Lecture Outline Biost 518 / Biost 515 Applied Biostatistics II / Biostatistics II

Transcription

1 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Lctur Outln Bost 58 / Bost 55 Ald Bostatstcs II / Bostatstcs II Scott S. Emrson, M.D., Ph.D. Profssor of Bostatstcs Unvrsty of Washngton Rgrsson wth Bnary Rsons Rsk dffrnc: lnar rgrsson Rsk rato: Posson rgrsson Odds rato: logstc rgrsson Sml Logstc Rgrsson Modls Sml Posson Rgrsson Modls Lctur 5: Sml Logstc Rgrsson Modl Sml Posson Rgrsson Modl January 5, 24 2 Bnary Random Varabls Rgrsson wth Bnary Rsons Many varabls can tak on only two valus For convnnc cod as or ndcator varabl Vtal status: Dad codd = alv = dad S: Fmal codd = mal = fmal Intrvnton: T codd = control = nw thray Somtms dchotomz varabls For scntfc rasons (statstcally lss rcs) Blood rssur lss than 6 mm Hg PSA lss than 4 ng/ml Srum glucos lss than 2 mg/dl 3 4 Bost 58 / 55 Ald Bostatstcs II WIN 24

2 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Brnoull Probablty Dstrbuton A bnary varabl Y must hav a Brnoull robablty dstrbuton A sngl aramtr: = Pr( Y = ) wth < < W wrt Y ~ B(, ) Probablty mass functon: Pr ( Y = y ) = y ( ) -y Man: E[ Y ] = Varanc: Var( Y ) = ( ) A man varanc rlatonsh If th man s dffrnt btwn two grous, th varanc must also b dffrnt Mamum varanc of.25 whn =.5 Rgrsson wth Bnary Rsons Conctually, thr should b no roblm modlng th roorton (whch s th man of th dstrbuton) Howvr, thr ar svral scntfc and tchncal rasons why w do not us lnar rgrsson vry oftn wth bnary rsons Th sum of n ndndnt Brnoull random varabls has a bnomal dstrbuton: S n =Y + + Y n ~ B( n, ) Man: E[ S n ] = n Varanc: Var( S n ) = n ( ) 5 6 Statstcal Hyothss Summary masurs of ntrst for a Brnoull random varabl ar rtty much lmtd to thr Th roorton (a man), or Th odds o = / (- ) Choc of Summary Masur / Contrast W thus hav thr chocs of rgrsson wth a bnary rsons RD: lnar rgrsson RR: Posson rgrsson OR: logstc rgrsson Contrasts usd to comar th dstrbuton of a Brnoull random varabl across suboulatons thus nclud Dffrnc n roortons: (rsk dffrnc (RD)) Rato of roortons : / (rsk rato (RR)) Odds rato : / o / o / (odds rato (OR)) 7 In choosng among ths rgrsson modls, w consdr scntfc ssus ncludng Any dsr to accntuat ublc halth mact Any dsr to accntuat dffrncs wth rar vnts Possblty of avodng major nonlnarts and ffct modfcaton Intrlay of How w want to vntually us rsults of th analyss Whch varabl do w dally want to condton on? And th way w samld our data 8 Wr saml szs n any suboulatons fd by dsgn? Bost 58 / 55 Ald Bostatstcs II WIN 24 2

3 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Choc of Summary Masur / Contrast Publc halth mact s tycally bst masurd by th dffrnc n roortons rhas rfr RD rgrsson RD, as a dffrnc n roortons, can stmat th numbr of affctd ol n a largr oulaton Wth rar vnts, th stnc of an assocaton s bst dmonstratd usng ratos rhas rfr RR rgrsson Though unlkly n thr cas, I am many mor tms lkly to wn th lottry by buyng a tckt than by fndng a wnnng tckt Choc of Summary Masur / Contrast Gratr ossblty of avodng major nonlnarts rhas rfr OR rgrsson Basd on fact that < <, but < o < In RD and RR rgrsson, constrants on dctat that thr strngth of assocaton btwn Y and s constrand rlatv to rang of ossbl valus that can hav, or th assocaton btwn Y and must b nonlnar 9 Choc of Summary Masur / Contrast Effct Modfcaton Eaml Gratr ossblty of avodng ffct modfcaton rhas rfr OR rgrsson Basd on fact that < <, but < o < Any covarat that s strongly assocatd wth outcom Y must modfy th ffcts of othr modratly assocatd random varabls on th RD or RR scal W wll dscuss ths furthr whn w consdr adjustd analyss, but th followng aml llustrats th gnral da Th rstrcton on rangs for robablts also mak t lkly that ffct modfcaton wll oftn b rsnt wth roortons Eaml: 2 Yr Rlas rats by nadr PSA > 4, bon scan scor (BSS) n hormonally tratd rostat cancr Both nadr PSA and bon scan scor show strong assocatons wth rlas If bon scan scor < 3: A dffrnc of.6 4% of mn wth nadr PSA < 4 rlas n 24 months % of mn wth nadr PSA > 4 rlas n 24 months If bon scan scor = 3: 7% of mn wth nadr PSA < 4 rlas n 24 months Thus mossbl for mn wth nadr PSA > 4 to hav an absolut dffrnc of.6 hghr 2 Bost 58 / 55 Ald Bostatstcs II WIN 24 3

