Cox Regression. Chapter 565. Introduction. The Cox Regression Model. Further Reading

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1 NCSS Sascal Sofware Chaer 565 Inroducon Ths rocedure erforms Cox (rooronal hazards) regresson analyss, whch models he relaonsh beween a se of one or more covaraes and he hazard rae. Covaraes may be dscree or connuous. Cox s rooronal hazards regresson model s solved usng he mehod of margnal lkelhood oulned n Kalbflesch (98). Ths roune can be used o sudy he mac of varous facors on survval. You may be neresed n he mac of de, age, amoun of exercse, and amoun of slee on he survval me afer an ndvdual has been dagnosed wh a ceran dsease such as cancer. Under normal condons, he obvous sascal ool o sudy he relaonsh beween a resonse varable (survval me) and several exlanaory varables would be mulle regresson. Unforunaely, because of he secal naure of survval daa, mulle regresson s no arorae. Survval daa usually conan censored daa and he dsrbuon of survval mes s ofen hghly skewed. These wo roblems nvaldae he use of mulle regresson. Many alernave regresson mehods have been suggesed. The mos oular mehod s he rooronal hazard regresson mehod develoed by Cox (972). Anoher mehod, Webull regresson, s avalable n NCSS n he Dsrbuon Regresson rocedure. Furher Readng Several books rovde n deh coverage of Cox regresson. These books assume a famlary wh basc sascal heory, esecally wh regresson analyss. Colle (994) rovdes a comrehensve nroducon o he subjec. Hosmer and Lemeshow (999) s almos comleely devoed o hs subjec. Therneau and Grambsch (2) rovde a comlee and u-o-dae dscusson of hs subjec. We found her dscusson of resdual analyss very useful. Klen and Moeschberger (997) rovdes a very readable accoun of survval analyss n general and ncludes a lucd accoun of Cox regresson. The Model Survval analyss refers o he analyss of elased me. The resonse varable s he me beween a me orgn and an end on. The end on s eher he occurrence of he even of neres, referred o as a deah or falure, or he end of he subjec s arcaon n he sudy. These elased mes have wo roeres ha nvaldae sandard sascal echnques, such as -ess, analyss of varance, and mulle regresson. Frs of all, he me values are ofen osvely skewed. Sandard sascal echnques requre ha he daa be normally dsrbued. Alhough hs skewness could be correced wh a ransformaon, s easer o ado a more realsc daa dsrbuon. The second roblem wh survval daa s ha ar of he daa are censored. An observaon s censored when he end on has no been reached when he subjec s removed from sudy. Ths may be because he sudy ended before he subjec s resonse occurred, or because he subjec whdrew from acve arcaon. Ths may be because he subjec ded for anoher reason, because he subjec moved, or because he subjec qu followng he sudy roocol. All ha s known s ha he resonse of neres dd no occur whle he subjec was beng suded. When analyzng survval daa, wo funcons are of fundamenal neres he survvor funcon and he hazard funcon. Le T be he survval me. Tha s, T s he elased me from he begnnng on, such as dagnoss of cancer, and deah due o ha dsease. The values of T can be hough of as havng a robably dsrbuon NCSS, LLC. All Rghs Reserved.

2 NCSS Sascal Sofware Suose he robably densy funcon of he random varable T s gven by f ( T ) funcon of T s hen gven by F ( T ) Pr( < T ) The survvor funcon, S ( T ), s he robably ha an ndvdual survves as T. Ths leads o S T f ( ) d ( T ) Pr( T ) F( T ). The robably dsrbuon The hazard funcon s he robably ha a subjec exerences he even of neres (deah, relase, ec.) durng a small me nerval gven ha he ndvdual has survved u o he begnnng of ha nerval. The mahemacal exresson for he hazard funcon s The cumulave hazard funcon ( T ) h ( T ) lm T lm T f S ( T ) ( T ) Pr F ( T < ( T + T ) T ) ( T + T ) F( T ) H s he sum of he ndvdual hazard raes from me zero o me T. The formula for he cumulave hazard funcon s H T T T ( T ) h( u)du Thus, he hazard funcon s he dervave, or sloe, of he cumulave hazard funcon. The cumulave hazard funcon s relaed o he cumulave survval funcon by he exresson or H S H ( T ) ( T ) e ( T ) ln( S( T )) We see ha he dsrbuon funcon, he hazard funcon, and he survval funcon are mahemacally relaed. As a maer of convenence and raccaly, he hazard funcon s used n he basc regresson model. Cox (972) exressed he relaonsh beween he hazard rae and a se of covaraes usng he model or ln [ h( T )] ln[ h ( T )] + x β x β h ( ) ( ) T h T e where x, x 2,, x are covaraes, β, β2,, β are regresson coeffcens o be esmaed, T s he elased me, and h ( T ) s he baselne hazard rae when all covaraes are equal o zero. Thus he lnear form of he regresson model s h ln h ( T ) ( ) T x β NCSS, LLC. All Rghs Reserved.

3 NCSS Sascal Sofware Takng he exonenal of boh sdes of he above equaon, we see ha hs s he rao beween he acual hazard rae and he baselne hazard rae, somemes called he relave rsk. Ths can be rearranged o gve he model h h ( T ) ( T ) ex xβ xβ x2β2 e e e The regresson coeffcens can hus be nerreed as he relave rsk when he value of he covarae s ncreased by one un. Noe ha unlke mos regresson models, hs model does no nclude an nerce erm. Ths s because f an h T nerce erm were ncluded, would become ar of ( ). Also noe ha he above model does no nclude T on he rgh-hand sde. Tha s, he relave rsk s consan for all me values. Ths s why he mehod s called rooronal hazards. An neresng arbue of hs model s ha you only need o use he ranks of he falure mes o esmae he regresson coeffcens. The acual falure mes are no used exce o generae he ranks. Thus, you wll acheve he same regresson coeffcen esmaes regardless of wheher you ener he me values n days, monhs, or years. x β Cumulave Hazard Under he rooronal hazards regresson model, he cumulave hazard s H T ( T, X ) h( u, X ) T h du x β ( u) e du xβ T e h H ( ) T e ( u) xβ Noe ha he survval me T s resen n H ( T ), bu no n e x. Hence, he cumulave hazard u o me T s reresened n hs model by a baselne cumulave hazard H ( T ) whch s adjused by he covaraes by mullyng β. by he facor e x Cumulave Survval Under he rooronal hazards regresson model, he cumulave survval s S ( T, X ) ex( H ( T, X )) ex H H ( T ) e [ e ] S ( ) e T xβ β ( T ) xβ e du x β NCSS, LLC. All Rghs Reserved.

