Lecture Outline. Biost 518 Applied Biostatistics II. Logistic Regression. Simple Logistic Regression

Transcription

1 Ald Bostatstcs II, WIN 28 January 23, 28 Bost 58 Ald Bostatstcs II Scott S. Emrson, M.D., Ph.D. Profssor of Bostatstcs Unvrsty of Washngton Lctur 5: Rvw of Sml Rgrsson II Lctur Outln Gnral Rgrsson Sttng Infrnc on Mans Infrnc about Gomtrc Mans Infrnc about Odds Infrnc about Rats Infrnc about Hazards January 23, , 23, 25 Scott S. Emrson, M.D., Ph.D. Sml Logstc Rgrsson Infrnc About th Odds Logstc Rgrsson Bnary rsons varabl Allows contnuous (or multl) groung varabls But s OK wth bnary groung varabl also Comars odds of rsons across grous Odds rato 3 4

2 Ald Bostatstcs II, WIN 28 January 23, Why not Lnar Rgrsson? Many msconctons about th advantags and dsadvantags of analyzng th odds Rasons that I consdr vald Scntfc bass Us of odds ratos n cas-control studs Plausblty of lnar trnds and no ffct modfrs Statstcal bass Man varanc rlatonsh (f not usng robust SE) 6 Sml Logstc Rgrsson Modlng odds of bnary rsons Y on rdctor ( ) ( ) log odds log odds log odds log logt Modl Pr Dstrbuton Y 7 Intrrtaton as Odds Eonntaton of rgrsson aramtrs ( ) odds odds odds Modl Pr Dstrbuton Y 8 Estmatng Proortons Proorton odds / ( odds) ( ) ( ) / Modl Pr Dstrbuton Y

3 Ald Bostatstcs II, WIN 28 January 23, 28 Sml Logstc Rgrsson Stata Intrrtaton of th modl Odds whn rdctor s Found by onntaton of th ntrct from th logstc rgrsson: () Odds rato btwn grous dffrng n th valu of th rdctor by unt Found by onntaton of th slo from th logstc rgrsson: () 9 logt rsvar rdvar, [robust] Provds rgrsson aramtr stmats and nfrnc on th log odds scal Intrct, slo wth SE, CI, P valus logstc rsvar rdvar, [robust] Provds rgrsson aramtr stmats and nfrnc on th odds rato scal Only slo wth SE, CI, P valus Eaml Prvalnc of strok (crbrovascular accdnt- CVA) by ag n subst of Cardovascular Halth Study Rsons varabl s CVA Bnary varabl: no hstory of ror strok, ror hstory of strok Prdctor varabl s Ag Contnuous rdctor Odds Ratos usng logstc.logstc cva ag, robust Logstc rgrsson Numbr of obs 735 LR ch2() 2.52 Prob > ch2.27 Log lklhood Psudo R2.5 cva Odds Rato StdErr z P> z [95% Conf Int] ag

4 Ald Bostatstcs II, WIN 28 January 23, 28 Eaml: Intrrtaton Logstc Rgrsson and χ 2 Tst From logstc rgrsson analyss, w stmat that for ach yar dffrnc n ag, th odds of strok s 3.4% hghr n th oldr grou, though ths stmat s not statstcally sgnfcant (P.3). A 95% CI suggsts that ths obsrvaton s not unusual f a grou that s on yar oldr mght hav odds of strok that was anywhr from.8% lowr or 7.8% hghr than th youngr grou. 3 Logstc rgrsson wth a bnary rdctor (two grous) corrsonds to famlar ch squard tst Thr ossbl statstcs from logstc rgrsson Wald: Th tst basd on th stmat and SE Scor: Corrsonds to ch squard tst, but not gvn n Stata outut Lklhood rato tst: Can b obtand usng ostrgrsson commands n Stata (nt quartr) 4 Sml Posson Rgrsson Infrnc About Rats 5 Count Data Somtms a random varabl masurs th numbr of vnts occurrng ovr som rgon of sac and ntrval of tm E.g., Numbr of olys rcurrng n a atnt s colon durng a 3 yar ntrval btwn colonoscos Numbr of actnc kratoss dvlong ovr a thr month rod on a atnt s lft arm Numbr of ulmonary acrbatons rncd by a cystc fbross atnt durng a yar 6 4