4 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Choc of Summary Masur / Contrast Cas-Control Studs Intrlay of How w want to vntually us rsults of th analyss Whch varabl do w dally want to condton on? And th way w samld our data Wr saml szs n any suboulatons fd by dsgn? Want to condton on osurs, but us cas-control samlng wth rar dsas rhas rfr OR rgrsson If w constran saml szs for dsasd vs non-dsas, analyss should n gnral b basd on Pr( Eosur Dsas) Th on cton s whn usng th odds rato, bcaus th odds rato basd on condtonng on dsas status must b qual to th odds rato whn condtonng on osur status 3 Scntfc ntrst: Dstrbuton of ffct across grous dfnd by caus E.g., how dos rsk of lung cancr dffr by smokng bhavor Common samlng schms Cohort study: Saml by osur Saml smokrs, nonsmokrs Estmat rsk of cancr n osur grous Cas-control study: Saml by outcoms Saml cancr atnts, controls In gnral: stmat rvalnc of smokng n dagnoss grous E.g., roorton (or odds) of smokrs among ol wth or wthout cancr 4 Us of Odds Ratos Cohort study Odds of cancr among smokrs : odds of cancr among nonsmokrs Cas-control study Odds of smokng among cancr : odds of smokng among noncancr Mathmatcally, th two odds ratos ar th sam Hnc, whn usng cas-control samlng, t s vald to stmat thr odd rato (But th ntrct may b comltly unntrrtabl basd on avalabl data) Eaml: Two Saml Studs Invstgat assocaton btwn mortalty and smokng n a oulaton of ldrly adults Dath wthn 4 yars of som sntnl vnt Smokng bhavor currnt at tm of th sntnl vnt Samlng schms that mght b consdrd Cross-sctonal samlng of 4,994 subjcts Cohort study of 4 smokrs and,2 nonsmokrs Cas-control study of 3 daths wthn 4 yars of sntnl vnt and 9 controls alv 4 yars aftr th sntnl vnt 5 6 Bost 58 / 55 Ald Bostatstcs II WIN 24 4

5 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Cross-sctonal Study (N Tot = 4,994) Vald stmats: Mortalty condtonng on smokng bhavor Vald stmats: Smokng bhavor condtonng on mortalty Cohort Study (N Smk = 4; N NS =,2) Vald stmats: Mortalty condtonng on smokng bhavor Vald stmats: Smokng bhavor condtonng on mortalty Dath w/n 4 Yr Pr (Dth = Smk) Odds (Dth = Smk) Dath w/n 4 Yr Pr (Dth = Smk) Odds (Dth = Smk) Smokng 3, Smokng, RD RR RD RR Pr (Smk = Dth) Dth Smk:.2 Dth Smk:.27 Pr (Smk = Dth) Dth Smk:.33 Dth Smk:.45 Smk Dth:.25 Smk Dth:.2 Smk Dth:.29 Smk Dth:.8 OR OR Odds (Smk = Dth) Dth Smk:.246 Odds (Smk = Dth) Dth Smk:.63 Smk Dth:.246 Smk Dth: Cas-Control Study (N D = 3; N Surv = 9) Tak Hom Mssag Vald stmats: Mortalty condtonng on smokng bhavor Vald stmats: Smokng bhavor condtonng on mortalty Smokng Dath w/n 4 Yr Pr (Dth = Smk) Odds (Dth = Smk) RD RR Pr (Smk = Dth) Dth Smk:.9 Dth Smk:.44 Smk Dth:.67 Smk Dth:.5 OR Odds (Smk = Dth) Dth Smk:.59 Smk Dth:.59 9 Th corrsondng vald stmats from ach study dsgn ar stmatng th sam quantty E.g., both th cross-sctonal and cohort studs can b usd to stmat th oulaton Pr[ D w/n 4 yars Smok] I cratd a sngl cohort dsgn and a sngl cas-control dsgn by samlng from th cross-sctonal dsgn. Thr s of cours lss rcson n thos drvd dsgns, bcaus th saml szs ar smallr Furthrmor, th cross-sctonal dsgn dd not hav all that much rcson Th nfrnc from th cross-sctonal study stmatd an odds rato of.246, wth a 95% CI of.94 to.63 Th stmatd OR from th cohort and cas-control studs (whch wr.63 and.59, rsctvly) wr consstnt wth that lack of rcson 2 Bost 58 / 55 Ald Bostatstcs II WIN 24 5