4 NCSS Sascal Sofware Noe ha he survval me T s resen n ( T ) β. S, bu no n e x A Noe On Usng e The dscusson ha follows uses he erms ex(x) and e x. These erms are dencal. Tha s ex ( x) e x ( ) x The decson as o whch form o use deends on he conex. The referred form s e x. Bu ofen, he exresson used for x becomes so small ha canno be rned. In hese suaons, he ex(x) form wll be used. One oher on needs o be made whle we are on hs subjec. Peole ofen wonder why we use he number e. Afer all, e s an unfamlar number ha canno be exressed exacly. Why no use a more common number lke 2, 3, or? The answer s ha does maer because he choce of he base s arbrary n ha you can easly swch from one base o anoher. Tha s, s easy o fnd consans a, b, and c so ha e a b c 2 3 In fac, a s /ln(2).4427, b s /ln(3).92, and c s /ln() Usng hese consans, s easy o swch from one base o anoher. For examle, suose a calculae only comues x and we need he value of e 3. Ths can be comued as follows. ( ) e (. ) The on s, s smle o change from base e o base 3 o base. The number e s used for mahemacal convenence. Maxmum Lkelhood Esmaon Le,, M ndex he M unque falure mes T, T 2,..., T M. Noe ha M does no nclude dulcae mes or censored observaons. The se of all falures (deahs) ha occur a me T s referred o as D. Le c and d,, m ndex he members of D. The se of all ndvduals ha are a rsk mmedaely before me T s referred o as R. Ths se, ofen called he rsk se, ncludes all ndvduals ha fal a me T as well as hose ha are censored or fal a a me laer han T. Le r,, n ndex he members of R. Le X refer o a se of covaraes. These covaraes are ndexed by he subscrs, j, or k. The values of he covaraes a a arcular falure me T d are wren x, x,, x or x d n general. The regresson coeffcens o be esmaed are β, β,, β. d 2d d 2 The Log Lkelhood When here are no es among he falure mes, he log lkelhood s gven by Kalbflesch and Prence (98) as LL ( β ) M r R M x β ln x β ln ( G ) R ex x r β NCSS, LLC. All Rghs Reserved.

5 NCSS Sascal Sofware where G R ex x r β r R The followng noaon for he frs-order and second-order aral dervaves wll be useful n he dervaons n hs secon. A H jkr jr GR β j x ex jr r R GR β j βk H jr β 2 k x x ex x rβ jr kr r R x rβ The maxmum lkelhood soluon s found by he Newon-Rahson mehod. Ths mehod requres he frs and second order aral dervaves. The frs order aral dervaves are U j ( β ) LL β j M H jr x j GR The second order aral dervaves, whch are he nformaon marx, are I jk M G R A jkr H jr H G When here are falure me es (noe ha censor es are no a roblem), he exac lkelhood s very cumbersome. NCSS allows you o selec eher he aroxmaon roosed by Breslow (974) or he aroxmaon gven by Efron (977). Breslow s aroxmaon was used by he frs Cox regresson rograms, bu Efron s aroxmaon rovdes resuls ha are usually closer o he resuls gven by he exac algorhm and s now he referred aroxmaon (see for examle Homer and Lemeshow (999). We have ncluded Breslow s mehod because of s oulary. For examle, Breslow s mehod s he defaul mehod used n SAS. Breslow s Aroxmaon o he Log Lkelhood The log lkelhood of Breslow s aroxmaon s gven by Kalbflesch and Prence (98) as LL ( β ) M d D ( G ) R d d D r R M x d β m x β m ln ln R kr ex x r β NCSS, LLC. All Rghs Reserved.

6 NCSS Sascal Sofware NCSS, LLC. All Rghs Reserved. where R r G R x r ex β The maxmum lkelhood soluon s found by he Newon-Rahson mehod. Ths mehod requres he frs-order and second-order aral dervaves. The frs order aral dervaves are ( ) M R jr D d jd j j G H m x LL U β β The negave of he second-order aral dervaves, whch form he nformaon marx, are M R kr jr jkr R jk G H H A G m I Efron s Aroxmaon o he Log Lkelhood The log lkelhood of Efron s aroxmaon s gven by Kalbflesch and Prence (98) as ( ) M D d D d D R d M D d D d D c c R r r d G m d G x x m d x x LL ln ex ex ln β β β β β The maxmum lkelhood soluon s found by he Newon-Rahson mehod. Ths mehod requres he frs and second order aral dervaves. The frs aral dervaves are ( ) M m d D R jd jr M D d jd M D d D R jd jr jd j j G m d G H m d H x G m d G H m d H x LL U β β

7 NCSS Sascal Sofware NCSS, LLC. All Rghs Reserved. The second aral dervaves rovde he nformaon marx whch esmaes he covarance marx of he esmaed regresson coeffcens. The negave of he second aral dervaves are ( ) M m d D R kd kr jd jr jkd jkr D R k j jk G m d G H m d H H m d H A m d A G m d G LL I 2 2 β β β Esmaon of he Survval Funcon Once he maxmum lkelhood esmaes have been obaned, may be of neres o esmae he survval robably of a new or exsng ndvdual wh secfc covarae sengs a a arcular on n me. The mehods roosed by Kalbflesch and Prence (98) are used o esmae he survval robables. Cumulave Survval Ths esmaes he cumulave survval of an ndvdual wh a se of covaraes all equal o zero. The survval for an ndvdual wh covarae values of X s ( ) ( ) ( ) ( ) ( ) [ ] x T S x X T H X T H X T S ex ex ex ex β β The esmae of he baselne survval funcon ( ) T S s calculaed from he cumulaed hazard funcon usng ( ) T T T S α where ( ) ( ) ( ) ( ) ( ) ( ) x T S T S T S T S T S T S θ β α ex where r x r ex β θ

8 NCSS Sascal Sofware The value of α, he condonal baselne survval robably a me T, s he soluon o he condonal lkelhood equaon d D α θ d θd When here are no es a a arcular me on, D conans one ndvdual and he above equaon can be solved drecly, resulng n he soluon ˆ α r R ˆ θ ˆ θr R When here are es, he equaon mus be solved eravely. The sarng value of hs erave rocess s r θ r ˆ θ m ˆ α ex ˆ θr r R Baselne Hazard Rae Hosmer and Lemeshow (999) esmae he baselne hazard rae h ( T ) as follows h ( T ) α They menon ha hs esmaor wll ycally be oo unsable o be of much use. To overcome hs, you mgh smooh hese quanes usng lowess funcon of he Scaer Plo rogram. Cumulave Hazard H derved from relaonsh beween he cumulave hazard and he cumulave survval. The esmaed baselne survval s An esmae of he cumulave hazard funcon ( T ) ( ) ( T ) Sˆ ( T ) ˆ ln H Ths leads o he esmaed cumulave hazard funcon s Cumulave Survval Hˆ ( ) ( T ) ex x ˆ β ln Sˆ ( T ) The esmae of he cumulave survval of an ndvdual wh a se of covaraes values of X s ( ) ( ) x Sˆ ex T X S T ˆ ˆ β NCSS, LLC. All Rghs Reserved.

