Omnibus Tests for Comparison of Competing Risks with Covariate Effects via Additive Risk Model

Transcription

1 Georga Sae Unversy Georga Sae Unversy Mahemacs Theses Deparmen of Mahemacs and Sascs Omnbus Tess for Comparson of Compeng Rsks wh Covarae Effecs va Addve Rsk Model Duyrac Vu Nguyen Follow hs and addonal works a: hps://scholarworks.gsu.edu/mah_heses Par of he Mahemacs Commons Recommended Caon Nguyen, Duyrac Vu, "Omnbus Tess for Comparson of Compeng Rsks wh Covarae Effecs va Addve Rsk Model." Thess, Georga Sae Unversy, 27. hps://scholarworks.gsu.edu/mah_heses/25 Ths Thess s brough o you for free and open access by he Deparmen of Mahemacs and Sascs a Georga Sae Unversy. I has been acceped for ncluson n Mahemacs Theses by an auhorzed admnsraor of Georga Sae Unversy. For more nformaon, please conac scholarworks@gsu.edu.

2 OMNIBUS TESTS FOR COMPARISON OF COMPETING RISKS WITH COVARIATE EFFECTS VIA ADDITIVE RISK MODEL by DUYTRAC VU NGUYEN Under he drecon of Ychuan Zhao ABSTRACT I s of neres ha researchers sudy compeng rsks n whch subecs may fal from any one of K causes. Comparng any wo compeng rsks wh covarae effecs s very mporan n medcal sudes. Ths hess develops omnbus ess for comparng cause-specfc hazard raes and cumulave ncdence funcons a specfed covarae levels. In he hess, he omnbus ess are derved under he addve rsk model, ha s an alernave o he proporonal hazard model, wh by a weghed dfference of esmaes of cumulave cause-specfc hazard raes. Smulaneous confdence bands for he dfference of wo condonal cumulave ncdence funcons are also consruced. A smulaon procedure s used o sample from he null dsrbuon of he es process n whch he graphcal and numercal echnques are used o deec he sgnfcan dfference n he rsks. A melanoma daa se s used for he purpose of llusraon. INDEX WORDS: Cause-specfc hazard raes, Addve rsk model, Cumulave ncdence funcon, Confdence bands, Compeng rsks.

3 OMNIBUS TESTS FOR COMPARISON OF COMPETING RISKS WITH COVARIATE EFFECTS VIA ADDITIVE RISK MODEL by DUYTRAC VU NGUYEN A Thess Submed n Paral Fulfllmen of he Requremens for he Degree of Maser of Scence n he College of Ars and Scences Georga Sae Unversy 27

4 Copyrgh by Duyrac Vu Nguyen 27

5 OMNIBUS TESTS FOR COMPARISON OF COMPETING RISKS WITH COVARIATE EFFECTS VIA ADDITIVE RISK MODEL by DUYTRAC VU NGUYEN Maor Professor: Dr. Ychuan Zhao Commee: Dr. Yu-Sheng Hsu Dr. Xu Zhang Dr. Gengsheng Jeff Qn Elecronc Verson Approval: Offce of Graduae Sudes College of Ars and Scences Georga Sae Unversy May 27

6 v ACKNOWLEDGEMENTS I would lke o acknowledge hose who have helped me n he compleon of hs hess. Frs of all, I mus hank Dr. Zhao who had been an exraordnary advsor, a grea professor, and a grea supporer. Ths hess canno be fnshed whou hm. I has been a grea learnng experence for me o work and ake hs classes for almos wo years. I would lke o acknowledge he oher members of my commee, Dr. Yu-Sheng Hsu, Dr. Gengsheng Jeff Qn, and Dr. Xu Zhang for akng he me o read hs hess and provde useful commens. I d also lke o hank all oher professors n he Mahemacs & Sascs Deparmen a Georga Sae Unversy for her suppors and encouragemen. They have been a grea helper and very generous. I hank you for everyhng. Lasly, I wan o acknowledge he suppor of my parens and my fancé who endured hs long process wh me, always offerng suppor and love.

7 v TABLE OF CONTENTS Acknowledgemens.....v Ls of Tables..... v Ls of Fgures......v Ls of Abbrevaon...v Chaper One: Inroducon... 1 Chaper Two: Tes Procedure Cumulave cause-specfc hazard funcon Tes Process Confdence Bands Chaper Three: Smulaon Sudy Smulaon example Smulaon example Smulaon example Chaper Four: Applcaon....2 Chaper Fve: Concluson References Appendx... 36

8 v Ls of Tables Table 3.1: Powers of es for equaly of condonal cause-specfc hazard raes based on D 2 and D 3 a nomnal level.5 wh z =.5,.5, CR = Table 3.2: Powers of es for equaly of condonal cause-specfc hazard raes based on D 2 and D 3 a nomnal level.5 wh z =.5,.5, CR = Table 3.3: Powers of es for equaly of condonal cause-specfc hazard raes based on D 2 and D 3 a nomnal level.5 wh z =.3,.3, CR = Table 4.1: Sascs es under addve rsk model for falure caused by dsease and falure caused by oher causes. H 2 and H 3 as alernave hypoheses Table 4.2: Sascs es under Cox regresson model for falure caused by dsease and falure caused by oher causes. H 2 and H 3 as alernave hypoheses...22 Table 4.3: Rsk facors n Addve rsk model for falure me o deah occurrng from dfferen causes of 25 malgnan melanoma paens Table 4.4: Rsk facors n Cox regresson model for falure me o deah occurrng from dfferen causes of 25 malgnan melanoma paens... 25

9 v Ls of Fgures Fgure 4.1a: Cumulave ncdence funcons plong Fgure 4.1b: Hazard rae plong Fgure 4.2: The es process L usng W 1 plong Fgure 4.3: The es process L usng W 2 plong Fgure 4.4: The es process L usng W 3 plong Fgure 4.5: The es process L usng W 4 plong Fgure 4.6: The smulaneous confdence bands for he dfference of cumulave ncdence funcons

