Logistic Regression Sara Vyrostek Senior Exercise November 16, 2001

Transcription

1 ogstc Rgrsso Sara Vrostk Sor Ercs Novmbr 6,

2 Itroducto: I th modlg of data, aalsts dvlop rlatoshps basd upo th obsrvd valus of a st of prdctor varabls ordr to dtrm th pctd valu of th rspos varabl of trst, a tchqu kow as rgrsso. A vr commo form of rgrsso aalss s lar rgrsso, whch ams th ffct of a ut chag a prdctor varabl o th rspos varabl. owvr, thr ar ma cass, whthr t s du to pattrs th data or th dsg of th data tslf, whr a lar aalss s approprat. O partcular cas whr th lar aalss dos ot provd a adquat ft occurs wh th rspos varabl s bar, mag t has ol two possbl valus. For ths scaro, a rlatd tchqu kow as logstc rgrsso s usd to modl th data. Th logstc modl provds a curvlar ft wth asmptots at both zro ad o, th two possbl valus for th rspos varabl. Fgur dsplas a graph of a logstc modl ft to a tpcal bar data st, vsuall dsplag ths charactrstcs. Fgur : A Tpcal ogstc Modl Ntr, Joh, Wllam Wassrma, Chrstophr J. Nachtshm, ad Mchal. Kutr, Appld ar Rgrsso Modls, 3 rd d. Chcago: McGraw-ll, 996 p. 57.

3 Th logstc rgrsso tchqu s wdl usd a ara whch th rspos varabl tds to tak o th bar form. For ampl, halth rsarch t s commo for th rspos varabl to hav th form s or o or rspodd to mdcato vrsus dd ot rspod to mdcato, stuatos whch logstc rgrsso would b mplod. Not ol wll ths papr srv as a troducto to rgrsso th logstc sttg, but t wll also stp through th procsss of paramtr stmato ad trprtato, th buldg of a logstc rgrsso modl, ad th aalss of ft. Throughout th papr, a ogog aalss of a data st b mas of both had calculatos as wll as th trprtato of SAS output wll provd th opportut to work through all of th topcs dscussd. Th data st, a subst of varabls ad obsrvatos tak from a largr stud do at th Uvrst of Massachustts Ads Rsarch Ut ordr to am drug us assocato wth IV, was foud osmr ad mshow s Appld ogstc Rgrsso. Th varabls th data st ar as follows: Fgur : Dscrpto of Varabls Varabl Dscrpto Cods/Valus Nam Idtfcato Cod -575 ID Ag at Erollmt Yars ag 3 Bck Dprsso Scor at.-54. bck Admsso 4 IV Drug Us stor at Nvr, Prvous, IV Admsso 3Rct 5 Numbr of Pror Drug -4 prors Tratmts 6 Subjct s Rac Wht, Othr rac 7 Tratmt Radomzato Short, og tratmt Assgmt 8 Tratmt St A, B st 9 Rmad Drug Fr for Moths Rmad Drug Fr, Othrws drugfr osmr, Davd W., ad Stal mshow, Appld ogstc Rgrsso Nw York: Joh Wl & Sos, pp

4 Smpl ogstc Rgrsso Modl: Workg th bar sttg, Y ca tak o ol o of two valus, ad thus ca b codd as follows:, Y, f f th vt occurs th vt dos ot occur I th smpl sttg wth ol o prdctor varabl, th logstc rgrsso modl wll tak o th followg gral form: whr, Y ε, E[ Y X ] ad ε s a radom rror trm. 3 Altratvl, ca b wrtt as Bcaus rprsts th probablt that Y gv th valu of X, ths mpls that rprsts th probablt that Y, gv th valu of X. Y s dstrbutd such that t has two possbl outcoms wth a probablt of succss, mag that th Y 's follow th Broull dstrbuto. Ulk th lar modl whr th rror trms ar ormall dstrbutd, th rror trms ths modl hav ol two possbl valus. If Y, th ε wth probablt, corrspodg to th abov formato. O th othr had, f Y whch occurs wth probablt, th ε. Th ma of th rror trms ca b foud b calculatg thr pctd valu. Sc ε ca ol tak o two possbl valus, th pctd valu s obtad through th followg formula: 3 Chattrj, Samprt, Al S. ad, ad Brtram Prc, Rgrsso Aalss b Eampl Nw York: Joh Wl & Sos, p. 3. 4

5 5 ] ][ [ ] [ P P E ε ε ε ε ε ε ε Th varac of th rror trms ca b dtrmd a smlar mar: ] [ } ] ]{[ [ ] [ ] [ ] [ ] [ ] [ P P E E Var ε ε ε ε ε ε ε ε ε Ths formato provds a basc udrstadg of th dstrbutoal sttg of ths aalss. 4 avg dfd ad dscrbd th logstc rgrsso fucto, th t task volvs stmatg th paramtrs ad for th modl wh fttg t to a actual data st. B th vr atur of th probablt fucto dscrbd abov, th stmato procss ca b smplfd b formg a lar quato volvg ths two paramtrs. Wth dotg th probablt that Y, th odds of Y occurrg ca b dfd as th probablt that Y dvdd b th probablt that Y, or Y. I othr words, odds. Th odds ar mportat for a bar data st bcaus th offr a rato for th chacs of a vt occurrg as opposd to a vt ot occurrg. For ampl, f th probablt of th su comg out o a gv da s two-thrds, mag that th probablt of ot sg th su o a gv da s o-thrd, th th odds of a su da ar 3 / 3 /, or a two to o rato. Ths 4 Chattrj, p. 3.