9 NCSS Sascal Sofware Sascal Tess and Confdence Inervals Inferences abou one or more regresson coeffcens are all of neres. These nference rocedures can be reaed by consderng hyohess ess and/or confdence nervals. The nference rocedures n Cox regresson rely on large samle szes for accuracy. Two ess are avalable for esng he sgnfcance of one or more ndeenden varables n a regresson: he lkelhood rao es and he Wald es. Smulaon sudes usually show ha he lkelhood rao es erforms beer han he Wald es. However, he Wald es s sll used o es he sgnfcance of ndvdual regresson coeffcens because of s ease of calculaon. These wo esng rocedures wll be descrbed nex. Lkelhood Rao and Devance The Lkelhood Rao es sasc s -2 mes he dfference beween he log lkelhoods of wo models, one of whch s a subse of he oher. The dsrbuon of he LR sasc s closely aroxmaed by he ch-square dsrbuon for large samle szes. The degrees of freedom (DF) of he aroxmang ch-square dsrbuon s equal o he dfference n he number of regresson coeffcens n he wo models. The es s named as a rao raher han a dfference snce he dfference beween wo log lkelhoods s equal o he log of he rao of he wo lkelhoods. Tha s, f L full s he log lkelhood of he full model and L subse s he log lkelhood of a subse of he full model, he lkelhood rao s defned as LR 2 subse l 2 ln l [ L L ] Noe ha he -2 adjuss LR so he ch-square dsrbuon can be used o aroxmae s dsrbuon. The lkelhood rao es s he es of choce n Cox regresson. Varous smulaon sudes have shown ha s more accurae han he Wald es n suaons wh small o moderae samle szes. In large samles, erforms abou he same. Unforunaely, he lkelhood rao es requres more calculaons han he Wald es, snce requres he fng of wo maxmum-lkelhood models. Devance When he full model n he lkelhood rao es sasc s he sauraed model, LR s referred o as he devance. A sauraed model s one whch ncludes all ossble erms (ncludng neracons) so ha he redced values from he model equal he orgnal daa. The formula for he devance s subse full full [ L ] D 2 L Reduced Sauraed The devance n Cox regresson s analogous o he resdual sum of squares n mulle regresson. In fac, when he devance s calculaed n mulle regresson, s equal o he sum of he squared resduals. The change n devance, D, due o excludng (or ncludng) one or more varables s used n Cox regresson jus as he aral F es s used n mulle regresson. Many exs use he leer G o reresen D. Insead of usng he F dsrbuon, he dsrbuon of he change n devance s aroxmaed by he ch-square dsrbuon. Noe ha snce he log lkelhood for he sauraed model s common o boh devance values, D can be calculaed whou acually fng he sauraed model. Ths fac becomes very moran durng subse selecon. The formula for D for esng he sgnfcance of he regresson coeffcen(s) assocaed wh he ndeenden varable X s DX Dwhou X Dwh X [ Lwhou X LSauraed] [ Lwh X LSauraed] [ Lwhou X Lwh X] NCSS, LLC. All Rghs Reserved.

10 NCSS Sascal Sofware Noe ha hs formula looks dencal o he lkelhood rao sasc. Because of he smlary beween he change n devance es and he lkelhood rao es, her names are ofen used nerchangeably. Wald Tes The Wald es wll be famlar o hose who use mulle regresson. In mulle regresson, he common -es for esng he sgnfcance of a arcular regresson coeffcen s a Wald es. In Cox regresson, he Wald es s calculaed n he same manner. The formula for he Wald sasc s where s b j z j b s j b j s an esmae of he sandard error of b j rovded by he square roo of he corresondng dagonal ˆ V β I. elemen of he covarance marx, ( ) Wh large samle szes, he dsrbuon of z j s closely aroxmaed by he normal dsrbuon. Wh small and moderae samle szes, he normal aroxmaon s descrbed as adequae. The Wald es s used n NCSS o es he sascal sgnfcance of ndvdual regresson coeffcens. Confdence Inervals Confdence nervals for he regresson coeffcens are based on he Wald sascs. The formula for he lms of a ( α )% wo-sded confdence nerval s ± b j zα / 2 s bj R 2 Hosmer and Lemeshow (999) ndcae ha a he me of he wrng of her book, here s no sngle, easy o nerre measure n Cox regresson ha s analogous o R 2 n mulle regresson. They ndcae ha f such a measure mus be calculaed hey would use ex n ( L L ) R 2 2 where L s he log lkelhood of he model wh no covaraes, n s he number of observaons (censored or no), and L s he log lkelhood of he model ha ncludes he covaraes. Subse Selecon Subse selecon refers o he ask of fndng a small subse of he avalable regressor varables ha does a good job of redcng he deenden varable. Because Cox regresson mus be solved eravely, he ask of fndng he bes subse can be me consumng. Hence, echnques whch look a all ossble combnaons of he regressor varables are no feasble. Insead, algorhms ha add or remove a varable a each se mus be used. Two such searchng algorhms are avalable n hs module: forward selecon and forward selecon wh swchng. Before dscussng he deals of hese wo algorhms, s moran o commen on a coule of ssues ha can come u. The frs ssue s wha o do abou he bnary varables ha are generaed for a caegorcal ndeenden varable. If such a varable has sx caegores, fve bnary varables are generaed. You can see ha wh wo or hree caegorcal varables, a large number of bnary varables may resul, whch grealy ncreases he oal number of varables ha mus be searched. To avod hs roblem, he algorhms used here search on model erms raher han on he ndvdual varables. Thus, he whole se of bnary varables assocaed wh a gven erm are consdered ogeher for ncluson n, or deleon from, he model. I s all or none. Because of he me 565- NCSS, LLC. All Rghs Reserved.