10 v Ls of Abbrevaon MM: CR: Malgnan Melanoma Censor Rae

11 1 Chaper One: Inroducon Compeng rsks has been arsng n many applcaons n medcne nowadays. In compeng rsks people can sudy he cause of rsk ha each subec may fal n. I s mporan o dscover whch cause affecs he deah of he subec because wll help researchers have a good way of reamen o paens or somemes hey can desgn a new vaccne for a dsease. Each subec s observed wh a me o falure and an ndcaor ha ells whch of he compeng rsks causes he falure. The classc example of compeng rsks s compeng causes of deah, such as cancer, hear dsease, AIDS, ec. For example, n cancer clncal rals, paens beng followed up for relapse may de before relapse occurs, so relapse and deah whou aneceden relapse are assocaed as compeng falure ypes. Anoher example, n clncal sudy ha nvolves 25 paens operaed for malgnan melanoma, 71 paens ded n whch 57 paens were recorded as havng ded of he dsease and 14 paens who ded from causes unrelaed o he dsease durng he follow-up Andersen e al., Ths hess develops a mehod ha s called omnbus ess n whch compeng rsks s compared va addve rsk model wh usng nonnegave wegh funcons. Comparson of cause-specfc hazard raes and cumulave ncdence funcons a specfed covarae levels are appled n hs mehod. As we consder a compeng rsks framework n whch subecs are a rsk from k ypes of falure and covarae measuremens on each subec are avalable. We wll propose a esng ha provdes graphcal and numercal mehods for comparng any wo of he k causespecfc hazard raes, or cumulave ncdence funcons, a a specfed covarae level. Suppose each subec has an underlyng falure me X ha may be subec o censorng. We assume here s an exsence of he laen falure me T correspondng o each falure

12 2 ype = 1,..., k. Then he observed me of falure s gven by X = mn T, whch may be rgh censored. When X s uncensored, he cause of falure δ { 1,..., k} s observed along wh p vecor Z represenng he covarae nformaon. We are neresed n focusng on he condonal cause-specfc hazard rae λ z lm P X +, = X, Z = z. In oher words, λ z s = δ / he nsananeous of falure of ype gven z and n he presence of he oher falure ypes. In he presence of dependen compeng rsks, he condonal cumulave ncdence funcon s defned by F z = P X, δ = Z = z. o o Accordng o McKeague, Glber, and Kank 21, a proposal of omnbus ess for comparson of compeng rsks wh covarae effecs has been developed under Cox proporonal hazards model Cox, Also, nonparamerc ess ha allow censorng and dependen compeng rsks bu no covarae have been consdered by Aly, Kochar, and McKeague 1994, Lam 1998, among ohers. Alhough he Cox model has been consdered as a maor ool for he regresson analyss of censored survval daa, he proporonal hazards assumpon may no be approprae o some daa analyses. Therefore, s of neres for us o nvesgae an alernave approaches o modelng he assocaon beween falure mes and rsk facors hrough hazard funcons. Addve rsk model n varous forms whch has been developed and successfully ulzed by numerous auhors Aalen, 198, Cox and Oakes, 1984; Ln and Yng, 1994 s one such alernave ha has been used o descrbe dfferen aspecs of hese assocaons, and s suable n many applcaons. Under he addve rsk model, we specfy each λ z and assume ha he o censorng s condonally ndependen of he laen falure mes gven Z. From hese

13 3 assumpons, we develop a graphcal mehod, along wh a formal procedure, for esng he null hypohess H : 1 2 z λ z = λ, τ, where zs a specfed p-vecor of covarae levels and [, τ ] s he me nerval of neres. Then, he followng hypoheses wll be consdered: H 1 : 1 2 z λ z λ, τ H F z F, τ H 2 : 1 2 z 3 : 1 2 z λ z λ, τ, wh src nequaly for some [, τ ] n H 2 and H 3. The hypohess H s equaly of he correspondng cumulave ncdence funcons over [, τ ], because o F z = S u z λ u z du Kalbflesch and Prence, 198; Gay, 1988, where X o o S z = P X > Z = z s he condonal survval funcon of X. The hypoheses H 2 X o o and H 3 are alernave expressng ha cause 2 s more serous han cause 1, wh H 3 beng he more resrcve alernave. H 2 s approprae for a comparson n erms of absolue rsk and H 3 n erms of rsks nensy. We develop omnbus ess ha are conssen agans all deparures from H n he drecons of H 1, H 2, and H 3. The res of hs hess s organzed as follows. Chaper wo ncludes a bref nroducon o he es procedure n whch a formula wll be proposed for usng comparson based on a comparson of he cumulave cause-specfc hazard esmaors a he specfed covarae level z. A samplng from he null dsrbuon of he es process ha s used o smulae and deec deparure from he null hypohess s dscussed n hs chaper. Choce of

14 4 wegh process ha wll help he es leads o graphcal procedure s also covered. Confdence band for dfference of wo cumulave ncdence funcons s consruced n Chaper wo. Smulaon resuls, comparng cause-specfc hazard raes and cumulave ncdence funcons are presened n Chaper hree. In Chaper four, we apply he proposed es o he real malgnan melanoma daa for comparng he rae of deah. The hess concludes wh a summary of resuls and poenal research.

15 5 Chaper Two: Tes Procedure 2.1 Cumulave cause-specfc hazard funcon As shown n Ln and Yng 1994, he regresson coeffcen esmaor n he semparamerc addve rsk model has an explc form, whch leads o a smpler mahemacal soluon n comparson o ha of he Cox model. In conras o he Cox model, he addve rsk model specfes he hazard funcon for a covarae vecor Z n he form of T λ Z = λ + β Z, 2.1 where λ. s an unspecfed baselne hazard funcon and β s an unknown p-vecor of regresson parameer for he h cause of falure. Here, Z s a p-vecor of possbly mevaryng covaraes and o be me ndependen. Ln and Yng 1994 appled he sandard counng process marngale echnques and showed ha he esmaor s asympocally normal wh mean β and a varance-covarance marx. Le C denoe he censorng me. The compeng rsks model daa are assumed o be ~ ~ ~ ~ gven by n ndependen replcaed of X, δ, Z, where X = mn X, C, δ = δi X C, and I. s ndcaor funcon. We also assume ha C s condonally ndependen of T 1,,T k gven covaraes Z. The laen falure me T do no have o be ndependen, bu we do requre ha P T T = for l. The cause of falure me δ = when X=T. Le = l ~ ~ = I δ, and N = I X = 1,, n be a counng process for ndcang = ~ wheher an even of ype has been observed for h subec by me. Le Y = I X denoe he predcable process ndcang wheher or no he h subec s a rsk us before

16 6 me. Le τ sasfy PX > τ >. Ln and Yng 1994 proposed he followng esmaon funcon U n τ β { d}, 2.2 ˆ T = Z dn Y dλ β, Y β Z = 1 where Λˆ ˆ, β = n τ { ˆ T dn u Y u β Z u du} n Y u = 1. = s equvalen o U n τ T { }{ dn Y β Z d}, = Z Z β 2.3 = 1 where Z n Y Z. Y = 1 n = = 1 The regresson coeffcens are esmaed by solvng he equaon U ˆ β =. Tha s n τ = 1 The resulng esmaor of β s Y n { } ˆ d = { Z Z } dn. Z Z 2 τ β 2.4 = n τ ˆ n τ β = Y { Z Z } d { Z Z } dn, 2.5 = 1 = 1 where a 2 = aa '. The confdence regon for he regresson parameer β has been developed by Zhao and Hsu, 25 n whch he emprcal lkelhood mehod under he semparamerc addve rsk model wh rgh censorng had been used.