6 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Tak Hom Mssag 2 OR Intrrtaton n Cas-Control Studs All study dsgns ar stmatng th odds rato comarng th odds of dath wthn 4 yars for smokrs to th odds of dath wthn 4 yars for nonsmokrs Th cross-sctonal and cohort studs can do ths drctly Th cas-control study can do ths ndrctly from a scntfc standont But bcaus ths s tru scntfcally, and bcaus th OR s mathmatcally th sam n thr drcton, w can actually ft th rvrs logstc rgrsson modl and gt th sam answr for th slo (though th ntrct n that modl s not stmatng a oulaton-basd odds) Ths rorty s an advantag of lookng at OR, bcaus wth rar vnts, cas-control samlng s mor fasbl and conomcal 2 Th odds rato s asly ntrrtd whn tryng to nvstgat rar vnts Odds = rob / ( rob) Rar vnt: ( rob) s aromatly Odds s aromatly th robablty Odds rato s aromatly th rsk rato Rsk ratos ar asly undrstood Cas-control studs ar tycally usd whn vnts ar rar Not that n th rvous aml, th robablty of dath was on th ordr of % n th cross-sctonal study, so th OR and th RR ar only aromatly qual. 22 Statstcs: Man-Varanc Thr s also a tchncal roblm wth usng classcal lnar rgrsson Classcal lnar rqurs qual varancs n ach rdctor grou n ordr for CI and valus to b vald But wth bnary Y, th varanc wthn a grou dnds on th man Man: E[ Y ] = Varanc: Var( Y ) = ( ) In th rsnc of an assocaton btwn rsons and POI, w wll dfntly hav htroscdastcty Sml Logstc Rgrsson Infrnc About th Odds Whn usng th Hubr-Wht sandwch stmat of robust standard rrors, ths roblm not such a lmtaton Modrat saml szs ar ndd Bost 58 / 55 Ald Bostatstcs II WIN 24 6

7 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Bost 58 / 55 Ald Bostatstcs II WIN 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 26 Sml Logstc Rgrsson Modlng odds of bnary rsons Y on rdctor log odds log odds log odds log logt Modl Pr Dstrbuton Y 27 Intrrtaton as Odds Eonntaton of rgrsson aramtrs odds odds odds Modl Pr Dstrbuton Y 28 Estmatng Proortons Proorton = odds / ( + odds) / Modl Pr Dstrbuton Y

8 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Paramtr Intrrtaton Intrrtaton of th logstc rgrsson aramtrs basd on odds 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: ( ) Smlarty to Othr Rgrssons Logstc rgrsson uss mamum lklhood stmaton to fnd aramtr stmats If a saturatd modl s ft, th stmatd odds of vnt n ach grou wll agr actly wth th saml odds In larg samls, th rgrsson aramtr stmats ar aromatly normally dstrbutd P valus and CI that ar dslayd for ach aramtr stmat ar Wald- basd stmats 95% CI : ( stmat) ( crt valu) ( std rr) ˆ z / 2 sˆ ˆ 29 Tst stat : ( stmat) ( null) Z ( std rr) ˆ Z sˆ ˆ 3 Tchncal Dtals Unlk lnar rgrsson, thr s no closd form rsson to fnd th logstc rgrsson aramtr stmats Instad, comutr rograms us an tratv sarch Ths sarch may fal n saturatd or narly saturatd modls f som aramtr corrsonds to a grou havng all vnts or no vnts In ths sttng, logstc rgrsson aramtrs modlng th log odds ar tryng to stmat ostv or ngatv nfnty Th saml sz s too small for th modl Stata 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 3 32 Bost 58 / 55 Ald Bostatstcs II WIN 24 8