6 mpls that a su da s twc as lkl as a cloud da. Bcaus th quato for th odds rsults a dffcult fucto to ft, h ca b dfd as th atural log of th odds, also kow as th logt fucto: h l l l l Ths fucto, h, s of th lar form, provdg a much asr modl to ft. 5 Th paramtrs ad ca b stmatd through th mthod of mamum lklhood stmato. Ths mthod solats th valus of a paramtr that mamz th lklhood fucto, whr th lklhood fucto, s dfd as th jot probablt dstrbuto for all of th data pots. 6 Sc th Y 's hav a Broull dstrbuto, th probablt dst fucto ca b dfd as P Y f [ ], whr or, ad,,. Bcaus th Y 's ar dpdt, th lklhood fucto ca b dfd as follows:, g,..., f [ ] 5 osmr, Davd W., ad Stal mshow, Appld ogstc Rgrsso Nw York: Joh Wl & Sos, 989 p osmr, p. 8. 6

7 7 I ordr to mamz ths fucto, th drvatv must b tak wth rspct to ach of th paramtrs. Th, th rsultg quatos would b st qual to zro ad solvd smultaousl. Ths procss ca b smplfd b prformg th sam aalss o th atural log of th lklhood fucto, bg that mamzg th atural log of th fucto would rsult th sam valu as mamzg th lklhood fucto tslf. Obtag th log-lklhood fucto: [ ] { } { } { } [ ] [ ] { } { } l l l l l ] l[ ] l[ ] l[ ] l[ l ] [ l, l Now takg th drvatv, frst wth rspct to ad th wth rspct to ad sttg ach qual to zro, th followg lklhood quatos ar formd: ] [, l ] [, l Bcaus th lklhood quatos ar ot lar, solvg ths quatos smultaousl rqurs a tratv procdur that s ormall lft to a softwar packag. 7 7 Ntr, pp

8 t us cosdr th data st o drug us. Suppos a smpl modl s dsrd usg ol o prdctor varabl, t, whr t rprsts th tp of tratmt admstrd. W wll dsgat th rspos varabl Y as a dcator for whthr or ot th subjct rmad drug fr for twlv moths. Th basc logstc modl s: Y ε. t t ε t Th accompag logt fucto s: l t t. h t t Usg SAS to obta th approprat paramtr stmats, th followg output s producd: Dspla : Smpl ogstc Rgrsso--Paramtr Estmats Th OGISTIC Procdur Aalss of Mamum klhood Estmats Stadard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Itrcpt <. tratmt Odds Rato Estmats Pot 95% Wald Effct Estmat Cofdc mts tratmt Dspla rports th valu for b, th paramtr stmat for, as.978, ad th valu for b, th paramtr stmat for, as.437. Thus, th stmatd logt fucto would rad h t t, wth th paramtr stmats for th modl havg b dtrmd through th mthod of mamum lklhood stmato. Th valu for th log-lklhood fucto s also producd th SAS output: Dspla : Smpl ogstc Rgrsso--Th og-klhood Fucto Modl Ft Statstcs Itrcpt Itrcpt ad 8

9 Crtro Ol Covarats AIC SC og B trprtg th output Dspla, t s vdt that th log-lklhood fucto s qual to / Th output also rports th valu of th log-lklhood fucto for th modl wth just th trcpt, whch s qual to / Sc th logt fucto s wrtt a lar form wth rprstg th slop, b rprsts th chag h for a o-ut chag. Thrfor, b h h l l l odds l odds odds l odds. Takg th potal of both sds, w gt: odds odds b, whch ca b dfd as th odds rato. Thrfor, aftr fdg th paramtr stmats, th b valu wll rprst th prctag cras th probablt that Y for ach ut cras X. 8 Ths valu for th drug data s s Dspla, wth a pot stmat for th odds rato of.548. Ths dcats that b chagg th tratmt prod from short to log, th probablt of a subjct rmag drug fr for a twlv-moth prod aftr th tratmt crass b 54.8%. Dspt havg ft th modl ad trprtd th coffcts, ths formato s uslss ulss th prdctor varabl s sgfcat to th modl. O mthod for dtrmg whthr or 8 Ntr, p

10 ot th coffct s sgfcat s through th lklhood rato tst. Assumg that w hav obsrvatos o ach varabl, th dvac of th modl, D, wll b dfd as follows: lklhood of th currt Modl D l lklhood of th saturatd Modl Th saturatd modl rprsts th modl cotag paramtrs, such that th modl prfctl prdcts th obsrvd data st. I othr words, th prdctd valus for ths modl ar qual to th obsrvd valus th data st. Th dvac provds a mas of comparg th lklhood of th modl that has b ft, or th probablt of obtag th obsrvd data st gv th currt modl, to that of th saturatd modl. 9 B mapulatg th abov dfto, w fd: [ l lklhood of th currt Modl l lklhood of th saturatd ] D Modl Rcall that th gral modl s of th form Y ε. For th currt modl, ˆ srvs as th stmator for. owvr, th saturatd modl grats th complt data st such that srvs as th stmator for. Also rcallg th drvato of th lklhood fucto, w kow that: l, l [ ] { l[ ] l[ ]}. Ths quato ca b substtutd to th formula for th dvac ad th mapulatd ordr to gt th followg workg quato:. 9 osmr, p. 3.