17 7 Under model 2.1, for each, he counng process N. can be decomposed unquely so ha for every h subec a, N = M + Y u dλ u Z ;, where M. s a local square negrable marngale Ln and Yng, The h cumulave cause-specfc hazard funcon for T gven Z = z akes he form where where β values are compued from 2.5 and 2.2 Tes process Λ Λ o T z = Λ o + β zdu, 2.6 n T 1 1 = dn u β Z u du, 2.7 n = 1 Y u n 1 Y = Y. n We use an approach es ha s based on a comparson of he cumulave causespecfc hazard esmaors a he specfed covarae level z, usng general predcable locally bounded nonnegave wegh processes W., = 1 Λˆ du z Λˆ du z L = W u The wegh process provdes a flexble way o conrol he relave mporance aached o dfferences o he cause-specfc hazards a dfferen mes and s useful for conrollng nsably n he als. Thus, s mporan o choose he wegh funcon so ha he funcon 2.8 gves a beer and nomnal level. Aly, Kochar, and McKeague 1994 used wegh funcon for comparng cumulave ncdence funcons and cause-specfc hazard raes, bu

18 8 he covarae was absen. Then, McKeague, Glber, and Kank 21 appled wegh funcons for comparng cause-specfc hazard raes and cumulave ncdence funcons a specfed covarae levels. Noe ha he varous choce of wegh funcon was under Cox hazards proporonal model. There are many ways o choose he wegh process. Frs, he smples choce s 1/ 2 W 1 = n, whch reduces he es process o he normalzed dfference of he esmaed cumulave cause-specfc hazard funcons, Λˆ z Λˆ L = n 2 1 z. 2.9 Ths choce s good and easy o nerpre, bu he varance of L ncreases wh, so s preferable o use a decreasng wegh process ha gves less wegh o he al, such as where. = 1 I when X C ~ W n, 2.1 n 2 = I X / = 1, oherwse I. =. Anoher choce ha gves more sophscaed wegh process and has a smlar effec under he addve rsk model s W = Sˆ 3 1/ 2 1/ 2 2 X z, 2.11 o S where S o n = Y, = 1 and k S ˆ X z = exp Λˆ z s he condonal survval funcon of X. In hs case, = 1 he asympoc dsrbuon of L s he addve hazards model based on he esmaor of a relavely smple form, and he varance funcon of V smplfes o he condonal

19 9 cumulave ncdence funcon F z. For k = 2 and no covaraes, W reduces o a wegh process consdered by Aly e al., 1994 and makes he es sascs asympocally dsrbuon free. 2 The las relavely smple choce s W = n 1/ Sˆ X, whch gves he 4 z normalzed dfference of he esmaed cumulave ncdence funcons, where F z = Sˆ u z Λˆ du z. o X Fˆ z Fˆ L = n 2 1 z, 2.12 We use a plong mehod o demonsrae 2.8 n lookng for possble deparure from H, wh a endency for large absolue values under H 1, large posve values under H 2, and an ncreasng rend under H 3. Ths can be seen from he deny o F z = S u z λ u z du. However, hese plos can be dffcul o nerpre when X o o he es process ha occur even under he hypohess of equal cause-specfc hazards. McKeague, Glber, and Kank 21 showed ha L from 2.8 converges n dsrbuon 3 under H o a zero mean Gaussan process provded 1/ 2 W / n converges unformly n probably over [,τ ] o a bounded funcon. Moreover, he lmng covarance s complcaed. Therefore, we develop a smulaon mehod for approxmaely samplng from he null dsrbuon of L under he addve rsk model. Ths procedure was presened n Song, Jeong, and Song 1997 for fndng confdence bands for survval curve whou usng wegh funcon; n Shen and Cheng 1999 for confdence bands for cumulave ncdence curves.

20 1 In order o sample he null dsrbuon of he es process, we derve he lmng dsrbuon of he cumulave hazard esmaor. Le { }, ˆ 2 1/ z z n L Λ Λ = and { }, 1 ; nf = = H τ where H. s he dsrbuon funcon of he observed falure me X ~. Le { } = du u Z u z u W z G,, = = n Y n Y 1 1, { } = = n du u Z u Y u Z u Z n C 1 ' 1 1, where = = = n n Y Z Y Z 1 1. Under H, he form 2.8 s also equvalen o:. ˆ ˆ ˆ ˆ Λ Λ Λ Λ = Λ Λ = z du z du u W z du z du u W z du z du u W L Then, he process L s asympocally equvalen o he process L 2 L 1, where { } = = + = n n u dm u Y u W n u dm u Z u Z n C z G L 1 1 ' ' 1,, 2.13 where Λ = Z u d u Y N M,.

21 11 The process o smulae L s defned by L * n whch we replace u n he frs and ~ second erms of L by G N u and G N u, respecvely, where ~ { G, G : 1,2; = 1 n} = are ndependen sandard normal varables. Realzaons of,..., L * are approxmae draws from he null dsrbuon of he es process. In oher words, under H, he condonal dsrbuon of L * gven he observed daa s he same n he lm as he uncondonal dsrbuon of L McKeague, Glber, and Kank, 21. The mehod works essenally because M has mean zero and varance { N } E. Wh he samplng from he null dsrbuon and he proposal of he es, we have formal procedures ha deec deparures from H n he drecon of H 1, H 2, and H 3 : M D = sup L, D = sup L, D = sup { L L }, 1 τ 2 τ 3 s s τ respecvely. 2.3 Confdence Bands In hs secon, we consruc confdence bands for he dfference of wo condonal cumulave ncdence funcons snce confdence nervals are used for nference or for makng sascal saemens abou esmaes. Gray 1988, Shen and Cheng 1999, developed he confdence bands for cumulave ncdence funcon for each ype of falure. However, McKeague, Glber, and Kank, 21 consruced confdence bands for he dfference n wo condonal cumulave ncdence funcons usng he wegh funcon W 4 under Cox regresson model. Consder he dfference δ = F z F can be 2 1 z obaned from he L 2.12 n whch he wegh funcon W 4 s used. Snce he process n 1/ 2 ˆ δ δ converges n dsrbuon o he null lmng dsrbuon of L, we can use

22 12 ˆ ˆ ˆ 1/ 2 δ F2 z F1 z = L / n o esmae he dfference of wo condonal cumulave ncdence funcons. The earler Mone Carlo procedure based on L * can be used o esmae an upper α / 2 quanle, c α, of he lmng dsrbuon of L. An / 2 ˆ 1 α / 2 2 approxmae 11 α% ponwse confdence band for δ s gven by δ ± c n. A smulaneous band for δ can be found by suably scalng he ponwse band. For ˆ 1 α / 2 2 example, we can use δ ± ac n, [, τ ], where 1 n ˆ δ δ a and s he crcal value of sup * L = sup τ c τ c α / 2 α / 2. Mone Carlo procedure s used o adus he consan a o furnsh he desred 11 α% smulaneous confdence level.

