€not€know€their€date€ofÏinfection€and€ñ ñthsñ ññ ñthusñ ñ€they€will€be€left€censored.€€If€they€are€still€alive,€they€are€also€rightÏcensored.€ÌÓ àÓSometimes€it€is€also€useful€to€distinguish€between€Type€I€censoring€and€Type€IIÏcensoring.€€Type€I€censoring€occurs€when€the€experiment€or€observations€must€end€at€aÏcertain€time,€and€certain€€subjects€will€not€have€experienced€the€exit€event€(death,€departureÏfrom€school,€etc.).€€Type€II€censoring€occurs€when€the€researcher€stops€collectingÏobservations€after€a€certain€NUMBER€of€exit€€events,€for€example€after€30€faculty€have€leftÏthe€school.ÌÌÓ àÓ(4)€€Mathematically,€the€Cox€proportional€hazards€model€assumes€that€the€hazard€rate€(h),Ïis€a€function€of€time€(t)€and€a€vector€of€righthand€side€variables€(x)€multiplied€by€a€vector€ofÏslope€coefficients€(b).€€That€is,€€h(ñññ ñœñ ñññññññœññññññœññt,xññ›ññññññ›ñññññññ ñ›ñ ñññ)€=€hòòoóó(t)exp(xb),€where€hòòoóó(t)€is€the€baseline€hazardÐ ¨*`'( Ðfunction€(the€underlying€hazard€function€which€applies€to€all€members€of€the€population),Ð  ,X)* Ðand€exp()€is€the€exponential€function.€€Large€positive€values€of€x€and€b,€for€example€wouldÏcause€the€hazard€rate€h(ñññ ñœñ ñññññññœññññññœññt,xññ›ññññññ›ñññññññ ñ›ñ ñññ)€to€increase,€raising€the€attrition€rate.€€Large€negative€values€ofÏx€and€b€would€cause€the€hazard€rate€to€become€smaller€(but€still€positive€„„€hazard€rates€haveÏto€always€be€nonnegative€by€definition).ÌÓ à=–ÓÌÌÓ àÓÔ‡¼.;¼ŠŠs£ÔBIBLIOGRAPHYÔ#†Šs£Š¼¼.;+š#ÔÐ Ð  ÐÌÓÓà0  àà ° àAshenfelter,€Orley€and€Card,€David.€€ð ðFaculty€Retirement€in€the€Post„Mandatory€Era:€Early€ÏFindings€from€the€Princeton€Retirement€Survey,ðð€Princeton€Conference€on€HigherÏEducation,€March€1996.Ð (#(# ÐÌÓ àšÓÓšÓÓÓà0  àà ° àCox,€David€R.€€and€Snell,€E.€€J.€€ð ðA€General€Definition€of€Residuals,ðð€òòJournal€of€the€Royal€óóÐ (à ÐòòStatistical€Societyóó,€Vol.€ñññ ñœñ ñññññññœññññññœññB30ññ›ññññññ›ñññññññ ñ›ñ ñññ€(May/Aug.€€1968),€pp.€€248„275.Ð$Ü(#(# ÐÌà0  àà ° àCrowley,€John€and€Hu,€Marie.€€ð ðCovariance€Analysis€of€Heart€Transplant€Survival€Data,ðð€òòÐ Ô ÐJournal€of€the€American€Statistical€Associationóó,€Vol.€72,€No.€€357€(March€1977),€pp.Ð Ð Ð27„36.Ð (#(# ÐÌà0  àà ° àDwyer,€Mary€M.;€Flynn,€Arlene€A.;€and€Inman,€Patricia€S.€€ð ðDifferential€Progress€of€WomenÏFaculty:€Status€1980„1990,ðð€òòHigher€Education:€Handbook€of€Theory€and€Researchóó,Ð À ÐVol.