11 D l[ ˆ ] l[ ˆ ] l l ˆ ˆ l l. l[ ˆ ] l l[ ˆ ] l Esstall, th dvac taks th lklhood of th currt modl whr a lmt of rror s prst ad subtracts th lklhood of th saturatd modl whch thr s o rror trm prst, ad th sums ovr ths dffrc. Thus, th dvac th logstc sttg srvs th sam purpos as th rsdual sums of squars th lar sttg. I ordr to dtrm whthr or ot th paramtr s sgfcat to th modl, th dvac of th modl cotag th paramtr must b compard wth th dvac of th modl wthout th paramtr. Thrfor, th tst statstc, G, s: G Dfor th modl wthout th varabl - Dfor th modl wth th varabl lklhood of currt Modl wthout G l lklhood of saturatd Modl lklhood of currt Modl wthout G l lklhood of currt Modl wth lklhood of currt Modl wth l lklhood of saturatd Modl G l lklhood of currt Modl wthout l lklhood of currt Modl wth W hav alrad foud th lklhood fucto for th modl wth th sgl prdctor varabl. For th lklhood fucto for th modl wthout th sgl prdctor varabl, th probablt that Y wll b qual to th avrag umbr of zros th sampl, whl th probablt that Y wll b qual to th avrag umbr of os th sampl. Thrfor, osmr, p. 3. osmr, pp. 3-4.

12 Gv that lklhood fucto, w ca dtrm th approprat quato for th tst statstc G: G ˆ ˆ l. G follows th ch-squard dstrbuto wth o dgr of frdom. I chckg th sgfcac of th coffct, th followg ull ad altratv hpothss ar to b tstd: : : a. For ths tst, th dcso rul rqurs that f ; χ α > G, s to b rjctd, mag that th coffct would b dmd sgfcat. O th othr had f ; χ α G, th o must fal to rjct, cocludg that th coffct s sgfcat. SAS rports th tst statstc ad th corrspodg p-valu, whr th p-valu s qual to th probablt of obsrvg a tst statstc at last as trm as th obsrvd valu assumg that th ull hpothss s tru. For ths lklhood rato tst, th p-valu PG th obsrvd G-valu, whr G ~ ; α χ.

13 Bcaus th α-lvl rprsts th probablt of rjctg th ull hpothss wh th ull hpothss s tru, as log as th p-valu s lss tha th chos α-lvl, th ull hpothss ca b safl rjctd. For all of th futur tsts ths papr, w wll us a α-lvl of.5. For th drug data st, G ca b calculatd from th SAS output two mars. Frst, kowg th formula for G, w s that: G l lklhood of currt Modl wthout l lklhood of currt Modl wth. Th valus for *log-lklhood trs from Dspla ca b srtd to calculat th tst statstc. Usg ths formato, G Thrfor wth a alpha lvl of.5 ad χ.5; , th ull hpothss ca b rjctd, mag that th paramtr stmat for th tp of tratmt admstrd s sgfcat to th modl. Th p- valu for ths tst rprsts th probablt of obsrvg a valu for G of at last 5.78 such that th p-valu P G Bcaus ths p-valu s lss tha th dsrd alpha lvl of.5, th ull hpothss would b rjctd hr, ladg to th sam cocluso. Aothr scto of th SAS output actuall computs G, or th lklhood rato tst statstc, ad calculats th approprat p-valu for th tst of th abov hpothss: Dspla 3: Smpl ogstc Rgrsso--Th klhood Rato Tst Tstg Global Null pothss: BETA Tst Ch-Squar DF Pr > ChSq klhood Rato Scor Wald Th row radg lklhood rato rports th tst statstc G 5.78, th sam as that calculatd abov, wth o dgr of frdom, ad a p-valu of.9. osmr, pp

14 Aothr mthod for computg th sgfcac of a coffct s through th Wald tst, bk whr ordr to tst th sam hpothss as abov, th tst statstc W s usd. Sc s b ths tst statstc s appromatl ormal, f W z α / th o must fal to rjct th ull hpothss, whl f W > z α /, th th ull hpothss ca b rjctd at th gv alpha k lvl. 3 For ths tst, th p-valu ca b dfd as follows: p valu P W > th obsrvd tst statstc P W > th obsrvd tst statstc,.437 whr W ~ z α /. For th drug data st, W.636. Usg a alpha lvl of.93.5, th crtcal valu s z , ad th corrspodg p-valu s P W >.636 * Sc W > z ad th corrspodg p-valu s lss tha.5, th ull hpothss wll stll b rjctd ad sam cocluso of varabl sgfcac ca b rachd. Ths tst also ca b wrtt a altratv mar. Bcaus th squarg a ormal radom varabl wll rsult a ch-squar radom varabl wth o dgr of frdom 4, th bk Wald tst statstc ca b wrtt as W whr W ~ χ α;. I accordac s bk wth ths chag, th dcso rul must b adjustd such that th ull hpothss s rjctd wh W > χ α;. kws, th p-valu wll b rdfd so that th p valu P W > th obsrvd tst statstc. ookg back at Dspla, w ca s that SAS rports ths tst statstc, rathr tha thos prvousl dscrbd. For th sgl prdctor 3 Ntr, pp