23 13 Chaper Three: Smulaon Sudy In hs chaper, we desgned smulaon o sudy he performance of he proposed es procedure. We consder f he nomnal sze accuraely maches observed levels a moderae sample sze and whch wegh funcon ha gves he bes performance n erms of power. We also wan o know wheher covarae level and censorng rae have effec on he powers of es. Frs, we consder a smple addve rsk model wh he baselne hazard λ = α, =1,,k where α s a varous choces of consan, and se β β =... = β 1, hen model 1 = 2 p = 2.1 becomes [ Z + Z +,..., Z ] λ z = α p. 3.1 Smply we denoe as where W λ z = α W, = Zβ, Z = Z 1,Z 2,,Z p covarae vecor, and β = 1,1,...,1 T. Le T denoe falure mes, C denoe he censorng mes. We smulae compeng rsk daa ses wh rgh censorng survval me as followng seps. 1. Smulang covaraes Z p, = 1,..., n. Z s are drawn from unform dsrbuon U,1. Snce Z 1, Z 2,..., Z p are random ndependen unform varables, we randomly choose a marx wh n rows and p columns from unform dsrbuon as he smulaed compeng rsk daa.

24 14 2. Smulang falure me T, = 1,..., k To smulae falure me T, we assume he condonal dsrbuon of T gven Z n model 3.1 s exponenal dsrbuon wh parameer α + W. The survval funcon relaed wh exponenal dsrbuon s derved as S x = e α + W x. Suppose he hazard funcon for falure me T s known, say λ z = α W, we can derve he survval funcon S and + he dsrbuon funcon F for T. Snce λ = d ln d S ' S =, we have S S z = e λ s z ds = e α + W ds = e α + W, 3.2 F = 1 S = 1 e α + W, 3.3 ln1 F = W + α. 3.4 Gven a dsrbuon funcon F, we can smulae falure me samples T dsrbued wh such a cumulave funcon F. Suppose U s drawn from unform dsrbuon U,1, hen he falure me T s obaned from T ln1 U = W + α Smulang censorng me C, Censorng me = 1,..., n C s are drawn from exponenal dsrbuon wh decay parameer adused so 15-3% of he observaons were censored n [, τ ]. Fnally, he smulaed observaons from he addve hazard rsk model 3.2 are he rples ~ ~ ~ δ, n p = 1,...,, where X = mn X, C, δ = δi X C and δ = when X = T. ~ as X,, W

25 Smulaon Example 1 In hs smulaon example, we generae a compeng rsk daa se wh censorng rae CR.3, a p = 2 dmensonal covarae Z. We se covarae level z =.5,.5, and he sample sze n = 5, 1, 2, and 4, respecvely. The sze and power of es based on D 2 and D 3 a he nomnal.5 level were esmaed from 1 ndependen samples, wh crcal values obaned n each sample from 1 realzaons of L *. Snce we assumed he alernave hypoheses H 2 and H 3 from he proposal es ha are expressng he noon of cause rsk 2 more serous han cause 1, we choose α = 1 1 and varous choces of α 2 1, 1.5, 2., 2.5, 3. n order o generae T from 3.5. Then we es all wegh funcons, W 1, W 2, W 3, and W 4, respecvely.

26 16 Table 3.1 Observed levels and powers of es for equaly of condonal cause-specfc hazard raes based on D 2 and D 3 a nomnal level.5, z =.5,.5, CR =.3. n α D 2 D 3 α 1 α 2 W 1 W 2 W 3 W 4 W 1 W 2 W 3 W The resuls n Table 3.1 show ha he observed levels for D 2 and D 3 que accuraely mach he.5 nomnal level of he es. Snce we smulae falure mes from 3.5 and gve value α1and α 2 ha represen rsk-1 and rsk-2, respecvely, he falure me of rsk-2 s always bgger han he falure me of rsk-1. Moreover, f he value of α 2 s much bgger

27 17 han he value of α 1, hen he powers of es s larger. Tha means he es sascs s much sgnfcan. In he smulaon, we fnd ha sample sze and wegh funcons also have apprecable effec on accuracy. For example, f he sample sze s ncreasng, he powers of es are also ncreasng. For example, he powers of es of sample sze n = 2 are bgger han he powers of es of sample sze n = 1 cf., Table 3.1. Especally, when sample sze n = 4 and α = 2.5, he power of es ends o be one. The wegh funcons W 2 2 and W 3 gave beer performance overall han W 1 and W 4 n erms of power. Ths makes sense because boh W 1 and W 4 do no down wegh observaons n he al, where here ends o be sharp ncrease n he varance of he cumulave baselne hazard esmae. 3.2 Smulaon Example 2 In hs smulaon example we wan o consder f censorng rae have effec on powers of es. Dong he same as smulaon n example 1, we generae a compeng rsk daa se wh seng censorng rae.15, wo covaraes, covarae level z =.5,.5, and he sample sze n = 5, 1, 2, respecvely. The resuls, repored n Table 3.2 show ha he censorng rae has effec on powers of es. As he censorng rae decreasng, he powers of es seem o be ncreasng, bu appears no so much dfference.

28 18 Table 3.2 Observed levels and powers of es for equaly of condonal cause-specfc hazard raes based on D 2 and D 3 a nomnal level.5, z =.5,.5, CR =.15. n CR α D 2 D α 1 2 α W 1 W 2 W 3 W 4 W 1 W 2 W 3 W

29 Smulaon Example 3 Now we consder he effec of covarae level. We wan o adus he covarae level by seng z =.3,.3; wo covaraes; censorng rae.3; he sample sze n = 5 and 1. Then, we observe he resul of powers of es as shown n Table 3.3. The resuls show he powers of es seem o be changed a lle. Table 3.3 Observed levels and powers of es for equaly of condonal cause-specfc hazard raes based on D 2 and D 3 a nomnal level.5, z =.3,.3, CR =.3. n z α D 2 D α 1 2 α W 1 W 2 W 3 W 4 W 1 W 2 W 3 W