11 NCSS Sascal Sofware consumng naure of he algorhm, hs s he only feasble way o deal wh caegorcal varables. If you wan he subse algorhm o deal wh hem ndvdually, you can generae he se of bnary varables manually and desgnae hem as Numerc Varables. Herarchcal Models A second ssue s wha o do wh neracons. Usually, an neracon s no enered n he model unless he ndvdual erms ha make u ha neracon are also n he model. For examle, he neracon erm A*B*C s no ncluded unless he erms A, B, C, A*B, A*C, and B*C are already n he model. Such models are sad o be herarchcal. You have he oon durng he search o force he algorhm o only consder herarchcal models durng s search. Thus, f C s no n he model, neracons nvolvng C are no even consdered. Even hough he oon for non-herarchcal models s avalable, we recommend ha you only consder herarchcal models. Forward Selecon The mehod of forward selecon roceeds as follows.. Begn wh no erms n he model. 2. Fnd he erm ha, when added o he model, acheves he larges value of R-squared. Ener hs erm no he model. 3. Connue addng erms unl a rese lm on he maxmum number of erms n he model s reached. Ths mehod s comaravely fas, bu does no guaranee ha he bes model s found exce for he frs se when fnds he bes sngle erm. You mgh use when you have a large number of observaons so ha oher, more me consumng mehods, are no feasble, or when you have far oo many ossble regressor varables and you wan o reduce he number of erms n he selecon ool. Forward Selecon wh Swchng Ths mehod s smlar o he mehod of Forward Selecon dscussed above. However, a each se when a erm s added, all erms n he model are swched one a a me wh all canddae erms no n he model o deermne f hey ncrease he value of R-squared. If a swch can be found, s made and he canddae erms are agan searched o deermne f anoher swch can be made. When he search for ossble swches does no yeld a canddae, he subse sze s ncreased by one and a new search s begun. The algorhm s ermnaed when a arge subse sze s reached or all erms are ncluded n he model. Dscusson These algorhms usually requre wo runs. In he frs run, you se he maxmum subse sze o a large value such as. By sudyng he Subse Selecon reors from hs run, you can quckly deermne he omum number of erms. You rese he maxmum subse sze o hs number and make he second run. Ths wo-se rocedure works beer han relyng on some F-o-ener and F-o-remove ess whose roeres are no well undersood o begn wh. Resduals The followng resenaon summarzes he dscusson on resduals found n Klen and Moeschberger (997) and Hosmer and Lemeshow (999). For a more horough reamen of hs oc, we refer you o eher of hese books. In mos sengs n whch resduals are suded, he deenden varable s redced usng a model based on he ndeenden varables. In hese cases, he resdual s smly he dfference beween he acual value and he redced value of he deenden varable. Unforunaely, n Cox regresson here s no obvous analog hs acual mnus redced. Realzng hs, sascans have looked a how resduals are used and hen, based on hose uses, 565- NCSS, LLC. All Rghs Reserved.

12 NCSS Sascal Sofware develoed quanes ha mee hose needs. They call hese quanes resduals because hey are used n lace of resduals. However, you mus remember ha hey are no equvalen o usual he resduals ha you see n mulle regresson, for examle. In he dscusson ha follows, he formulas wll be smlfed f we use he subsuon θ ex r x r β Cox-Snell Resduals The Cox-Snell resduals were used o assess he goodness-of-f of he Cox regresson. The Cox-Snell resduals are defned as r H B ( T ) θ where here b s are he esmaed regresson coeffcens and H ( ) baselne hazard funcon. Ths value s defned as follows H B T ( T ) T T s Breslow s esmae of he cumulave m θ j j R T The Cox-Snell resduals were he frs o be roosed n he leraure. They have snce been relaced by oher yes of resduals and are now only of hsorcal neres. See, for examle, he dscusson of Marubn and Valsecch (996) who sae ha he use of hese resduals on dsrbuonal grounds should be avoded. Marngale Resduals Marngale resduals can no be used o assess goodness-of-f as are he usual resduals n mulle regresson. The bes model need no have he smalles sum of squared marngale resduals. Marngale resduals follow he un exonenal dsrbuon. Some auhors suggesed analyzng hese resduals o deermne how close hey are o he exonenal dsrbuon, hong ha a lack of exonealy ndcaed a lack of f. Unforunaely, jus he oose s he case snce n a model wh no useful covaraes, hese resduals are exacly exonenal n dsrbuon. Anoher dagnosc ool for n regular mulle regresson s a lo of he resduals versus he fed values. Here agan, he marngale resduals canno be used for hs urose snce hey are negavely correlaed wh he fed values. So of wha use are marngale resduals? They have wo man uses. Frs, hey can be used o fnd oulers ndvduals who are oorly f by he model. Second, marngale resduals can be used o deermne he funconal form of each of he covaraes n he model. Fndng Oulers The marngale resduals are defned as M c r where c s one f here s a falure a me T and zero oherwse. The marngale resdual measures he dfference beween wheher an ndvdual exerences he even of neres and he execed number of evens based on he model. The maxmum value of he resdual s one and he mnmum ossble value s negave nfny. Thus, he resdual s hghly skewed. A large negave marngale resdual ndcaes a hgh rsk ndvdual who sll had a long survval me NCSS, LLC. All Rghs Reserved.

13 NCSS Sascal Sofware Fndng he Funcon Form of Covaraes Marngale resduals can be used o deermne he funconal form of a covarae. To do hs, you generae he Marngale resduals from a model whou he covaraes. Nex, you lo hese resduals agans he value of he covarae. For large daases, hs may be a me consumng rocess. Therneau and Grambsch (2) sugges ha he marngale resduals from a model wh no covaraes be loed agans each of he covaraes. These los wll reveal he arorae funconal form of he covaraes n he model so long as he covaraes are no hghly correlaed among hemselves. Devance Resduals Devance resduals are used o search for oulers. The devance resduals are defned as ( M ) 2 [ M + c ( c M )] DEV sgn ln or zero when M s zero. These resduals are loed agans he rsk scores gven by ex x b When here s slgh o moderae censorng, large absolue values n hese resduals on o oenal oulers. When here s heavy censorng, here wll be a large number of resduals near zero. However, large absolue values wll sll ndcae oulers. Schoenfeld s Resduals A se of Schoenfeld resduals s defned for each noncensored ndvdual. The resdual s mssng when he ndvdual s censored. The Schoenfeld resduals are defned as follows where r c x c x w r r R r R x r R r R x rθr θr R xrwr Thus hs resdual s he dfference beween he acual value of he covarae and a weghed average where he weghs are deermned from he rsk scores. These resduals are used o esmae he nfluence of an observaon on each of he regresson coeffcens. Plos of hese quanes agans he row number or agans he corresondng covarae values are used o sudy hese resduals. r r θ θ r r NCSS, LLC. All Rghs Reserved.