15 varabl tratmt, th Wald ch-squar tst statstc s 5.66 wth a corrspodg p-valu of.36, aga dcatg that ths prdctor varabl s sgfcat to th modl. Multpl ogstc Rgrsso Modl: W ca td th abov aalss for th smpl logstc rgrsso modl whr w hav ol o prdctor varabl to a sttg wth mor tha o prdctor varabl. I ths sttg, th vctor... rprsts th collcto of p prdctor varabls for ths modl. p Th formulas for th probablt that Y,, as wll as for th logt trasformato, h, ca b tdd whr:... p p.. p p ad h... p p. Th ol tm that th modl wll dffr from ths gral formula s f o of th prdctor varabls s a dscrt, catgorcal varabl wth mor tha two lvls. If o or mor of ths s prst ad f th umbr of varabl catgors s qual to k, th k- dsg varabls must b cratd. 5 Ths dsg varabls ar just bar varabls mat to srv as a dcator for o of th lvls of th assocatd varabl. Usg our drug data varabl IV as a ampl, ths varabl would hav to b rcodd bcaus t s a dscrt varabl wth thr lvls: rct, prvous, or o hstor of IV drug us. Bcaus IV currtl has thr dscrt lvls, two dcator varabls would hav to b cratd ad substtutd to th modl. Ths two w varabls mght b IV ad IV, whr, IV 3 IV, ad, IV 3, IV IV., IV 4 ogg, Robrt V., ad All T. Crag, Itroducto to Mathmatcal Statstcs odo: Th MacMlla Compa, 97 p osmr, pp

16 W would ol b rqurd to clud two dcator varabls bcaus th thrd valu of th prdctor varabl s mplctl cludd th modl. I ths cas, wh IV whch mas that IV, ths dcats that th subjct has o hstor of drug us. kws, wh IV whch mas that ow IV, ths mas that th subjct has a hstor of prvous drug us. owvr, f both of ths dcator varabls qual zro, ths would mpl that th subjct has a hstor of thr o drug us or prvous drug us. Istad, ths would dcat that th subjct has a hstor of rct drug us. Th paramtrs th multpl sttg ar oc aga dtrmd through mamum lklhood stmato. Bcaus Y stll rmas a Broull varabl wth th sam probablt dstrbuto, th drvato of th mamum lklhood stmators rmas th sam, wth th cpto of th cluso of mor paramtrs. Thus, th log-lklhood quato would tak th form: l... p p {... p p l },,..., p I th sam mar as bfor, th quatos rsultg from takg th drvatv of th loglklhood quato wth rspct to ach of th paramtrs ad th sttg ach drvatv qual to zro would b solvd smultaousl ordr to obta th stmats. Bcaus ths procdur s v mor computatoall tsv wth multpl paramtrs, th stmato oc aga s lft to computr softwar. 6 Eamg multpl logstc rgrsso b mas of th drug us data st, w ca dvlop a modl cludg thr chos prdctor varabls: ag, rac, ad tratmt. Th modl wll tak th form:. 6 Ntr, p. 7. 6

17 Y ε, whr 33 33, wth rprstg ag, rac, ad 3 tratmt. Th logt wll tak th form: h 33 Usg SAS to stmat th approprat paramtrs, th followg formato s prtd th output: Dspla 4: Multpl ogstc Rgrsso--Paramtr Estmats Aalss of Mamum klhood Estmats Stadard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Itrcpt ag rac tratmt Odds Rato Estmats Pot 95% Wald Effct Estmat Cofdc mts ag rac tratmt Usg ths valus, th stmatd logt quato s: h Th trprtato of th paramtrs s th sam hr as for th smpl modl. Th odds rato stmat, or b, rprsts th prctag cras or dcras th probablt that a drug usr rmas drug fr moths aftr th cocluso of th tratmt program du to a o-ut chag th varabl, holdg all ls costat. That prctag chag probablt s qual to th odds rato stmat mus o. Accordg to th pot stmats for th odds ratos gv th output abov, th lklhood of a subjct rmag drug fr upo th cocluso of tratmt crass b.% for ach addtoal ar ag. Also, th odds that a o-wht prso rmas drug fr ar 5.7% gratr tha th odds of a wht prso rmag drug fr.. 7