30 2 Chaper Four: Applcaon The daa conss of measuremens made on paens wh malgnan melanoma. Each paen had her umors removed by surgery a he Deparmen of Plasc Surgery, Unversy Hospal of Odense, Denmark durng he perod 1962 o The surgery conssed of complee removal of he umors ogeher wh abou 2.5cm of he surroundng skn. Among he measuremens aken were he hckness of he umors and wheher was ulceraed or no. These are hough o be mporan prognosc varables n ha paens wh a hck and/or ulceraed umors have an ncreased chance of deah from melanoma. Paens were followed unl he end of There were 25 paens nvolvng operaed for malgnan melanoma, 71 paens ded, of whom 57 were recorded as des of he dsease and 14 from causes unrelaed o he dsease durng he follow-up. The remanng 134 paens were censored a he closure of he sudy snce hey were sll alve a ha pon n me Andersen e al., For me-varyng covarae effecs o cause of rsk were also colleced, ncludng sex, age, year enry sudy, he sze of umor, and ulceraon saus a he me of operaon. The laer facon s dchoomous and s scored as presen f he surface of he melanoma shows sgns of ulcers and as absen oherwse. Falure me s defned as he me snce operaon. For comparson, we use he proposed es o compare he rae of deah by he dsease and he rae of deah from oher causes. Le he falure caused by he dsease, melanoma be rsk-2 and he falure caused by oher facors be rsk-1. We assume rsk-2 s more serous han rsk-1. The follow-up me for he es s en years. Then we have hypoheses as follow: H : 1 2 z λ z = λ, 1, where z s a specfed covarae levels of a four-vecor covarae,.e., sex, age, he sze of umor, and ulceraon saus. The alernave hypoheses wll be consdered:

31 21 H F z F, 1 H 2 : 1 2 z 3 : 1 2 z λ z λ, 1 In our applcaon, he covarae age has mean 52 years and sandard devaon 17 years, and umor hckness covarae has mean 2.9 mm and sandard devaon 3 mm. Snce he predcon of he cumulave ncdence based on he sandardze covaraes has reduced sandard errors, we specfed he z levels as z = {ulceraon, 2.9 mm, male, 52 years} for he covaraes ulceraon, umor sze, sex, and age, respecvely. Table 4.1 summarzes he resul of es sascs and s p-value n whch we used he smulaon mehod for samplng from he null dsrbuon of L. We also used he four wegh funcons, W 1, W 2, W 3, and W 4 as descrbed n Secon 2.2 o deec deparures from H. Table 4.1 Sascal es under addve rsk model for falure caused by dsease and falure resulng by oher causes. H 2 and H 3 as alernave hypoheses. W 1 W 2 W 3 W 4 Tes sascs H 2 P-value <.1 <.1 Tes sascs H 3 P-value <.1 <.1

32 22 Table 4.2 Sascal es under Cox regresson model for falure caused by dsease and falure resulng by oher causes. H 2 and H 3 as alernave hypoheses. W 1 W 2 W 3 W 4 Tes sascs H 2 P-value Tes sascs H 3 P-value From Tables 4.1 and 4.2, he p-values of W 1, W 2, W 3, and W 4 n boh alernave hypoheses, H 2 and H 3 show he es s very conssen and sgnfcan under boh addve rsk model and Cox regresson model. Tha means he falure caused by he dsease s overall hgher han he falure resulng by oher causes unrelaed o he dsease. The graphcal mehod s also appled n hese alernave hypoheses es. Applyng esmaon of he cumulave ncdence probably drecly by he subdsrbuon approach Gay, 1988; Fne and Gray, 1999 o esmae he cumulave ncdence funcon of rsk-1 and rsk-2, and nonparamerc Epanechnkov kernel esmaes hazard raes. As shown n Fgures 4.1a and 4.1b, n whch he cumulave ncdence of rsk-2 malgnan melanoma overally exceeded he cumulave ncdence of rsk-1 and he hazard rae of rsk-2 s also hgher han he hazard rae of rsk-1. Thus, he es s sasfed for boh alernave hypoheses, H 2 and H 3.

33 23 Fgure 4.1a. Malgnan melanoma cohor sudy. No adusmen for covaraes. Esmaed he dsease and oher causes cumulave ncdence funcons wh 95% ponwse confdence lms. Plo Hazard funcon Esmaed Hazard rae Oher cause MM cause Tmeyears snce operaon Fgure 4.1b. Nonparamerc Epanechnkov kernel esmaes of he dsease and oher causes hazard raes based on he smoohed hazard rae esmae wh 95% ponwse confdence lms calculaed by ransformng he symmerc confdence lms.

34 24 Accordng o Andersen e al., 1993, he number deah paens who nvolvng operaed for malgnan melanoma s hgher han he number deah paens from causes unrelaed o he dsease durng he follow-up. The resuls n Tables 4.1,4.2 and Fgures 4.1a, 4.1b show he es sasfes he alernave hypoheses and our assumpon. We also wan o consder he effec of rsk facors on he rae of deah under he addve rsk models. A he.5 sgnfcance level, unvarable addve rsk models denfed umor hckness and ulceraon saus are hghly sgnfcan rsk facors for deah caused by malgnan melanoma, whereas boh covaraes have lle effec on he nensy of dyng from oher causes Tables 4.3,4.4. In fac, he covarae age has he mos sgnfcan nfluence on deah caused by oher facors. In addon, s worh menonng ha ceran ransformaons of he connuous covaraes may mprove he effcency of esmaors under he addve rsk model Shen and Cheng, In our applcaon, he covaraes age mean: 52 years; sandard devaon: 17 years and umor hckness mean: 2.92 mm; sandard devaon: 3 mm are sandardzed as gven n Tables 4.3 and 4.4. We also f he cause-specfc nensy funcon by he Cox proporonal hazards model Cox, 1972 n order o compare s resuls wh he addve rsk model. Alhough he magnude of each parameer esmae appears o dffer beween he wo model fngs, s noable ha here s good agreemen beween he es sascs for subec covarae effecs under boh models.