9 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Eaml Prvalnc of strok (crbrovascular accdnt- CVA) by ag n subst of Cardovascular Halth Study Lowss Smooth of CVA vs Ag Scattr lot s rtty uslss Rsons varabl s CVA Bnary varabl: = no hstory of ror strok, = ror hstory of strok Prdctor varabl s Ag Contnuous rdctor Pror Hstory of Strok (= Ys, = No) Ag (yars) 34 Charactrzaton of Plot Clarly th scattrlot (vn wth surmosd smooth) s rtty uslss wth a bnary rsons (Not that w ar stmatng roortons not odds wth ths lot, so w can not vn judg lnarty for logstc rgrsson) Eaml: Rgrsson Modl Answr quston by assssng lnar trnds n log odds of strok by ag Estmat bst fttng ln to log odds of CVA wthn ag grous logodds CVA Ag Ag An assocaton wll st f th slo ( ) s nonzro In that cas, th odds (and robablty) of CVA wll b dffrnt across dffrnt ag grous Bost 58 / 55 Ald Bostatstcs II WIN 24 9

10 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Paramtr Estmats. logt cva ag (traton nfo dltd) Numbr of obs = 735 LR ch2() = 2.45 Prob > ch2 =.75 Log lklhood = Psudo R2 =.5 Intrrtaton of Stata Outut Rgrsson modl for CVA on ag Intrct s labld by _cons Estmatd ntrct: Slo s labld by varabl nam: ag Estmatd slo:.336 Estmatd lnar rlatonsh: log odds CVA by ag grou gvn by cva Cof StdErr z P> z [95% Conf Int] ag _cons log odds CVA Ag Intrrtaton of Intrct log odds CVA Ag Intrrtaton of Slo log odds CVA Ag Estmatd log odds CVA for nwborns s Odds of CVA for nwborns s =.92 Probablty of CVA for nwborns Us rob = odds / (+odds):.92 / (+.92)=.9 Prtty rdculous to try to stmat W nvr samld anyon lss than 67 In ths roblm, th ntrct s just a tool n fttng th modl Estmatd dffrnc n log odds CVA for two grous dffrng by on yar n ag s.336, wth oldr grou tndng to hghr log odds Odds Rato:.336 =.34 For 5 yar ag dffrnc: =.34 5 =.83 (If a straght ln rlatonsh s not tru, w ntrrt th slo as an avrag dffrnc n log odds CVA r on yar dffrnc n ag) 39 4 Bost 58 / 55 Ald Bostatstcs II WIN 24

11 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Stata: logt vrsus logstc W ar rarly ntrstd n th ntrct by tslf W do hav to us t whn stmatng odds of an vnt n a sngl grou Gvn that w ar rarly ntrstd n th ntrct, w mght as wll us th logstc command It wll rovd nfrnc for th odds rato W don t hav to onntat th slo stmat Odds Ratos usng logstc.logstc cva ag Logstc rgrsson Numbr of obs = 735 LR ch2() = 2.45 Prob > ch2 =.75 Log lklhood = Psudo R2 =.5 cva Odds Rato StdErr z P> z [95% Conf Int] ag Eaml: Intrrtaton 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. Commnts on Intrrtaton I rss ths as a dffrnc btwn grou odds rathr than a chang wth agng W dd not do a longtudnal study To th tnt that th tru grou log odds hav a lnar rlatonsh, ths ntrrtaton als actly If th tru rlatonsh s nonlnar Th slo stmats th frst ordr trnd for th samld ag dstrbuton W should not rgard th stmats of ndvdual grou robablts / odds as accurat Bost 58 / 55 Ald Bostatstcs II WIN 24

12 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Logstc Rgrsson and 2 Tst 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 ost-rgrsson commands n Stata (covrd wth adjustng for covarats) Sgnal and Nos Not that th Sgnal and Nos da dos not aly so wll hr W do not tnd to quantfy an rror dstrbuton wth logstc rgrsson Count Data Sml Posson Rgrsson Infrnc About Rats Somtms a random varabl masurs th numbr of vnts occurrng ovr som rgon of sac and ntrval of tm 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 Bost 58 / 55 Ald Bostatstcs II WIN 24 2