18 Fall, a prso rcvg a log tratmt as opposd to a short tratmt would b about.5 tms mor lkl to rma drug fr as wll. Aga, bfor drawg a coclusos from th stmatd logstc rgrsso modl, th complt modl as wll as th dvdual coffcts should b tstd for sgfcac. Frst, usg th dvac prvousl dscrbd, th sgfcac of th modl ca b tstd. Th hpothss of trst ar: : 3 I ordr to tst ths, w wat to comput G whr A : at last oof th abov ' s G Dfor th modl cotag, or th rducd modl - Dfor th full modl lklhood of th rducd Modl lklhood l l lklhood of th saturatd Modl lklhood of of th full Modl th saturatd Modl lklhood of th rducd Modl l lklhood of th full Modl l lklhood of th rducd Modl l lklhood of th full Modl From th SAS output, Dspla 5: Multpl ogstc Rgrsso Sgfcac of Modl Modl Ft Statstcs Itrcpt Itrcpt ad Crtro Ol Covarats AIC SC og Thus, G Notc that th valus th trcpt ol colum ar dtcal to thos valus Dspla. Bcaus ths colum s dpdt of th umbr or tp of prdctor varabls put to th 7 osmr, p. 3. 8

19 modl, ths colum wll rma th sam for whatvr st of prdctor varabls ar cludd. Th computd G-statstc s chod Dspla 6 blow, aothr porto of th SAS output that cluds th lklhood rato tst: Dspla 6: Multpl ogstc Rgrsso--Th klhood Rato Tst Tstg Global Null pothss: BETA Tst Ch-Squar DF Pr > ChSq klhood Rato Scor Wald G wll oc aga follow th ch-squar dstrbuto, but ths cas wth thr dgrs of frdom bcaus th sgfcac of thr paramtrs s bg tstd. Th sam dcso ruls appl to ths sttg as to th smpl sttg abov. Thrfor, th ull hpothss wll b rjctd f G > χ α;3 ad th ull hpothss wll ot b rjctd f G χ α;3. Wth χ.95; ad th assocatd p-valu.34, thr s ough vdc to rjct th ull hpothss ad coclud that th paramtrs ar sgfcat. Kowg that th tr modl s sgfcat dos ot guarat th sgfcac of all of th paramtrs cludd th modl. Thrfor, th paramtrs should b tstd dvduall or small groups usg thr th lklhood-rato tst or th Wald tst, alrad dscrbd, ordr to chck for thr sgfcac. I usg th lklhood rato to tst th sgfcac of a group of varabls, th G statstc would b obtad b comparg th dvac of th full modl, or that cotag all of th paramtrs, to th dvac of th rducd modl, or that wth th paramtrs qusto havg b lmatd. Th rsultg tst statstc would th b compard wth th ch-squar crtcal valu basd upo dgrs of frdom qualg th umbr of paramtrs bg tstd. Th sam dcso ruls alrad dscrbd would b usd. For ampl, th thr varabl modl ft abov, th sgfcac of th dvdual paramtrs ca b tstd usg th Wald p-valus gv th SAS output from Dspla 4. Ol o paramtr 9

20 ag wth a p-valu of.57 appars to b sgfcat at th α.5 lvl. Kowg ths formato, ag could b lmatd from th st of prdctor varabls ad th modl could b rfttd. I tstg th sgfcac of a dscrt catgorcal varabl dvdd to varous sublvls, a clar dcso ca b mad ol wh th sam tst rsult occurs for all lvls of th varabl. Othrws, a larg lmt of ucrtat accompas a cocluso. 8 Buldg th Rgrsso Modl: Thr ar two ma statstcal mthods that ar oft mplod dtrmg whch varabls to clud a modl, th stpws rgrsso mthod ad th bst subst mthod. owvr, addto to th statstcal dvlopmt of th modl, t s mportat to hav a substatal amout of kowldg about th topc bg studd. I som stacs, a varabl that mght b cosdrd vr mportat trms of practcalt would b lmatd statstcall ad would hav to b forcd to th modl. Thrfor whl ths procdurs provd a good tool for th dvlopmt of a workg modl, t s a good da to cosdr th tr of all possbl varabls basd upo a workg kowldg of th data. 9 Th frst tchqu, stpws rgrsso, bgs wth a bas modl cotag ol th trcpt paramtr. It th adds varabls sgfcat to th modl utl thr ar o rmag sgfcat varabls lft to b addd. Th c fatur of ths procdur s that at ach stp aftr a varabl has b addd to th modl, all of th varabls cludd prvous stps ar rtstd ordr to s f th ar stll sgfcat. Th cluso ad tracto of varabls from th modl th stpws procdur s basd upo th lklhood-rato tst. Normall wh dog th lklhood-rato tst, a commol accptd alpha lvl such as. or.5 s chos as th crtcal valu for th tr of varabls to th modl. For th modl buldg procss, ths 8 osmr, p osmr, p. 83.