14 NCSS Sascal Sofware Scaled Schoenfeld s Resduals Hosmer and Lemeshow (999) and Therneau and Grambsch (2) sugges ha scalng he Schoenfeld resduals by an esmae of her varance gves quanes wh greaer dagnosc ably. Hosmer and Lemeshow (999) use he covarance marx of he regresson coeffcens o erform he scalng. The scaled Schoenfeld resduals are defned as follows r * m k where m s he oal number of deahs n he daase and V s he esmaed covarance marx of he regresson coeffcens. These resduals are loed agans me o valdae he rooronal hazards assumon. If he rooronal hazards assumon holds, he resduals wll fall randomly around a horzonal lne cenered a zero. If he rooronal hazards assumon does no hold, a rend wll be aaren n he lo. V r k Daa Srucure Survval daa ses requre u o hree comonens for he survval me: he endng survval me, he begnnng survval me durng whch he subjec was no observed, and an ndcaor of wheher he observaon was censored or faled. Based on hese hree comonens, varous yes of daa may be analyzed. Rgh censored daa are secfed usng only he endng me varable and he censor varable. Lef runcaed and Inerval daa are enered usng all hree varables. The able below shows survval daa ready for analyss. These daa are from a lung cancer sudy reored n Kalbflesch (98), age 223. These daa are n he LungCancer daase. The varables are TIME CENSOR STATUS MONTHS AGE THERAPY days of survval censor ndcaor erformance saus monhs from dagnoss age n years ror heray LungCancer daase (subse) TIME CENSOR STATUS MONTHS AGE THERAPY NCSS, LLC. All Rghs Reserved.

15 NCSS Sascal Sofware Procedure Oons Ths secon descrbes he oons avalable n hs rocedure. Varables, Model Tab Ths anel les you desgnae whch varables and model are used n he analyss. Varables Tme Varable Ths varable conans he lengh of me ha an ndvdual was observed. Ths may reresen a falure me or a censor me. Wheher he subjec acually ded s secfed by he Censor varable. Snce he values are elased mes, hey mus be osve. Zeroes and negave values are reaed as mssng values. Durng he maxmum lkelhood calculaons, a rsk se s defned for each ndvdual. The rsk se s defned o be hose subjecs who were beng observed a hs subjec s falure and who lved as long or longer. I may ake several rows of daa o secfy a subjec s hsory. Ths varable and he Enry Tme varable defne a erod durng whch he ndvdual was a rsk of falng. If he Enry Tme s no secfed, s value s assumed o be zero. Several yes of daa may be enered. These wll be exlaned nex. Falure Ths ye of daa occurs when a subjec s followed from her enrance no he sudy unl her deah. The falure me s enered n hs varable and he Censor varable s se o he faled code, whch s ofen a one. The Enry Tme Varable s no necessary. If an Enry Tme varable s used, s value should be zero for hs ye of observaon. Inerval Falure Ths ye of daa occurs when a subjec s known o have ded durng a ceran nerval. The subjec may, or may no, have been observed durng oher nervals. If hey were, hey are reaed as Inerval Censored daa. An ndvdual may requre several rows on he daabase o record her comlee follow-u hsory. For examle, suose he condon of he subjecs s only avalable a he end of each monh. If a subjec fals durng he ffh monh, wo rows of daa would be requred. One row, reresenng he falure, would have a Tme of 5. and an Enry Tme of 4.. The Censor varable would conan he falure code. A second row, reresenng he ror erods, would have a Tme of 4. and an Enry Tme of.. The Censor varable would conan he censor code. Censored Ths ye of daa occurs when a subjec has no faled u o he secfed me. For examle, suose ha a subjec eners he sudy and does no de unl afer he sudy ends 2 monhs laer. The subjec s me (365 days) s enered here. The Censor varable conans he censor code. Inerval Censored Ths ye of daa occurs when a subjec s known no o have ded durng a ceran nerval. The subjec may, or may no, have been observed durng oher nervals. An ndvdual may requre several rows on he daabase o record her comlee follow-u hsory NCSS, LLC. All Rghs Reserved.

16 NCSS Sascal Sofware For examle, suose he condon of he subjecs s only avalable a he end of each monh. If a subjec fals durng he ffh monh, wo rows of daa would be requred. One row, reresenng he falure, would have a Tme of 5. and an Enry Tme of 4.. The Censor varable would conan he falure code. A second row, reresenng he ror erods, would have a Tme of 4. and an Enry Tme of.. The Censor varable would conan he censor code. Tes Mehod The basc Cox regresson model assumes ha all falure mes are unque. When es exs among he falure mes, one of wo aroxmaon mehods s used o deal wh he es. When no es are resen, boh of hese mehods resul n he same esmaes. Breslow Ths mehod was suggesed frs and s he defaul n many rograms. However, he Efron mehod has been shown o be more accurae n mos cases. The Breslow mehod s only used when you wan o mach he resuls of some oher (older) Cox regresson ackage. Efron Ths mehod has been shown o be more accurae, bu requres slghly more me o calculae. Ths s he recommended mehod. Enry Tme Varable Ths oonal varable conans he elased me before an ndvdual enered he sudy. Usually, hs value s zero. However, n cases such as lef runcaon and nerval censorng, hs value defnes a me erod before whch he ndvdual was no observed. Negave enry mes are reaed as mssng values. I s ossble for he enry me o be zero. Censor Varable The values n hs varable ndcae wheher he value of he Tme Varable reresens a censored me or a falure me. These values may be ex or numerc. The nerreaon of hese codes s secfed by he Faled and Censored oons o he rgh of hs oon. Only wo values are used, he Falure code and he Censor code. The Unknown Tye oon secfes wha s o be done wh values ha do no mach eher he Falure code or he Censor code. Rows wh mssng values (blanks) n hs varable are omed from he esmaon hase, bu resuls are shown n any reors ha ouu redced values. Falure Ths value denfes hose values of he Censor Varable ha ndcae ha he Tme Varable gves a falure me. The value may be a number or a leer. We sugges he leer F or he number when you are n doub as o wha o use. A faled observaon s one n whch he me unl he even of neres was measured exacly; for examle, he subjec ded of he dsease beng suded. The exac falure me s known. Lef Censorng When he exac falure me s no known, bu nsead only an uer bound on he falure me s known, he me value s sad o have been lef censored. In hs case, he me value s reaed as f were he rue falure me, no jus an uer bound. So lef censored observaons should be coded as faled observaons NCSS, LLC. All Rghs Reserved.