5 Ald Bostatstcs II, WIN 28 January 23, 28 Evnt Rats Posson Probablty Modl Whn a rsons varabl masurs counts ovr sac and tm, w most oftn summarz th rsons across atnts by consdrng th vnt rat Evnt rat ctd numbr of vnts r unt of sac-tm Th rat s thus a man count In most statstcal roblms, w know th ntrval of tm and volum of sac samld 7 Frquntly: Assum counts ar Posson Th Posson dstrbuton can b drvd from th followng assumtons Th ctd numbr of vnts occurrng n an ntrval of tm s roortonal to th sz of th ntrval Th robablty that two vnts occur n an nfntsmally small ntrval of sac-tm s Th numbr of vnts occurrng n sarat ntrvals of sac-tm ar ndndnt (Assumton of a constant rat wth ndndnc ovr sarat ntrvals s rtty strong) 8 Posson Dstrbuton Counts th vnts occurrng at a constant rat λ n a scfd tm (and sac) t Indndnt ntrvals of tm and sac Probablty dstrbuton has aramtr λ > For k,, 2, 3, 4, λt k ( ) ( λt) PrY k k! Man E(Y) λt; varanc Var(Y) λt Posson aro to Bnomal for low 9 Rgrsson wth Counts Whn th rsons varabl rrsnts counts of som vnt, w tycally modl th (log) rat usng Posson rgrsson Comars rats of rsons r sac-tm (rson-yars) across grous Rat rato 2 5

6 Ald Bostatstcs II, WIN 28 January 23, 28 Why not Lnar Rgrsson? Prmarly statstcal: Th rat s n fact a man For Posson Y masurd ovr tm t and havng vnt rat λ E(Y) λt Var (Y) λt But Want to account for dffrnt aras or lngth of tm for masurmnt Nd to account for man-varanc rlatonsh (f Why a Multlcatv Modl? In Posson rgrsson, w tnd to us a log lnk whn modlng th vnt rat Thus w ar assumng a multlcatv modl Multlcatv modl comarsons btwn grous basd on ratos Addtv modl comarsons btwn grous basd on dffrncs Tchncal statstcal rorts: Log rat s th canoncal aramtr for th Posson not usng robust SE) 2 22 Posson Rgrsson Sml Posson Rgrsson Rsons varabl s count of vnt ovr sac-tm (oftn rson-yars) Offst varabl scfs sac-tm Allows contnuous (or multl) groung varabls But s OK wth bnary groung varabl also Offst varabl scfs sac-tm 23 Modlng rat of count rsons Y on rdctor λt k ( ) ( ) ( λt) Dstn Y ~ P λt PrY k T t k! Modl E( Y T, ) log( λt) log( T) logλ logλ logλ 24 6

7 Ald Bostatstcs II, WIN 28 January 23, 28 Intrrtaton as Rats Sml Posson Rgrsson Eonntaton of aramtrs Dstn Modl Y E ~ P ( λt ) Pr( Y k T t ) ( Y T, ) log( λt) log( T) λ λ λ ( λt ) λt k! k 25 Intrrtaton of th modl Rat whn rdctor s Found by onntaton of th ntrct from th Posson rgrsson: () Rat rato btwn grous dffrng n th valu of th rdctor by unt Found by onntaton of th slo from th Posson rgrsson: () 26 Eaml: Sttng Eaml: Varabls Chmosnstzrs for cancr chmothray In vtro valuaton of th ablty of som drugs to otntat th cytotoc ffcts of doorubcn Clls culturd n th laboratory ar osd to doorubcn at svral concntratons wth and wthout chmosnstzrs Ths aml: Only th control grou Rsons: Numbr of survvng cll colons Each rsumably arsng from a sngl cll Offst: Dfault valu of Sam volum of cultur usd for all samls Prdctor: Concntraton of doorubcn