21 cutoff valu for th tr or rmoval of a varabl should b crasd to aroud.. Ths wll hlp avodg possbl sgfcat varabls from bg ovrlookd or rmovd ucssarl from th modl. Usg th stpws procdur to buld a modl for our drug us data st, th procss ad rsults wll b dscussd wth th ad of SAS output. Th rspos varabl for ths modl wll b Y drugfr, whr Y s a dcator varabl for whthr or ot a subjct rmas drug fr for twlv moths. Th possbl prdctor varabls wll clud th complt pool of varabls avalabl, whr ag, bck, 3 IV, 4 IV, 5 prors, 6 rac, 7 tratmt, ad 8 st. Itall a modl s ft wth just th trcpt trm ad th th corrspodg lklhood,, s computd. Dspla 7: Stp -- Th trcpt s trd. Modl Covrgc Status Covrgc crtro GCONVE-8 satsfd. Aalss of Mamum klhood Estmats Stadard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Itrcpt <. Rsdual Ch-Squar Tst Ch-Squar DF Pr > ChSq <. SAS th fts ght dvdual rgrsso modls, ach cotag just o of th ght avalabl prdctor varabls, ad computs th lklhood, for ach of ths modls. Usg ths lklhood valus, a Scor ch-squar tst statstc s foud b mas of a farl volvd matr computato. It volvs a comparso of th full modl vrsus th rducd modl, whr th full modl cotas th paramtrs whos sgfcac ou ar tstg, whl th rducd modl dos ot clud ths paramtrs. Th dfto for th Scor statstc follows for a full modl cotag s paramtrs ad a rducd modl cotag t paramtrs: osmr, pp. 6-9.

22 ˆ ˆ ˆ U I U Scor whr t t t t t t t t t t t t I U,..., l,..., l,..., l,..., l,,..., l,...,,..., l,,...,, Th Scor statstc follows a ch-squar dstrbuto wth s - t dgrs of frdom. Ths statstc for all of th possbl varabls, as wll as th assocatd p-valus, ca b s Dspla 8: Dspla 8: Th Stpws Procdur--Aalss of th Effcts ot th Modl Scor Effct DF Ch-Squar Pr > ChSq ag bck IV IV prors rac tratmt st I gral, a hghr Scor statstc s bttr tha a lowr o. Wth prors havg th largst sgfcat tst statstc, ths varabl wll b trd to th modl. SAS outputs th paramtr stmats as wll as th lklhood valus for th w modl cotag o prdctor varabl, prors. Ths w formato appars th followg dspla. Dspla 9: Stp --Prors s trd to th modl Modl Covrgc Status Covrgc crtro GCONVE-8 satsfd. Modl Ft Statstcs Itrcpt Itrcpt ad Crtro Ol Covarats AIC SC og SAS, STAT Usr s Gud, Vrso 8, Volum Car, NC: SAS Publshg, 999, p. 948.

23 Tstg Global Null pothss: BETA Tst Ch-Squar DF Pr > ChSq klhood Rato Scor Wald Aalss of Mamum klhood Estmats Stadard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Itrcpt <. prors ookg at Dspla 9, t s vdt from th modl ft statstcs that l ad l, I addto, th rsults for th lklhood rato tst dcat that th w modl s sgfcat wth G.839 ad th p-valu.6. Th w paramtr stmats ar also gv, wth th Wald p-valu dcatg that th varabl prors s dd sgfcat to th modl. Bcaus th varabl prors has b trd, SAS rchcks th sgfcac of all trd paramtrs, ths cas ol th varabl prors, to mak sur that all of th cludd varabls ar stll sgfcat to th w modl. Nt, th lklhood of th sv modls cotag th trcpt, prors, ad o of th rmag varabls s amd. Aothr varabl s trd basd upo th calculato of th w Scor ch-squar statstcs. 3

24 Dspla : Aalss of Prors ad th Effcts ot th Modl Aalss of Effcts Modl Wald Effct DF Ch-Squar Pr > ChSq prors Aalss of Effcts Not th Modl Scor Effct DF Ch-Squar Pr > ChSq ag bck IV IV rac tratmt st B th formato gv Dspla, IV s ow trd to th modl, havg th largst Scor ch-squar statstc. Th rsultg formato rgardg lklhood valus, th lklhood rato tst, ad th paramtr stmats for th w modl s show Dspla. Dspla : Stp --IV s Etrd to th Modl Modl Covrgc Status Covrgc crtro GCONVE-8 satsfd. Modl Ft Statstcs Itrcpt Itrcpt ad Crtro Ol Covarats AIC SC og Tstg Global Null pothss: BETA Tst Ch-Squar DF Pr > ChSq klhood Rato 7.5. Scor Wald Aalss of Mamum klhood Estmats Stadard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Itrcpt <. IV prors Ths procss cotus, at ach stp addg aothr varabl ad th usg th dvac tst to mak sur that o varabls should b rmovd from th w modl. Oc SAS has fshd th stpws procdur, mag that o addtoal sgfcat varabls rma, a fal modl s outputtd. Th fal modl for our data st s show Dspla. Dspla : Th Stpws Fal Modl Effct Numbr Scor Wald Stp Etrd Rmovd DF I Ch-Squar Ch-Squar prors IV ag tratmt IV