35 25 Table 4.3 Esmaon resuls of rsk facors n addve rsk model for falure me o deah occurrng from dfferen causes of 25 malgnan melanoma paens. Covarae Esmae SE Esmae/SE P-value ulceraon Falure caused by melanoma hckness /3 sex age -52/ ulceraon Falure resulng from oher causes hckness /3 sex age -52/ Table 4.4 Esmaon resuls of rsk facors n Cox regresson model for falure me o deah occurrng from dfferen causes of 25 malgnan melanoma paens. Covarae Esmae SE Esmae/SE P-value ulceraon <.1 Falure caused by melanoma hckness /3 sex < age -52/ Falure resulng from oher causes ulceraon hckness /3 sex age -52/

36 26 We appled he es o compare cause-specfc hazard raes for he full model n whch four covaraes, ulceraon saus, umor hckness, sex, and age have been ncluded, and anoher model n whch only he sgnfcan rsk facors covaraes, ulceraon saus, umor hckness, and age sex was excluded because has lle effec on he nensy of dyng he dsease and oher causes cf., Tables 4.3, 4.4. Tes of H versus H 2 and H 3 appled a covarae level z = {ulceraon, 2.9 mm, male, 52 years} for he full model and z = {ulceraon, 2.9 mm, 52 years} for oher one, ndcaed a sgnfcanly greaer ype 1 hazard rae and cumulave ncdence funcon, and he wegh funcons W 2 and W 3 performed beer han he wegh funcon W 1 and W 4 as shown n Tables 4.1, 4.2 and Fgures n boh models. Fgures also show he model whou covarae sex performed beer han he full model. I s of neres o esmae he dfference beween he rsk-1 and rsk-2 cumulave ncdence funcons for varous covarae subgroups. The confdence bands dsplayed n Fgure 4.6 were compued va he procedure descrbed n Secon 2.3 for he same covarae subgroups as n Fgures , condons a and b, respecvely. The consan value a s esmaed by Mone Carlo procedure n he condon wh covarae sex s 1.4, and he condon whou covarae sex s 1.9 Based on he 9% smulaneous bands, we fnd he probably of rsk-2 malgnan melanoma s lower han he probably of rsk-1 n he frs wo years. However, he probably of rsk-2 s overall hgher han he probably of rsk-1 durng he follow-up.

37 27 Fgure 4.2. The es process L sold lne and 1 realzaons of L * dash lnes for he wegh process W 1. a. Condons wh covarae {ulceraon saus, umor hckness, sex, and age}. b. Condons wh covarae {ulceraon saus, umor hckness, and age}.

38 28 Fgure 4.3. The es process L sold lne and 1 realzaons of L * dash lnes for he wegh process W 2. a. Condons wh covarae {ulceraon saus, umor hckness, sex, and age}. b. Condons wh covarae {ulceraon saus, umor hckness, and age}..

39 29 Fgure 4.4. The es process L sold lne and 1 realzaons of L * dash lnes for he wegh process W 3. a. Condons wh covarae {ulceraon saus, umor hckness, sex, and age}. b. Condons wh covarae {ulceraon saus, umor hckness, and age}.

40 3 Fgure 4.5. The es process L sold lne and 1 realzaons of L * dash lnes for he wegh process W 4. a. Condons wh covarae {ulceraon saus, umor hckness, sex, and age}. b. Condons wh covarae {ulceraon saus, umor hckness, and age}.

41 31 Fgure 4.6. Nney-fve percen ponwse and 9% smulaneous confdence bands for he dfference n condonal rae of deah cumulave ncdence funcons. a. Condons wh covarae {ulceraon saus, umor hckness, sex, and age} a level z = {ulceraon, 2.9 mm, male, 52 years}. b. Condons wh covarae {ulceraon saus, umor hckness, and age} a level z = {ulceraon, 2.9 mm, 52 years}.

42 32 Chaper Fve: Concluson The man purpose of hs hess was o develop omnbus ess for comparson of compeng rsks wh covarae effecs va semparamerc addve rsk model ha s used for alernave of he Cox regresson model. I has shown how useful s o compare causespecfc hazard rae and cumulave ncdence n compeng rsks daa. The graphcal mehod s also appled for llusraon of he comparson. The smulaon resuls and he appled problem verfy ha he addve rsk model s o be f wh compeng rsk daa. The developng of he smulaon mehod for approxmaely samplng he null dsrbuon of L under he addve rsk models o deec he hypoheses s conssed. Besde formal omnbus ess ha are conssen agans all deparures from H n he drecons of H 1, H 2, and H 3, we used graphcal mehod as shown n Fgures 4.1a - 4.1b and Fgures o llusrae he comparson rsk-1 and rsk-2. An adusmen of covarae level and censorng rae are also used n smulaon of hs hess. I s very mporan o know whch covarae has effec on he nensy of dyng because wll help he researcher o deermne he cause of deah and s rae of deah, or n he desgnng of a new vaccne. For example, McKeague, Glber, and Kank, 21 have menoned abou desgnng a new vaccne for he human mmunodefcency vrus HIV n whch hey needed o compare he nfecon rae HIV ype1 and HIV ype2 wh adusmen for covarae effecs because n order for an HIV vaccne o be effcacous n a parcular geographc regon, may be necessary o mach he genoypes of he HIV angens conaned n he vaccne o he local HIV genoype ha pose he greaes rsk of HIV nfecon. Ths hess has shown o be a relable and useful ool for comparson of compeng rsk daa where covarae has possbly me-varyng.

43 33 Fuure research could exend hs o comparng of more han wo compeng rsks under he addve rsk model or he Cox regresson model. Sun 21 suded he nonparamerc es procedures for comparng mulple cause-specfc hazard raes. We wll nvesgae he challengng problem n he fuure. Anoher area of he research could be o develop a goodness-of-f es for he model n whch a comparson beween he addve rsk model and he Cox regresson model s consdered.

44 34 References Aalen, O.O A Model for Non-paramerc Regresson Analyss of Counng Processes. In Lecure Noes on Mahemacal Sascs and Probably, 2, W. Kloneck, A. Kozek, and J. Rossk, eds. New York: Sprnger-Verlag, pp Andersen, P. K., Borgan, O., Gll, R. D,. and Kedng, N Sascal Models Based on Counng Processes. New York: Sprnger-Verlag. Aly, E.-E., Kochar, S. C., and McKeague, I. W Some ess for comparng cumulave ncdence funcons and cause-specfc hazard raes. Journal of he Amercan Sascal Assocaon 89, Cheng, M., Hall, P., and Tu, D. 26. Confdence bands for hazard raes under random censorshp. Bomerka 93, Cox, D. R Regresson models and lfe ables wh dscusson. Journal of he Royal Sascal Socey, Seres B, 34, Cox, D. R and Oakes, D Analyss of Survval Daa. London: Chapman & Hall. Fne, J. P. and Gray, R. J A Proporonal Hazards Model for he Subdsrbuon of a Compeng Rsk. JASA. 94, Gray, R. J A class of k-sample ess for comparng he cumulave ncdence of a compeng rsk. Annals of Sascs 16, Kalbflesch, J. D. and Prence, R. L The Sascal Analyss of Falure Tme Daa. New York: Wley. Klen, J.P., and Moesehberger, M.L. 23. Survval analyss echnques for censored and runcaed daa. New York: Sprng Verlag, 2 nd Edon. Lam, K. F A class of ess for he equaly of k cause-specfc hazard raes n a compeng rsks model. Bomerka, 85, Ln, D.Y. and Yng, Z.L Semparamerc analyss of he addve model. Bomerka, 81, Ln, D.Y. and Yng, Z.L Addve hazards regresson models for survval daa. In: proceedng of he Frs Seale Symposum n bosascs: Survval Analyss. Marnussen, T. and Scheke, T. H. 26. Dynamc Regresson Models for Survval Daa. Sprnger.