8 Ald Bostatstcs II, WIN 28 January 23, 28 Scattrlot Clls vs Do Conc Charactrzaton of Scattrlot clls 243 Charactrzaton of scattrlot Doorubcn concntraton was samld on log scal Ths samlng schm was usd bcaus t was known that roorton of clls klld s mor or lss lnar n log concntraton Mchals-Mntn kntcs: Actually S shad n log concntraton, but wll aromatd lnarly ovr a rang of doss. 5 conc 29 3 Scattrlot: Clls vs log (Conc) Charactrzaton of Scattrlot clls 243 Lowss smoothr, bandwdth.8 Outlrs: Non obvous Frst ordr trnd: Dcrasng cll survval wth ncrasng log concntraton Scond ordr trnd: Hnt of S-shad curv, but counts farly wll aromatd by straght ln Wthn grou varablty: lconc 3 Dcrasng varanc for lowr grou mans (not smallr saml sz n frst grou) 32 8

9 Ald Bostatstcs II, WIN 28 January 23, 28 Stata Commands Sam form as for othr rgrsson modls Ecton: If th obsrvd counts ar masurd ovr dffrnt amounts of tm or sac, w must scfy th lngth of osur osson rsvar rdvar, osur(tm) [robust] Eosur can also b gvn as th offst, whch s just th log of th osur tm osson rsvar rdvar, Estmaton of Rgrsson Modl. osson clls lconc (Itraton nformaton omttd) Numbr of obs 282 LR ch2() Prob > ch2. Psudo R clls Cof. StErr. z P> z [95% CI] lconc _cons offst(logtm)[robust] Intrrtaton of Stata Outut log rat lconc Rgrsson modl for clls on log concntraton Intrct s labld by _cons Estmatd ntrct: 3.75 Slo s labld by varabl nam: lconc Estmatd slo: Intrrtaton of Intrct log rat lconc Estmatd count rat for lconc s found by onntaton: (3.75) 42.5 lconc corrsonds to a concntraton of. Ths was th hghst concntraton samld In ths roblm, th ntrct s of ntrst f th lnar rlatonsh btwn log concntraton and log rat s corrct

10 Ald Bostatstcs II, WIN 28 January 23, 28 Intrrtaton of Slo Rol of Lnarty log rat lconc Estmatd rato of rats for two grous dffrng by n log concntraton s found by onntaton slo: (-.366).694 Grou on log unt hghr has survval rat only.694 as larg (69.4% as larg) log unt 2.78 tms hghr concntraton fold ncras n concntraton tnds to caus a survval rat only as larg 56.9% dcras n survval rat 37 W hav to b carful n ntrrtng ths modl f th lnar rlatonsh dos not hold Scattrlot suggstd lnar rlatonsh btwn cll counts and log concntraton was rasonabl But w modld th log rat vrsus log concntraton 38 Fttd Rgrsson Modl. rdct fclls. grah clls fclls lconc, s(ot) c(.l) clls rdctd numbr of vnts 243 Sml Proortonal Hazards Rgrsson Infrnc About Hazards lconc 39 4

11 Ald Bostatstcs II, WIN 28 January 23, 28 Rght Cnsord Data A scal ty of mssng data: th act valu s not always known Som masurmnts ar known actly Som masurmnts ar only known to cd som scfd valu (rhas dffrnt for ach subjct) Tycally rrsntd by two varabls An obsrvaton tm: Tm to vnt or cnsorng, whchvr cam frst An ndcator of vnt: Tlls us whch wr Statstcal Mthods In th rsnc of cnsord data, th usual dscrtv statstcs ar not arorat Saml man, saml mdan, sml roortons, saml standard dvaton should not b usd Pror dscrtvs should b basd on Kalan- Mr stmats Smlarly, scal nfrntal rocdurs ar ndd wth cnsord data obsrvd vnts 4 42 Notaton Proortonal Hazards Modl Unobsrvd : Tru tms to vnt : Cnsorng Tms : Obsrvd data : Obsrvaton Tms : Evnt ndcators : { T, T2,, Tn } { C, C,,C } 2 T mn ( T, C ) ft T D othrws n 43 Instantanous rat of falur at ach tm among subjcts who hav not fald Proortonal hazards assums that th rato of ths nstantanous falur rats s constant n tm btwn two grous Proortonal hazards (Co) rgrsson trats th survval dstrbuton wthn a grou smaramtrcally A sm-aramtrc modl: Th hazard rato s th aramtr, thr s no ntrct 44