25 Summar of Stpws Slcto Stp Pr > ChSq Varabl abl.8 Pror Drug Tratmts.4 Idcator for Rct IV Drug Us 3. Ag at Erollmt 4.96 Tratmt: Short, og 5.75 Idcator for Prvous IV Drug Us I ths ampl, th fv varabls foud to b sgfcat to th modl ar prors, IV, ag, tratmt, ad IV. ookg at th paramtr stmats ad p-valus gv for th modl cotag ths fv varabls Dspla 3, a quato ca b stmatd for th bst modl as dtrmd b th stpws procdur. Dspla 3: Stpws Fal Paramtr Estmats Aalss of Mamum klhood Estmats Stadard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Itrcpt <. ag IV IV prors tratmt Thus, h Aga, f othr varabls ar dmd practcall mportat to th modl, ths could b cludd as wll. Aothr popular procdur for fttg ths logstc modls s th bst subst procdur. Thr ar svral dffrt crtro usd to dtrm bst for ths computatos. SAS uss th Scor ch-squar statstc, whch was dscussd a prvous scto. I gral, whchvr modl has th hghst Scor statstc s cosdrd th bst modl. owvr, bcaus a dgr of frdom s lost for vr addtoal varabl addd to a modl, th bst modl would ot ol hav a rlatvl hgh Scor ch-squar statstc, but also a small umbr of prdctor varabls. Callg up th bst substs procdur SAS, th two bst modls for vr lvl of q s show th followg output, wth q rprstg th umbr of prdctor varabls: 5

26 Dspla 4: Bst Subst b th Scor Crtro Numbr of Scor Varabls Ch-Squar Varabls Icludd Modl prors IV IV prors prors tratmt 3.75 ag IV prors ag IV IV ag IV IV prors ag IV prors tratmt ag IV IV prors tratmt ag IV prors rac tratmt ag IV IV prors rac tratmt ag IV IV prors tratmt st ag IV IV prors rac tratmt st ag bck IV IV prors rac tratmt ag bck IV IV prors rac tratmt st ookg at th abov output, w s that th modl wth all ght varabls provds th hghst Scor statstc, as would b pctd. owvr, thr ar svral othr modls wth fwr prdctor varabls that hav Scor statstcs almost as hgh as ths modl. ookg at th substs for th modls wth fv prdctor varabls, w s that th frst modl has a Scor statstc of 3.565, ol slghtl smallr tha th mamum valu. I addto, t ol has fv varabls. Thrfor ths modl, whch s th sam modl slctd b our stpws mthod, would b a good choc for th bst subst of prdctor varabls. I ths cas, th modl dcso basd upo th bst subst mthod s quvalt to th modl dcso th stpws procdur, but t should b otd that ths procdurs wll ot alwas produc th sam rsults. Adquac of th Modl: B mas of both mthods, th strogst modl wth whch to ft th data s th followg logt modl: h Aftr fttg ths modl, a fw dagostc masurs ca b usd to look at th ft of th modl. Th stmatd modl s as follows: 6

27 Dspla 5: Fal Modl Stadard Wald Paramtr DF Estmat Error Ch-Squar Pr > ChSq Itrcpt <. ag IV IV prors tratmt Thrfor, w aga fd that Odds Rato Estmats Pot 95% Wald Effct Estmat Cofdc mts ag IV IV prors tratmt h B lookg at th odds rato stmats Dspla 5, w s that a addtoal ar ag crass th probablt of rmag drug fr b 5.4%. Rct IV drug us dcrass that probablt b 55.3% whl prvous IV drug us dcrass that probablt b 46.4%. W also ca ot that for vr addtoal pror drug tratmt to whch th subjct was posd, th probablt of rmag drug fr falls b 6.%. Fall, udrgog th log tratmt procss as opposd to th short tratmt procss crass th chacs of rmag drug fr b 57%. A graphc mag of ths ft s dpctd blow: Fgur : Fal ogstc Ft. 7

28 O wa of assssg th ft of ths modl s b mas of th osmr-mshow Goodss of Ft tst. I ordr to prform ths tst, th data must b dvdd to g groups, whr t s ormall rcommdd that g s qual to t. For all of th goodss-of-ft tsts, th hpothss ar: A : E{ Y} : E{ Y} Th assocatd tst statstc s: Cˆ, or th modl ft s approprat, or th modl ft s approprat. g [ ] k k k k k k [ k ] whr Ĉ s a ch-squar radom varabl wth g dgrs of frdom. Thrfor, th ull hpothss wll b rjctd f Cˆ > χ α; g whl th ull hpothss wll ot b rjctd ˆ f C χ α; g. SAS prforms ths goodss of ft tst usg g. Accordg to SAS, Dspla 6: osmr ad mshow Goodss of Ft Tst drugfr drugfr Group Total Obsrvd Epctd Obsrvd Epctd osmr ad mshow Goodss-of-Ft Tst Ch-Squar DF Pr > ChSq Ra, pp

29 Wth Ĉ 3.3 ad a p-valu of.935, th ull hpothss wll ot b rjctd, rsultg th cocluso that, accordg to th osmr ad mshow Goodss-of-Ft tst, th modl dos fact provd a good ft to th data. A smlar goodss-of-ft tst ca b prformd o th sam st of hpothss as abov, whr ths cas th tst would b basd upo th modl dvacs. Th tst statstc hr would b th dvac of th modl that has b ft, as dfd abov, or whr D l l ˆ ˆ b b... bp p b b... b ˆ. p p Th ull hpothss should b rjctd f DEV > χ α; p ad th ull hpothss should ot b rjctd f DEV χ α; p. 3 Gv th drug data, th dvac s lstd as whl χ.5; Modl Ft Statstcs Itrcpt Itrcpt ad Crtro Ol Covarats AIC SC og Bcaus th calculatd dvac s lss tha th ch-squard crtcal valu, oc aga th ull hpothss wll ot b rjctd, furthr supportg th adquac of th ft of ths modl to th data. Cocluso 3 Ntr, p