17 NCSS Sascal Sofware Censor Ths value denfes hose values of he Censor Varable ha ndcae ha he ndvdual recorded on hs row was censored. Tha s, he acual falure me occurs someme afer he value of he Tme Varable. We sugges he leer C or he number when you are n doub as o wha o use. A censored observaon s one n whch he me unl he even of neres s no known because he ndvdual whdrew from he sudy, he sudy ended before he ndvdual faled, or for some smlar reason. Noe ha does no maer wheher he censorng was Rgh or Inerval. All you need o ndcae here s ha hey were censored. Oher Censor Ths oon secfes wha he rogram s o assume abou observaons whose censor value s no equal o eher he Falure code or he Censor code. Noe ha observaons wh mssng censor values are always reaed as mssng. Censored Observaons wh unknown censor values are assumed o have been censored. Faled Observaons wh unknown censor values are assumed o have faled. Mssng Observaons wh unknown censor values are assumed o be mssng and hey are removed from he analyss. Numerc X s Secfy he numerc (connuous) ndeenden varables. By numerc, we mean ha he values are numerc and a leas ordnal. Nomnal varables, even when coded wh numbers, should be secfed as Caegorcal Indeenden Varables. Alhough you may secfy bnary (-) varables here, hey are beer analyzed when you secfy hem as Caegorcal Indeenden Varables. If you wan o creae owers and cross-roducs of hese varables, secfy an arorae model n he Cusom Model feld under he Model ab. If you wan o creae hazard values for values of X no n your daabase, add he X values o he boom of he daabase and leave her me and censorng blank. They wll no be used durng esmaon, bu varous hazard and survval sascs wll be generaed for hem and dslayed n he Predced Values reor. Caegorcal X s Secfy caegorcal (nomnal or grou) ndeenden varables n hs box. By caegorcal we mean ha he varable has only a few unque, numerc or ex, values lke, 2, 3 or Yes, No, Maybe. The values are used o denfy caegores. Regresson analyss s only defned for numerc varables. Snce caegorcal varables are nomnal, hey canno be used drecly n regresson. Insead, an nernal se of numerc varables mus be subsued for each caegorcal varable. Suose a caegorcal varable has G caegores. NCSS auomacally generaes he G- nernal, numerc varables for he analyss. The way hese nernal varables are creaed s deermned by he Recodng Scheme and, f needed, he Reference Value. These oons can be enered searaely wh each caegorcal varable, or hey can secfed usng a defaul value (see Defaul Recodng Scheme and Defaul Reference Value below). The synax for secfyng a caegorcal varable s VarName(CTye; RefValue) where VarName s he name of he varable, CTye s he recodng scheme, and RefValue s he reference value, f needed NCSS, LLC. All Rghs Reserved.

18 NCSS Sascal Sofware CTye The recodng scheme s enered as a leer. Possble choces are B, P, R, N, S, L, F, A,, 2, 3, 4, 5, or E. The meanng of each of hese leers s as follows. B for bnary (he grou wh he reference value s sked). Examle: Caegorcal varable Z wh 4 caegores. Caegory D s he reference value. Z B B2 B3 A B C D P for Polynomal of u o 5h order (you canno use hs oon wh caegory varables wh more han 6 caegores. Examle: Caegorcal varable Z wh 4 caegores. Z P P2 P R o comare each wh he reference value (he grou wh he reference value s sked). Examle: Caegorcal varable Z wh 4 caegores. Caegory D s he reference value. Z C C2 C3 A B C D N o comare each wh he nex caegory. Examle: Caegorcal varable Z wh 4 caegores. Z S S2 S S o comare each wh he average of all subsequen values. Examle: Caegorcal varable Z wh 4 caegores. Z S S2 S L o comare each wh he ror caegory. Examle: Caegorcal varable Z wh 4 caegores. Z S S2 S NCSS, LLC. All Rghs Reserved.

19 NCSS Sascal Sofware F o comare each wh he average of all ror caegores. Examle: Caegorcal varable Z wh 4 caegores. Z S S2 S A o comare each wh he average of all caegores (he Reference Value s sked). Examle: Caegorcal varable Z wh 4 caegores. Suose he reference value s 3. Z S S2 S o comare each wh he frs caegory afer sorng. Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D 2 o comare each wh he second caegory afer sorng. Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D 3 o comare each wh he hrd caegory afer sorng. Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D 4 o comare each wh he fourh caegory afer sorng. Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D NCSS, LLC. All Rghs Reserved.

20 NCSS Sascal Sofware 5 o comare each wh he ffh caegory afer sorng. Examle: Caegorcal varable Z wh 5 caegores. Z C C2 C3 C4 A B C D E E o comare each wh he las caegory afer sorng. Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D RefValue A second, oonal argumen s he reference value. The reference value s one of he caegores. The oher caegores are comared o, so s usually a baselne or conrol value. If neher a baselne or conrol value s evden, he reference value s he mos frequen value. For examle, suose you wan o nclude a caegorcal ndeenden varable, Sae, whch has four values: Texas, Calforna, Florda, and NewYork. Suose he recodng scheme s secfed as Comare Each wh Reference Value wh he reference value of Calforna. You would ener Sae(R;Calforna) Defaul Recodng Scheme Selec he defaul ye of numerc varable ha wll be generaed when rocessng caegorcal ndeenden varables. The values n a caegorcal varable are no used drecly n regresson analyss. Insead, a se of numerc varables s auomacally creaed and subsued for hem. Ths oon allows you o secfy wha ye of numerc varable wll be creaed. The oons are oulned n he secons below. The conras ye may also be desgnaed whn arenheses afer he name of each caegorcal ndeenden varable, n whch case he defaul conras ye s gnored. If your model ncludes neracons of caegorcal varables, hs oon should be se o Conras wh Reference or Comare wh All Subsequen' n order o mach GLM resuls for facor effecs. Bnary (he grou wh he reference value s sked). Examle: Caegorcal varable Z wh 4 caegores. Caegory D s he reference value. Z B B2 B3 A B C D Polynomal of u o 5h order (you canno use hs oon wh caegory varables wh more han 6 caegores. Examle: Caegorcal varable Z wh 4 caegores. Z P P2 P NCSS, LLC. All Rghs Reserved.