12 Ald Bostatstcs II, WIN 28 January 23, 28 Borrowng Informaton Sml PH Rgrsson Modl Us othr grous to mak stmats n grous wth sars data Borrows nformaton across rdctor grous E.g., 67 and 69 yar olds would rovd som rlvant nformaton about 68 yar olds Borrows nformaton ovr tm Rlatv rsk of an vnt at ach tm s rsumd to b th sam undr Proortonal Hazards 45 Basln hazard functon s unscfd Smlar to an ntrct Modl log ( λ( t )) log( λ ( t) ) log hazard at t log log hazardat t log log hazardat t log ( λ( t) ) ( λ( t) ) ( λ( t) ) 46 Modl on Hazard scal Eonntatng aramtrs Modl λ ( t ) λ ( t) hazard at t λ hazard at t λ hazard at t λ ( t) ( t) ( t) 47 Intrrtaton of th Modl No ntrct Gnrally do not look at basln hazard But can b stmatd Slo aramtr Hazard rato btwn grous dffrng n th valu of th rdctor by unt Found by onntaton of th slo from th roortonal hazards rgrsson: () 48 2

13 Ald Bostatstcs II, WIN 28 January 23, 28 Stata Eaml stco obsvar vntvar, [robust] Provds rgrsson aramtr stmats and nfrnc on th hazard rato scal Only slo wth SE, CI, P valus Prognostc valu of nadr PSA rlatv to tm n rmsson PSA data st: 5 mn who rcvd hormonal tratmnt for advancd rostat cancr Followd at last 24 months for clncal rogrsson, but act tm of follow-u vars Nadr PSA: lowst lvl of srum rostat scfc antgn achvd ost tratmnt 49 5 Estmaton of Rgrsson Modl. stst obstm rlas. stco nadr Co rgrsson -- Brslow mthod for ts No. of subj 5 No. of obs 5 No. fal 36 Tm at rsk 423 Wald ch2() 6.79 Log lklhood -3.3 Prob > ch2.8 Robust _ t HzRat StdErr z P> z [95% Conf Int] nadr Intrrtaton of Stata Outut Scntfc ntrrtaton of th slo Hazard rato nadr. 5 Estmatd hazard rato for two grous dffrng by n nadr PSA s found by onntaton slo (Stata only rorts th hazard rato): Grou on unt hghr has nstantanous vnt rat.5 tms hghr (.5% hghr) Grou unts hghr has nstantanous vnt rat.5.62 tms hghr (6.2% hghr) 52 3

14 Ald Bostatstcs II, WIN 28 January 23, 28 Statstcal Valdty of Infrnc Infrnc (CI, P vals) about assocatons rqurs thr gnral assumtons Assumtons about aromat normal dstrbuton for aramtr stmats Assumtons about ndndnc of obsrvatons Assumtons about varanc of obsrvatons wthn grous Normally Dstrbutd Estmats Assumtons about aromat normal dstrbuton for aramtr stmats Classcally or Robust SE: Larg saml szs Dfnton of larg dnds on undrlyng robablty dstrbuton Indndnc / Dndnc Wthn Grou Varanc Assumtons about ndndnc of obsrvatons for lnar rgrsson Classcally: All obsrvatons ar ndndnt Robust standard rror stmats: Allow corrlatd obsrvatons wthn dntfd clustrs 55 Assumtons about varanc of rsons wthn grous for roortonal hazards rgrsson Classcally: Man varanc rlatonsh for bnary data Proortonal hazards consdrs odds of vnt at vry tm Nd roortonal hazards and lnarty of rdctor Robust standard rror stmats: Allow unqual varancs across grous (Do not nd roortonal hazards or lnarty) 56 4