30 I th modlg of a bar rspos varabl, th logstc rgrsso tchqu srvs as a k lmt to a aalss. B th vr atur of th rspos varabl, th rsults of th fttd modl ca asl b trprtd trms of th probablt that th vt of trst occurs. Th modl fttg procdur of mamum lklhood stmato s farl computatoall tsv, but wth so ma statstcal programs avalabl, th rsults ca b asl computd. Ths tchqu s vr mportat for statstcal aalss, partcularl aras such as halth scc whr utl th dvlopmt of logstc rgrsso, thr was o wa of accuratl fttg a modl to th data. Wth th currt focus mdc lg toward fdg tratmts ad v curs for ma of th halth ssus plagug our soct, such as cacr, AIDS, ad v th commo cold, logstc rgrsso wll srv as a good tool for dtrmg whthr or ot ths proposd tratmts ar ffctv. Th growg dsr of th gral huma populato to ga mor kowldg mdcal ad pschologcal aras, as wll as ma othr aras, wll caus th cotud us of ths rgrsso tchqu. 3

31 SAS Cod Usd for Aalss OPTIONS S75; FIENAME data ':\Comps\drugData.dat'; DATA drugus; INFIE data; INPUT ID ag bck IV prors rac tratmt st drugfr; obsum_n_; ABE IDIdtfcato Cod agag at Erollmt bckbck Dprsso Scor IVIV Drug Us stor prorspror Drug Tratmts rac'rac: Wht, Othr' tratmt'tratmt: Short, og' st'st: A, B' drugfr'rmad Drug Fr, Othrws' ; DATA drugus; SET drugus; IF IV3 TEN IV; ESE IV; IF IV TEN IV; ESE IV; ABE IVIdcator for Rct IV Drug Us IVIdcator for Prvous IV Drug Us ; /*Smpl ogstc Rgrsso Modl*/ PROC OGISTIC DATAdrugUs dscdg; MODE drugfrtratmt/lklogt; OUTPUT OUTsmpl Phat RESDEVdvRsd; RUN; PROC GPOT DATAsmpl; POT drugfr*tratmt hat*tratmt/ovrla; smbol vcrcl lo cblack; smbol vdot lspl cblack; /*Multpl ogstc Rgrsso Modl*/ PROC OGISTIC DATAdrugUs dscdg; MODE drugfrag rac tratmt/ lklogt Rsquar; OUTPUT OUTmultpl Phat RESDEVdvRsd; RUN; /*Modl Buldg*/ /*Stpws*/ PROC OGISTIC DATAdrugUs dscdg; MODE drugfrag bck IV IV prors rac tratmt st/ lklogt slctostpws dtals sltr. slsta.; OUTPUT OUTstpws Phat; RUN; /*Modl Buldg*/ /*Bst Subst*/ PROC OGISTIC DATAdrugUs dscdg; MODE drugfrag bck IV IV prors rac tratmt st/ slctoscor dtals; OUTPUT OUTbst Phat; RUN; /*Modl Buldg*/ /*mtd Bst Subst*/ PROC OGISTIC DATAdrugUs dscdg; MODE drugfrag bck IV IV prors rac tratmt st/ slctoscor bst dtals; OUTPUT OUTtopbst Phat; RUN; /*Tmporar Modl*/ PROC OGISTIC DATAdrugUs dscdg; MODE drugfrag IV IV prors tratmt/rsquar Ifluc ackft Iplots; OUTPUT OUTtmp Phat RESDEVdvRsd RESCIchRsd hat DIFDEVdvChg DIFCISQchChg DFBETAS_A_; RUN; PROC GPOT DATAtmp; POT drugfr*obsum hat*obsum/ovrla; smbol vcrcl o cblack; smbol vdot o cblack; RUN; 3

32 Sourcs Chattrj, Samprt, Al S. ad, ad Brtram Prc, Rgrsso Aalss b Eampl Nw York: Joh Wl & Sos, ogg, Robrt V., ad All T. Crag, Itroducto to Mathmatcal Statstcs odo: Th MacMlla Compa, 97 osmr, Davd W., ad Stal mshow, Appld ogstc Rgrsso Nw York: Joh Wl & Sos, 989 osmr, Davd W., ad Stal mshow, Appld ogstc Rgrsso Nw York: Joh Wl & Sos, 989 Mard, Scott, Appld ogstc Rgrsso Aalss Thousad Oaks, Calfora: Sag Publcatos, 995 Ntr, Joh, Wllam Wassrma, Chrstophr J. Nachtshm, ad Mchal. Kutr, Appld ar Rgrsso Modls, 3 rd d. Chcago: McGraw-ll, 996 Ra, Thomas P., Modr Rgrsso Mthods Nw York: Joh Wl & Sos, Ic., 997 SAS, STAT Usr s Gud, Vrso 8, Volum Car, NC: SAS Publshg, 999. SAS vrso 8. 3