€€12€(1991),€pp.€€173„222.Ð (#(# ÐÌà0  àà ° àGarcia,€Philip.€€ð ðCalifornia€Colleges€and€University€Enrollment€Demand:€1994„2005,ðð€CAIRÏAnnual€Conference,€Sacramento€CA,€November€9,€1995.Ð (#(# ÐÌà0  àà ° àHeckman,€James€J.€and€Borjas,€George€J.€€ð ðDoes€Unemployment€Cause€FutureÏUnemployment?€€Definitions,€Questions,€and€Answers€from€a€Continuous€TimeÏModel€of€Heterogeneity€and€State€Dependence,ðð€òòEconomicaóó,€Vol.€47,€No.€187Ð è& #$ Ð(March€1984),€pp.€€248„283.Ð (#(# ÐÌà0  àà ° àKalbfleisch,€John€D.€€and€Prentice,€Ross€L.€€òòThe€Statistical€Analysis€of€Failure€Time€Dataóó.€Ð Ü)”&' Ѐ€New€York:€John€Wiley€&€Sons,€Inc.€€1980.Ð (#(# Ðâ â€Ð Ô+Œ() Ðà0  àà ° àKiefer,€Nicholas.€€ð ðEconomic€Duration€Data€and€Hazard€Functions,ðð€òòJournal€of€Economic€Ð H Ðâ â€à0` (#(#àà  àLiteratureóó,€Vol.€€26,€No.€€8€(June€1988),€pp.€€646„679.ÐDü` (#` (# ÐÌà0  àà ° àMorita,€June€G.;€Lee,€Thomas€W.;€and€Mowday,€Richard€T.€€ð ðIntroducing€Survival€AnalysisÏto€Organizational€Researchers:€A€Selected€Application€to€Turnover€Research,ððÏòòJournal€of€ñ ñApplliedñ ññ ñAppliedñ ñ€Psychologyóó,€Vol.€74,€No.€€2€(April€1989),€pp.€€280„292.Ð4ì(#(# ÐÌà0  àà ° àRonco,€Sharron€L.€€ð ðHow€Enrollment€Ends:€Analyzing€the€Correlates€of€Student€Graduation,ÏTransfer€and€Dropout€with€a€Competing€Risks€Model,ðð€òòAIR€Professional€Fileóó,€€No.Ð ( à Ð61,€Summer€1996.Ð (#(# ÐÌà0  àà ° àRosenfeld,€R.€€A.,€and€Jones,€J.€€A.€€ð ðInstitutional€Mobility€Among€Academics:€The€Case€ofÏPsychologists,ðð€òòSociology€of€Educationóó,€Vol.€59€(1986),€pp.€€212„226.ÐÐ (#(# ÐÌà0  àà ° àSinger,€Judith€D.€€and€Willett,€John€B.€€ð ðññññœññModellingññññModelingññ€the€Days€of€Our€Lives:€Using€SurvivalÏAnalysis€When€Designing€Longitudinal€Studies€of€Duration€and€Timing€of€Events,ðð€òòÐ  Ä Ðà0` (#(#àà  àPsychological€Bulletinóó,€Vol.€110,€No.€2€(1991),€pp.€€268„290.ÐÀ` (#` (# ÐÌà0  àà ° àTuckman,€B.H.€and€Tuckman,€ññññœññññññœññH.P.ññ›ññññññ›ññññ€€ð ðWomen€as€Part„Time€Faculty€Members,ðð€òòHigherÐ ¸ ÐEducationóó,€Vol.€€10,€No.€€2€(1981),€pp.€€169„179.Ðü´(#(# ÐÌà0  àà ° àWillett,€John€B.€€and€Singer,€Judith€D.€€ð ðInvestigating€Onset,€Relapse,€and€Recovery:€WhyÏYou€Should,€and€How€You€Can,€Use€Discrete„Time€Survival€Analysis€to€ExamineÏEvent€Occurrence,ðð€òòJournal€of€Consulting€and€Clinical€Psychologyóó,€€Vol.€61,€No.€€6Ð ì¤ Ð(1993),€pp.€€952„965.