45 35 McKeague, I.W., Glber, P.B., and Kank, P.J. 21. Omnbus es for comparson of compeng rsks wh adusmen for covarae effecs. Bomercs 57, Shen, Y. and Cheng, S.C Confdence bands for cumulave ncdence curves under he addve rsk model. Bomercs 55, Song, M., Jeong, D., and Song, J Confdence bands for survval curve under he addve rsk model. Journal of he Korean, Sascal Socey, Vol. 26, No. 4. Sun, Y. 21. Generalzed Nonparamerc Tes Procedures for Comparng Mulple Cause-Specfc Hazard Raes. Journal of Nonparamerc Sascs, 13, Zhao, Y. and Hsu, Y.S. 25. Sem-paramerc analyss for addve rsk model va emprcal lkelhood. Communcaons n Sascs---Smulaon and Compuaon, 34,

46 36 Appendx A FORTRAN Code for comparson compeng rsk n smulaon c c Ths program s o mplemen ess of orderng c of wo condonal cause-specfc hazard funcons c and wo cumulave ncdence funcons. c Smulaed daa example. c c Tes H_: g_1 z=g_2 z on [1,2] c where z, 1 and 2 need o be specfed c versus c H_1: F_1 z<=f_2 z on [1,2] c OR c H_2: g_1 z<=g_2 z on [1,2] c wh src nequaly for some c Noe: H_1 and H_2 are denoed H_2 and H_3 n he paper. c program pvalues parameermxnsmp=1,mxncov=2 c c ncov : number of covaraes c c se mxncov=ncov n he subroune ESTP as well!! c c nsamp : sample sze c me : falure me for subec c censor : cause-of-falure = censored, 1 or 2 c covar,,k : covarae for subec a k-h even me c clevel : specfed level of covarae c nboo: number of smulaed es processes used c o fnd P-value nboo=1 gves good resuls c nsm: number of runs only needed for a smulaon sudy c o check accuracy of he ess; c neger ncov,nsamp,ndxmxnsmp,seeda,seedb,seedc,seedd, $ nsm,nboo real censormxnsmp,covarmxncov,mxnsmp,mxnsmp, $ memxnsmp,da14,ccensormxnsmp,covmxncov,mxnsmp, $ cmemxnsmp, smxnsmp, $ Umxncov,copymxnsmp,nre,nrea, $ U2mxncov,s1mxncov,mxnsmp,s2mxncov,mxncov, $ varmxncov,mxncov,beamxncov,base22,:mxnsmp, $ bea22,mxncov,s22,mxnsmp,clevelmxncov,ph:mxnsmp, $ s:mxnsmp,beav1,beav2,cbea,clev,cc,

47 37 $ s122,mxncov,mxnsmp,wmxnsmp,censor1mxnsmp, $ censor2mxnsmp,aa, bb:mxnsmp, phboomxnsmp, $ v1mxncov,mxncov,v2mxncov,mxncov, $ uumxncov,:mxnsmp,zmxnsmp,mxncov, $ gmxncov,:mxnsmp,g1,mxncov,mxnsmp,y,sumy, $ cvaluemxncov,mxncov,cempmxncov,mxnsmp, $ cnvmxncov,mxncov,cmxncov,mxncov, $ gcvalue1,mxncov,mxnsmp,zbarmxncov,mxnsmp, $ emp1mxncov,mxnsmp,emp2mxncov,mxnsmp, $ value11,value22, value22,:mxnsmp, $ value32,:mxnsmp, cumulave2,:mxnsmp,delamxnsmp, $ uu1mxncov,uu2mxncov,b1:mxnsmp,b2:mxnsmp c *** Gve some values o smulae daa *** c Use hese npu values for he numercal example: c seeda=42. seedb=37. seedc=45. seedd=51. beav1=1. beav2=1. cbea=.3 nsamp=2 clev=.5 nboo=1 1=. 2=.15 nsm=1 call rsarseeda,seedb,seedc,seedd ncov=2 do 5 =1,ncov clevel=clev 5 connue c rcouna=. rcoun=. do 1 sm=1,nsm c do 1 =1,nsamp zz=un zzz=un cov1,=zz cov2,=zzz

48 38 m1=-alogun/beav1*zz+beav1*zzz+1 m2=-alogun/beav2*zz+beav2*zzz+2.5 cc=-alogun*exp-cbea*zz-cbea*zzz f m1.l.m2.and.m1.l.cc hen censor=1. me=m1 endf f m2.l.m1.and.m2.l.cc hen censor=2. me=m2 endf f m1.ge.cc.and.m2.ge.cc hen censor=. me=cc endf 1 connue prn *, ' ' c *************************************************** c prn*,'daa smulaed' c *** order he daa by me *** c *************************************************** call ndexxnsamp,me,ndx do 2 =1,nsamp copy=me 2 connue do 4 =1,nsamp me=copyndx 4 connue do 6 =1,nsamp copy=censor 6 connue do 8 =1,nsamp censor=copyndx 8 connue do 95 =1,ncov do 9 =1,nsamp copy=cov, 9 connue do 95 =1,nsamp

49 39 cov,=copyndx do 94 =1,nsamp covar,,=cov, 94 connue 95 connue c c c Fnd ndces and corr o me > 1 and c frs falure me <= 2 c c =1 =1 do 1 =1,nsamp f me.l.1 hen =+1 endf f me.le.2 hen = endf 1 connue do 15 =1,nsamp copy=censor f censor.g.1.5 hen censor=. endf censor1=censor c prn*,, censor 15 connue c c c do 2 =1,ncov bea=. 2 connue c *** CALL SUBROUTINE esp o ge bea ha value *** call espnsamp,ncov,me,covar,censor,s,s1,s2,bea, $ CL,var,NDEAD,NMIS prn *, 'bea1=', bea,=1,ncov do 25 =1,ncov

50 4 bea21,=bea 25 connue prn*, ' ' c c Fnd he frs cumulave funcon c for frs cause-of-falure c do 21 =1,nsamp do 28 =1,ncov base2,=. cumulave,=. value2,=. value3,=. 28 connue 21 connue sar=. sumy=nsamp value11=. bb1=. empvalue1= do 215 =1,ncov bb1=bb1+bea21,*clevel 215 connue do 225 =1,nsamp s21,=s value11= value11 +censor1/sumy dff=me-sar do 22 =1,ncov s121,,=s1, Zbar,= S1,/S empvalue1=empvalue1 +bea21,*zbar,*dff 22 connue value21,=empvalue1 value31,=value31, + me*bb1 base21,=base21,-1+value11-value21, cumulave1,=cumulave1,+value11 -value21,+value31,