15 Ald Bostatstcs II, WIN 28 January 23, 28 Lnarty of Modl Eaml: Intrrtaton Assumton about adquacy of lnar modl for rdcton of grou odds of rsons wth logstc rgrsson Th log hazard rato across grous s lnar n th modld rdctor (W can modl transformatons of th masurd rdctor) 57 From roortonal hazards rgrsson analyss, w stmat that for ach ng/ml unt dffrnc n nadr PSA, th rsk of rlas s.6% hghr n th grou wth th hghr nadr. Ths stmat s hghly statstcally sgnfcant (P <.). A 95% CI suggsts that ths obsrvaton s not unusual f a grou that has a ng/ml hghr nadr mght hav rsk of rlas that was anywhr from.8% hghr to 2.3% hghr than th grou wth th lowr nadr. 58 Log Transformd NadrPSA Basd on ror rnc A constant dffrnc n PSA would not b ctd to confr sam ncras n rsk Comarng 4 ng/ml to ng/ml s not th sam as comarng 4 ng/ml to ng/ml A multlcatv ffct on rsk mght b bttr Sam ncras n rsk for ach doublng of nadr Us log transformd nadr PSA Estmaton of Rgrsson Modl. gnrat lnadr log(nadr). stco lnadr, robust Co rgrsson -- Brslow mthod for ts No. of subj 5 No. of obs 5 No. fal 36 Tm at rsk 423 LR ch2() 34.4 Log lklhood -7.3 Prob > ch2. _ t HzRat StdErr z P> z [95% Conf Int] lnadr

16 Ald Bostatstcs II, WIN 28 January 23, 28 Intrrtaton of Paramtrs PH Rgrsson and Logrank Tst Hazard rato s.54 for an -fold dffrnc n nadr PSA I can mor asly undrstand doublng, trlng, 5-fold, -fold ncrass For doublng: HR :.54 log(2).35 6 Proortonal hazards rgrsson wth a bnary rdctor (two grous) corrsonds to th logrank tst Thr ossbl statstcs from roortonal hazards rgrsson Wald: Th tst basd on th stmat and SE Scor: Corrsonds to logrank tst, but not gvn n Stata outut Lklhood rato tst: Can b obtand usng ostrgrsson commands n Stata (nt quartr) 62 Intrrtaton of Slos Addtv Modls Idntty lnk functon Mans: lnar rgrsson 63 θ 64 6

17 Ald Bostatstcs II, WIN 28 January 23, 28 Addtv Modls : Slo Intrrtaton of slo: : (Avrag) Dffrnc n summary masur btwn grous r unt dffrnc n : (Avrag) Dffrnc n summary masur btwn grous r unt dffrnc n Addtv Modls : log() Slo wth log transformd rdctor log(k) : (Avrag) Dffrnc n summary masur btwn grous r k-fold dffrnc n θ log( ) θ Multlcatv Modls Log lnk functon Gom mans: lnar rgrsson on logs Odds: logstc rgrsson Hazards: roortonal hazard rgrsson Mans: Posson rgrsson Mdans: accl falur tm rgrsson Multlcatv Modls : Slo Intrrtaton of slo: : (Avrag) Rato of summary masur btwn grous r unt dffrnc n ( ) : (Avrag) Rato of summary masur btwn grous r unt dffrnc n log ( ) ( ) θ θ log 7

18 Ald Bostatstcs II, WIN 28 January 23, 28 Multlcatv Modls : log() Slo wth log transformd rdctor log(k) k ( ) log(k) : (Avrag) Rato of summary masur btwn grous r k-fold dffrnc n ( θ ) log( ) log 69 8