21 NCSS Sascal Sofware Comare Each wh Reference Value (he grou wh he reference value s sked). Examle: Caegorcal varable Z wh 4 caegores. Caegory D s he reference value. Z C C2 C3 A B C D Comare Each wh Nex. Examle: Caegorcal varable Z wh 4 caegores. Z S S2 S Comare Each wh All Subsequen. Examle: Caegorcal varable Z wh 4 caegores. Z S S2 S Comare Each wh Pror Examle: Caegorcal varable Z wh 4 caegores. Z S S2 S Comare Each wh All Pror Examle: Caegorcal varable Z wh 4 caegores. Z S S2 S Comare Each wh Average Examle: Caegorcal varable Z wh 4 caegores. Suose he reference value s 3. Z S S2 S NCSS, LLC. All Rghs Reserved.

22 NCSS Sascal Sofware Comare Each wh Frs Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D Comare Each wh Second Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D Comare Each wh Thrd Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D Comare Each wh Fourh Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D Comare Each wh Ffh Examle: Caegorcal varable Z wh 5 caegores. Z C C2 C3 C4 A B C D E Comare Each wh Las Examle: Caegorcal varable Z wh 4 caegores. Z C C2 C3 A B C D NCSS, LLC. All Rghs Reserved.

23 NCSS Sascal Sofware Defaul Reference Value Ths oon secfes he defaul reference value o be used when auomacally generang ndcaor varables durng he rocessng of seleced caegorcal ndeenden varables. The reference value s ofen he baselne, and he oher values are comared o. The choces are Frs Value afer Sorng Ffh Value afer Sorng Use he frs (hrough ffh) value n alha-numerc sored order as he reference value. Las Value afer Sorng Use he las value n alha-numerc sored order as he reference value. Frequences Ths s an oonal varable conanng he frequency (observaon coun) for each row. Usually, you would leave hs oon blank and le each row receve he defaul frequency of one. If your daa have already been summarzed, hs oon les you secfy how many acual rows each hyscal row reresens. Regresson Model Terms Ths oon secfes whch erms (erms, owers, cross-roducs, and neracons) are ncluded n he regresson model. For a sragh-forward regresson model, selec -Way. The oons are -Way Ths oon generaes a model n whch each varable s reresened by a sngle model erm. No crossroducs, neracons, or owers are added. Use hs oon when you wan o use he varables you have secfed, bu you do no wan o generae oher erms. Ths s he oon o selec when you wan o analyze he ndeenden varables secfed whou addng any oher erms. For examle, f you have hree ndeenden varables A, B, and C, hs would generae he model: A + B + C U o 2-Way Ths oon secfes ha all ndvdual varables, wo-way neracons, and squares of numerc varables are ncluded n he model. For examle, f you have hree numerc varables A, B, and C, hs would generae he model: A + B + C + A*B + A*C + B*C + A*A + B*B + C*C On he oher hand, f you have hree caegorcal varables A, B, and C, hs would generae he model: A + B + C + A*B + A*C + B*C U o 3-Way All ndvdual varables, wo-way neracons, hree-way neracons, squares of numerc varables, and cubes of numerc varables are ncluded n he model. For examle, f you have hree numerc, ndeenden varables A, B, and C, hs would generae he model: A + B + C + A*B + A*C + B*C + A*B*C + A*A + B*B + C*C + A*A*B + A*A*C + B*B*C +A*C*C + B*C*C NCSS, LLC. All Rghs Reserved.

24 NCSS Sascal Sofware On he oher hand, f you have hree caegorcal varables A, B, and C, hs would generae he model: A + B + C + A*B + A*C + B*C + A*B*C U o 4-Way All ndvdual varables, wo-way neracons, hree-way neracons, and four-way neracons are ncluded n he model. Also ncluded would be squares, cubes, and quarcs of numerc varables and her crossroducs. For examle, f you have four caegorcal varables A, B, C, and D, hs would generae he model: A + B + C + D + A*B + A*C + A*D + B*C + B*D + C*D + A*B*C + A*B*D + A*C*D + B*C*D + A*B*C*D Ineracon Manly used for caegorcal varables. A sauraed model (all erms and her neracons) s generaed. Ths requres a daase wh no mssng caegorcal-varable combnaons (you can have unequal numbers of observaons for each combnaon of he caegorcal varables). No squares, cubes, ec. are generaed. For examle, f you have hree ndeenden varables A, B, and C, hs would generae he model: A + B + C + A*B + A*C + B*C + A*B*C Noe ha he dscusson of he Cusom Model oon dscusses he nerreaon of hs model. Cusom Model The model secfed n he Cusom Model box s used. Cener X s The values of he ndeenden varables may be cenered o mrove he sably of he algorhm. An value s cenered when s mean s subraced from. Cenerng does no change he values of he regresson coeffcens, exce ha he algorhm mgh rovde slghly dfferen resuls because of beer numercal sably. Cenerng does affec he values of he row-wse sascs such as XB, Ex(XB), S, H, and so on because changes he value of X n hese exressons. When he daa are cenered, he devaon from he mean (X-Xbar) s subsued for X n hese exressons. The oons are avalable: Unchecked The daa are no cenered. Checked All varables, boh numerc and bnary, are cenered. Relace Cusom Model wh Prevew Model (buon) When hs buon s ressed, he Cusom Model s cleared and a coy of he Prevew model s sored n he Cusom Model. You can hen ed hs Cusom Model as desred. Maxmum Order of Cusom Terms Ths oon secfes ha maxmum number of varables ha can occur n an neracon (or cross-roduc) erm n a cusom model. For examle, A*B*C s a hrd order neracon erm and f hs oon were se o 2, he A*B*C erm would no be ncluded n he model. Ths oon s arcularly useful when used wh he bar noaon of a cusom model o allow a smle way o remove unwaned hgh-order neracons NCSS, LLC. All Rghs Reserved.