13 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Evnt Rats 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 Posson Probablty Modl 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 sactm ar ndndnt Assumton of a constant rat wth ndndnc ovr sarat ntrvals s rtty strong 49 5 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, Pr Y k t t k! k 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 Man E(Y) = λt; varanc Var(Y)= λt Posson aro to Bnomal for low 5 52 Bost 58 / 55 Ald Bostatstcs II WIN 24 3

14 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Why not Lnar Rgrsson? Prmarly statstcal Th rat s n fact a man For Posson Y masurd ovr tm t and havng vnt rat E(Y) = t Var (Y) = t But Want to account for dffrnt aras or lngth of tm for masurmnt Nd to account for man-varanc rlatonsh (f not usng robust SE) 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 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 Sml Posson Rgrsson Modlng rat of count rsons Y on rdctor Dstn Modl Y E ~ P t PrY k T t Y T, log T logt log log log t t k! k Bost 58 / 55 Ald Bostatstcs II WIN 24 4

15 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Intrrtaton as Rats Eonntaton of aramtrs Sml Posson Rgrsson Intrrtaton of th modl Dstn Modl Y E ~ P t PrY k T t Y T, log T logt t t k! k 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: () Stata Commands Smlarty to Othr Rgrssons 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, offst(logtm) [robust] By dfault, Stata rorts stmats on th log man and log man rato scal Scfyng th oton rr wll caus Stata to surss outut of th ntrct and to rort ncdnc rat ratos 59 Posson rgrsson uss mamum lklhood stmaton to fnd aramtr stmats If a saturatd modl s ft, th stmatd man n ach grou wll agr actly wth th saml man In larg samls, th rgrsson aramtr stmats ar aromatly normally dstrbutd P valus and CI that ar dslayd for ach aramtr stmat ar Wald- basd stmats 95% CI : ( stmat) ( crt valu) ( std rr) Tst stat : ( stmat) ( null) Z ( std rr) ˆ z / 2 ˆ Z sˆ ˆ sˆ ˆ 6 Bost 58 / 55 Ald Bostatstcs II WIN 24 5

16 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Tchncal Dtals Unlk lnar rgrsson, thr s no closd form rsson to fnd th Posson rgrsson aramtr stmats Instad, comutr rograms us an tratv sarch Eaml: Sttng Chmosnstzrs for cancr chmothray In vtro valuaton of th ablty of som drugs to otntat th cytotoc ffcts of doorubcn Ths sarch may fal n saturatd or narly saturatd modls f som aramtr corrsonds to a grou havng no vnts In ths sttng, Posson rgrsson aramtrs modlng th log man ar tryng to stmat ngatv nfnty Th saml sz s too small for th modl Clls culturd n th laboratory ar osd to doorubcn at svral concntratons wth and wthout chmosnstzrs Ths aml: Only th control grou 6 62 Eaml: Varabls Rsons: Numbr of survvng cll colons Each rsumably arsng from a sngl cll Scattrlot Clls vs Do Conc 243 Offst: Dfault valu of Sam volum of cultur usd for all samls clls Prdctor: Concntraton of doorubcn conc 64 Bost 58 / 55 Ald Bostatstcs II WIN 24 6

17 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Charactrzaton of Scattrlot Scattrlot: Clls vs log (Conc) 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 243 Lowss smoothr, bandwdth =.8 Mchals-Mntn kntcs: Actually S shad n log concntraton, but wll aromatd lnarly ovr a rang of doss clls lconc 66 Charactrzaton of Scattrlot 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: Dcrasng varanc for lowr grou mans (not smallr saml sz n frst grou) Estmaton of Rgrsson Modl. osson clls lconc (Itraton nformaton omttd) Numbr of obs = 282 LR ch2() = Prob > ch2 =. Psudo R2 =.6242 clls Cof. StErr. z P> z [95% CI] lconc _cons Bost 58 / 55 Ald Bostatstcs II WIN 24 7