51 41 sar=me sumy=nsamp - do 225 k=1,ncov s121,k,=s1k, 225 connue c c Now consder second cause of falure: c do 3 =1,ncov bea=. 3 connue do 35 =1,nsamp f copy.ge.2. hen censor=1. else censor=. endf censor2=censor 35 connue c *** CALL SUBROUTINE esp o ge second cause of falure *** call espnsamp,ncov,me,covar,censor,s,s1,s2,bea, $ CL,var,NDEAD,NMIS prn *, 'bea2=', bea,=1,ncov do 31 =1,ncov bea22,=bea

52 42 31 connue c c Fnd he second cumulave funcon *** c value22= sar=. sumy=nsamp bb2=. empvalue2= do 315 =1,ncov bb2=bb2+bea22,*clevel 315 connue do 33 =1,nsamp s22,=s value22= value22 +censor2/sumy dff=me-sar do 32 =1,ncov s122,,=s1, Zbar,= S1,/S empvalue2=empvalue2 + bea22,*zbar,*dff 32 connue value22,=empvalue2 value32,=value32, + me*bb2 base22,=base22,-1+value22-value22, cumulave2,=cumulave2,+value22 -value22,+value32, sar=me sumy=nsamp- do 33 k=1,ncov s122,,=s1, 33 connue c c *******************************************************************

53 43 c s=1. s1=1. s2=1. s=1. ph=. do 35 =nsamp ph= 35 connue bb1=. bb2=. do 4 =1,ncov bb1=bb1+bea21,*clevel bb2=bb2+bea22,*clevel 4 connue sumy=nsamp do 45 =1, s=s s = s*exp-cumulave1,-cumulave1,-1 + $ cumulave2,-cumulave2,-1 s1 = s1*exp-cumulave1,-cumulave1,-1 s2 = s2*exp-cumulave2,-cumulave2,-1 f.ge. hen c for consan wegh funcon W_1 use: w=sqrfloansamp c for wegh funcon W_2 use c w=floansamp-+1/sqrfloansamp c for wegh funcon W_3 use c w=1./sqr2./sumy c for wegh funcon W_4 use c w=sqrfloansamp*s c ph=sqrfloansamp*s1-s2 ph=ph-1+cumulave2,-cumulave2,-1- $ cumulave1,-cumulave1,-1*w else ph=ph-1

54 44 endf sumy=nsamp- c prn*, me, cumulave1,,cumulave2,,s 45 connue c c ************************************************************ c **** compue he es sascs **** c c************************************************************* c saa=ph sa=ph do 46 =,-1 xx=ph f xx.g.saa hen saa=xx endf do 46 =+1, xxx=ph-ph f xxx.g.sa hen sa=xxx endf 46 connue c N.B. he sqrnsamp n fron of he es sasc c s merged no he wegh funcon; c hs avods havng o normalze s^{} c by nsamp n varous places. prn*, ' ' prn*, 'The oupu of sample sze ',nsamp prn *,'Tes sasc H_ vs H_1 =', saa prn *,'Tes sasc H_ vs H_2 =', sa prn*,' ' c c c **** smulaon o deermne P-value **** c b1-1= b2-1= nrea=. nre=. do 9 boo=1,nboo sumy=nsamp do 51 =nsamp do 55 =1,ncov uu1= uu2=

55 45 cemp,= g,= 55 connue 51 connue c c Fnd he second erm of L c c *** culculae G' value *** sar=. g1,= do 52 =1, dffvalue=me- sar do 515 =1,ncov g,=g,-1+ w*clevel- zbar,*dffvalue 515 connue sar=me 52 connue c sar2= do 58 =1,nsamp dffvalue2=me- sar2 do 525 =1,ncov g1,,=g, 525 connue c prn*,, g1,,, =1,2 c *** ranspose of Z value *** do 53 =1,ncov z,=covar,, 53 connue do 535 =1,ncov cemp,=cemp,+ covar,,-zbar,*dffvalue2 535 connue sar2=me do 55 K=1,ncov do 545 L=1,ncov emp1=zero do 54 M=1,1

56 46 emp1= emp1 + cempl,*z,k/floansamp 54 connue cvaluel,k=cvaluel,k+emp1 545 connue 55 connue 58 connue c c *** ge C nverse *** c call MATINVcvalue,ncov,ncov,IFLAG do 62 K = 1, ncov do 61 J = 1, ncov cnvk,j=cvaluek,j 61 connue 62 connue c c *** ge C ranspose *** c do 64 k=1,ncov do 63 L=1,ncov ck,l=cnvl,k 63 connue 64 connue c prn*, c1,1, c1,2 c prn*, c2,1, c2,2 do 68 =1,nsamp do 66 k=1,ncov emp15=zero do 65 =1,ncov emp15= emp15 + g1,,*c,k/floansamp 65 connue gcvalue1,k,=emp15 66 connue 68 connue c c c Fnd phboo hen p-value c do 75 =, b1=b1-1+w*rnor*censor1/sumy b2=b2-1+w*rnor*censor2/sumy a1= a2=

57 47 yy1=rnor*censor1 yy2=rnor*censor2 do 71 =1,ncov uu1=uu1+ yy1*covar,,-zbar, uu2=uu2+ yy2*covar,,-zbar, a1=a1 + gcvalue1,,*uu1 a2=a2 + gcvalue1,,*uu2 71 connue sumy=nsamp- phboo=b1+b2 + a1+a2/floansamp 75 connue c bsaa=phboo bsa=phboo do 8 =,-1 xx=phboo f xx.g.saa hen bsaa=xx endf do 8 =+1, xxx=phboo-phboo f xxx.g.bsa hen bsa=xxx endf 8 connue f bsaa.g.saa hen nrea=nrea+1. endf f bsa.g.sa hen nre=nre+1. endf 9 connue prn*, ' ' prn*, 'P-value vs H_1 = ', nrea/floanboo prn*, 'P-value vs H_2 = ', nre/floanboo prn *,'H_1 and H_2 are denoed H_2 and H_3 n he paper!' prn*,'compleed smulaon',sm, ', runs wh nboo = ', nboo prn*,'***************************************************' f nrea/floanboo.l..5 hen rcouna=rcouna+1.

58 48 endf f nre/floanboo.l..5 hen rcoun=rcoun+1. endf 1 connue prn*,'compleed smulaon ',nsm, ' runs wh nboo = ', nboo prn*,'re. a.5 level vs H_1= ',rcouna/floansm prn*,'re. a.5 level vs H_2= ',rcoun/floansm end c c ********************************************************************* c endng man program c *********************************************************************