25 NCSS Sascal Sofware Cusom Model Ths oons secfes a cusom model. I s only used when he Terms oon s se o Cusom. A cusom model secfes he erms (sngle varables and neracons) ha are o be ke n he model. Ineracons An neracon exresses he combned relaonsh beween wo or more varables and he deenden varable by creang a new varable ha s he roduc of he varables. The neracon beween wo numerc varables s generaed by mullyng hem. The neracon beween o caegorcal varables s generaed by mullyng each ar of ndcaor varables. The neracon beween a numerc varable and a caegorcal varable s creaed by generang all roducs beween he numerc varable and he ndcaor varables generaed from he caegorcal varable. Synax A model s wren by lsng one or more erms. The erms are searaed by a blank or lus sgn. Terms nclude varables and neracons. Secfy regular varables (man effecs) by enerng he varable names. Secfy neracons by lsng each varable n he neracon searaed by an asersk (*), such as Fru*Nus or A*B*C. You can use he bar ( ) symbol as a shorhand echnque for secfyng many neracons quckly. When several varables are searaed by bars, all of her neracons are generaed. For examle, A B C s nerreed as A + B + C + A*B + A*C + B*C + A*B*C. You can use arenheses. For examle, A*(B+C) s nerreed as A*B + A*C. Some examles wll hel o ndcae how he model synax works: A B A + B + A*B A B A*A B*B A + B + A*B + A*A + B*B Noe ha you should only reea numerc varable. Tha s, A*A s vald for a numerc varable, bu no for a caegorcal varable. A A B B (Max Term Order2) A + B + A*A + A*B + B*B A B C A + B + C + A*B + A*C + B*C + A*B*C (A + B)*(C + D) A*C + A*D + B*C + B*D (A + B) C (A + B) + C + (A + B)*C A + B + C + A*C + B*C Subse Selecon Search Mehod Ths oon secfes he subse selecon algorhm used o reduce he number of ndeenden varables ha used n he regresson model. Noe ha snce he soluon algorhm s erave, he selecon rocess can be very me consumng. The Forward algorhm s much qucker han he Forward wh Swchng algorhm, bu he Forward algorhm does no usually fnd as good of a model. Also noe ha n he case of caegorcal ndeenden varables, he algorhm searches among he orgnal caegorcal varables, no among he generaed ndvdual bnary varables. Tha s, eher all bnary varables assocaed wh a arcular caegorcal varable are ncluded or no hey are no consdered ndvdually. Herarchcal models are such ha f an neracon s n he model, so are he erms ha can be derved from. For examle, f A*B*C s n he model, so are A, B, C, A*B, A*C, and B*C. Sascans usually ado herarchcal models raher han non-herarchcal models. The subse selecon rocedure can be made o consder only herarchcal models durng s search. The subse selecon oons are: NCSS, LLC. All Rghs Reserved.

26 NCSS Sascal Sofware None No Search s Conduced No subse selecon s aemed. All secfed ndeenden varables are used n he regresson equaon. (Herarchcal) Forward Wh hs algorhm, he erm wh he larges log lkelhood s enered no he model. Nex, he erm ha ncreases he log lkelhood he mos s added. Ths selecon s connued unl all he erms have been enered or unl he maxmum subse sze has been reach. If herarchcal models are seleced, only hose erms ha wll kee he model herarchcal are canddaes for selecon. For examle, he neracon erm A*B wll no be consdered unless boh A and B are already n he model. When usng hs algorhm, you mus make one run ha allows a large number of erms o fnd he arorae number of erms. Nex, a second run s made n whch you decrease he maxmum erms n he subse o he number afer whch he log lkelhood does no change sgnfcanly. (Herarchcal) Forward wh Swchng Ths algorhm s smlar o he Forward algorhm descrbed above. The erm wh he larges log lkelhood s enered no he regresson model. The erm whch ncreases he log lkelhood he mos when combned wh he frs erm s enered nex. Now, each erm n he curren model s removed and he res of he erms are checked o deermne f, when hey are used nsead, he lkelhood funcon s ncreased. If a erm can be found by hs swchng rocess, he swch s made and he whole swchng oeraon s begun agan. The algorhm connues unl no erm can be found ha mroves he lkelhood. Ths model hen becomes he bes wo-erm model. Nex, he subse sze s ncreased by one, he bes hrd erm s enered no he model, and he swchng rocess s reeaed. Ths rocess s reeaed unl he maxmum subse sze s reached. Hence, hs model fnds he omum subse for each subse sze. You mus make one run o fnd an arorae subse sze by lookng a he change n he log lkelhood. You hen rese he maxmum subse sze o hs value and rerun he analyss. If herarchcal models are seleced, only hose erms ha wll kee he model herarchcal are canddaes for addon or deleon. For examle, he neracon erm A*B wll no be consdered unless boh A and B are already n he model. Lkewse, he erm A canno be removed from a model ha conans A*B. So search when number of erms reaches Once hs number of erms has been enered no he model, he subse selecon algorhm s ermnaed. Ofen you wll have o run he rocedure wce o fnd an arorae value. You would se hs value hgh for he frs run and hen rese aroraely for he second run, deendng uon he values of he log lkelhood. Noe ha he nerce s couned n hs number NCSS, LLC. All Rghs Reserved.

27 NCSS Sascal Sofware Ieraon Tab Ths anel les you conrol he maxmum lkelhood esmaon algorhm. Ieraon Oons These oons conrol he number of eraons used whle he algorhm s searchng for he maxmum lkelhood soluon. Maxmum Ieraons Ths oon secfes he maxmum number of eraons used whle fndng a soluon. If hs number s reached, he rocedure s ermnaed remaurely. Ths s used o reven an nfne loo and o reduce he runnng me of lenghy varable selecon runs. Usually, no more he 2 eraons are needed. In fac, mos runs converge n abou 7 or 8 eraons. Durng a varable selecon run, may be advsable rese hs value o 4 or 5 o seed u he varable selecon. Usually, he las few eraons make lle dfference n he esmaed values of he regresson coeffcens. Convergence Zero Ths oon secfes he convergence arge for he maxmum lkelhood esmaon rocedure. The algorhm fnds he maxmum relave change of he regresson coeffcens. If hs amoun s less han he value se here, he maxmum lkelhood rocedure s ermnaed. For large daases, you mgh wan o ncrease hs value o abou. so ha fewer eraons are used, hus decreasng he runnng me of he rocedure. Regresson Coeffcen Sarng Values These oons conrol he sarng regresson coeffcen values (he B s). Sar B s a Selec a sarng value (or ener a ls of ndvdual sarng values) for he regresson coeffcens (he B s). Alhough he B s can be any numerc value, he ycal range s beween - and. So sarng values n hs range usually allow he algorhm o converge o a useful soluon. The Cox regresson algorhm solves for he maxmum lkelhood esmaes of he regresson coeffcens by he erave Newon-Rahson algorhm. Ths algorhm begns a a se of sarng values for he regresson coeffcens and, a each eraon, modfes he B s n a way ha leads o a local maxmum of he lkelhood funcon. Somemes he algorhm does no converge o he global maxmum, so a dfferen se of sarng values mus be red. Tycally, a good soluon s one wh all B s less han 5 n absolue value. If your soluon has one or more B s ha are large (over,), you should rerun he algorhm wh a dfferen se of sarng values. Ls of Sarng B s (If Sar B s a Ls ) Ener a ls of sarng values. If no enough values are enered, he las value wll be used over and over. Alhough he B's can be any numerc value, he ycal range s beween - and. So sarng values n hs range usually allow he algorhm o converge o a useful soluon NCSS, LLC. All Rghs Reserved.