18 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Intrrtaton of Stata Outut log rat lconc Rgrsson modl for clls on log concntraton Intrct s labld by _cons Estmatd ntrct: 3.75 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 Slo s labld by varabl nam: lconc Estmatd slo: In ths roblm, th ntrct s of ntrst f th lnar rlatonsh btwn log concntraton and log rat s corrct 69 7 Intrrtaton of Slo 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 =.43 as larg 56.9% dcras n survval rat 95% CI for fold ncras n conc: 56.3% to 57.5% dcras =.425, -.36 = Eaml: Intrrtaton From Posson rgrsson analyss, w stmat that for ach fold ncras n concntraton of doorubcn, th robablty of cll survval dcrass by 56.9%, hghly statstcally sgnfcant obsrvaton (P <.). A 95% CI suggsts that ths obsrvaton s not unusual f a cll cultur osd to a -fold hghr concntraton of doorubcn mght hav a cll survval rat that was dcrasd anywhr from 56.3% to 57.5% comard to th lowr concntraton grou. 72 Bost 58 / 55 Ald Bostatstcs II WIN 24 8

19 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Rol of Lnarty W hav to b carful n ntrrtng ths modl f th lnar rlatonsh dos not hold Fttd Rgrsson Modl. rdct fclls. grah clls fclls lconc, s(ot) c(.l) clls rdctd numbr of vnts Scattrlot suggstd lnar rlatonsh btwn cll counts and log concntraton was rasonabl 243 But w modld th log rat vrsus log concntraton lconc Rol of Lnarty Th tru goal of ths rmnt was to stmat th concntraton at whch 5% of clls mght b ctd to d Th LC 5 Thr sgnfcanc of th slo was takn for grantd Th lack of lnarty mans that w cannot trust th Posson rgrsson wth a lnar trm to rovd th stmat w want W ar tryng to mak stmat of ach grou s man In ral lf, I ft a nonlnar curv to ths data That curv was basd on th S-shad curv that Mchals- Mntn kntcs would suggst Dsas Incdnc Rats Posson rgrsson s frquntly usd to nvstgat dsas ncdnc rats Incdnc rat ratos ar thn th targt of nfrnc For a scfd combnaton of covarats, data mght consst of Total rson-yars of obsrvaton whl at rsk Numbr of vnts obsrvd wthn that tm rod Oftn such data s groud nto ag ntrvals, wth a artcular ndvdual contrbutng rson-yars to multl strata Ths can b usd as a cws onntal survval analyss Bost 58 / 55 Ald Bostatstcs II WIN 24 9

20 Lctur 5: Sml Logstc, Posson Rgrsson January 5, 24 Wdr Us of Posson Rgrsson Mor gnrally, Posson rgrsson can b usd to modl mans on a multlcatv scal n a wd varty of sttngs Eonntatd slo stmats ar man ratos (cf: lnar rgrsson on log Y as gomtrc man ratos) Bcaus classcal Posson rgrsson rsums a artcular man-varanc rlatonsh, robust SE should usually b usd to rmov that assumton Wth Brnoull data havng low vnt rats, th Posson aromaton to th Bnomal justfs th us of Posson rgrsson, and th man-varanc rlatonsh s corrct Brnoull: Var (Y) = ( ) Posson: Var (Y) = Eaml: CVA vs Ag W can comar logstc rgrsson to Posson rgrsson on analyss nvstgatng an assocaton btwn crbrovascular accdnts (CVA strok) and ag Logstc rgrsson has odds rato (OR) as targt of nfrnc Posson rgrsson has rsk rato (RR) as targt of nfrnc In ths data, 75 of 735 subjcts hav had a hstory of CVA Ovrall stmatd robablty s.36 Ovrall stmatd odds s.36 / ( -.36) = 282 Ths ncdnc s not so low as to rgard that OR and RR wll b actly th sam I us robust SE n th Posson rgrsson modl Eaml: CVA vs Ag. osson cva ag, robust rr Posson rgrsson Numbr of obs = 735 Wald ch2() = 2.6 Prob > ch2 =.64 Log sudolklhood = Psudo R2 =.44 Robust cva IRR Std. Err. z P> z [95% Conf. Intrval] Ag logstc cva ag Logstc rgrsson Numbr of obs = 735 LR ch2() = 2.45 Prob > ch2 =.75 Log lklhood = Psudo R2 =.5 cva Odds Rato Std. Err. z P> z [95% Conf. Intrval] ag Bost 58 / 55 Ald Bostatstcs II WIN 24 2