Marginal Distribution Plots for Proportional Hazards Models with Time-Dependent Covariates or Time-Varying Regression Coefficients

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1 Ope Jural f Statstcs ISSN Ole: ISSN Prt: 26-78X Margal Dstrbut Plts fr Prprtal Hazards Mdels wth Tme-Depedet Cvarates r Tme-Varyg Regress Ceffcets Qqg u Juy Dg Gerge Wg 2 Departmet f Mathematcal Sceces SUN Bghamt N USA 2 Departmet f Itegratve Medce Mut Sa Beth Israel New rk N USA Hw t cte ths paper: u QQ Dg J ad Wg G (27 Margal Dstrbut Plts fr Prprtal Hazards Mdels wth Tme-Depedet Cvarates r Tme- Varyg Regress Ceffcets Ope Jural f Statstcs Receved: Nvember Accepted: February Publshed: February Cpyrght 27 by authrs ad Scetfc Research Publshg Ic Ths wrk s lcesed uder the Creatve Cmms Attrbut Iteratal Lcese (CC B 4 Ope Access Abstract Gve a sample f regress data frm ( Z a ew dagstc plttg methd s prpsed fr checkg the hypthess H : the data are frm a gve Cx mdel wth the tme-depedet cvarates Z It cmpares tw estmates f the margal dstrbut F f Oe s a estmate f the mdfed express f F uder H based a csstet estmate f the parameter uder H ad based the basele dstrbut f the data The ther s the Kapla-Meer-estmatr f F tgether wth ts cfdece bad The ew plt called the margal dstrbut plt ca be vewed as a test fr testg H The ma advatage f the test ver the exstg resdual tests s the case that the data d t satsfy ay Cx mdel r the Cx mdel s ms-specfed The the ew test s stll vald but t the resdual tests ad the resdual tests fte make type II errr wth a very large prbablty Keywrds Cx s Mdel Tme-Depedet Cvarate Sem-Parametrc Set-Up Dagstc Plt Itrduct I ths paper we prpse a ew dagstc plttg methd fr the prprtal hazards (PH mdel [] wth ctuus survval tme whch may be rght cesred ad wth pssble tme-depedet cvarates Z r tme-varyg regress ceffcets β The ew methd has sme advatages ver the exstg DOI: 4236/js2778 February 27 27

2 Q Q u et al methds the lterature especally whe the data d t satsfy ay PH mdel r whe the PH mdel s ms-specfed It prvdes a alteratve dagstc plt fr mdel checkg Dete the cdtal hazard fuct ad survval fuct f fr gve Z by hz( ad SZ( respectvely r smply h( ad S ( Let ( Z ( Z be d cpes f ( Z wth the dstrbut fuct F Z The PH mdel was frst defed by htz = exp ( β z h where h s the basele hazard fuct z s a p cvarate vectr β s a p parameter vectr h ad β are ukw ad p des t deped The mdel s referred as the tme-depedet cvarate PH (TIPH mdel Ths mdel has bee exteded tw ways: the cvarate s tme-depedet e z = z s a fuct f tme t ; 2 the regress ceffcet s tme-varyg e β = β s a fuct f tme t Fr the tme-depedet cvarates PH (TDPH mdel Kalbflesch ad Pretce [2] dstgush the exteral es frm the teral es It s fte that the exteral tme-depedet cvarate Z ca be wrtte as Z = Z = ( U g where ( s a fuct U U are d cpes frm the tme-depedet radm vectr U ad g ( s a fuct f tme t A smple example f ( U g s wu g( t where w( s a fuct t depedg tme t Wthut lss f geeralty (WLOG we ca assume wu = U Tw smple examples f g ( t are (a g = ( t a ad (b g = ( t a( t a where a s a cstat but may deped subject [] The PH mdel case (b s referred as the learly-depedet PH mdel (LDPH mdel hereafter We als csder ther frms f g ( fr the exteral case Fr the tme-varyg regress ceffcets PH mdel addt t the assumpts made the TDPH mdel e further assumes β zt htz ( = e h A specal case f ths mdel s the pecewse PH (PWPH mdel wth k cut-pts a ak [3] A mprtat step the data aalyss uder the PH mdel s t check whether the mdel s deed apprprate fr the data T ths ed t s desrable t have sme dagstc plttg methds fr the PH mdel I the lterature sme dagstc plttg methds uder the sem-parametrc set-up are desged t spect whether the data fllw the TIPH mdel htz = h exp ( β z (e g( t s cstat Oe well-kw dagstc methd fr the PH mdel s the lg-mus-lg plts (lg-lg plts Several ther graphcal methds usg resduals t check the PH mdel assumpt have bee prpsed the lterature [4] By plttg the estmatr f the cumulatve hazard fucts evaluated at each Cx-Sell resduals agast the resduals e expects a straght le patter f the data set satsfes the PH mdel Hwever as meted by Baltazar-Aba ad Pea [5] ths plttg methd has serus defect Ths methd may fal t detfy the pr fttg may crcumstaces Martgale resduals ad ts cumulatve sums are trduced t exame the fuctal frm f the cvarate [4] [6] [7] By plttg the Martgale resduals wthut a cvarate say Z agast ths mssg 93

3 Q Q u et al cvarate e ca bserve the fuctal frm f Z Ad by plttg the scre prcess versus fllw-up tme e ca exame the vlat f the prprtal hazards assumpt Scheke ad Martusse [8] later exted ths methd t the tme-varyg mdel Devace resduals [9] are the trduced These methds prvde mre symmetrc plts ad are useful detfyg the utlers Mrever e may use Schefeld resduals r the cumulatve sums frm [] The resdual plttg methds als duce several resdual tests fr the PH mdels We prvde a ew dagstc plttg methd fr the PH mdel The ma dea s t plt the Kapla-Meer estmatr (KME f S agast a prper estmatr f the margal dstrbut f uder the selected mdel Thus t s called the margal dstrbut (MD plt The MD plt ca be descrbed as a 5-step prcedure: Ft the Cx mdel yu have md t bta the regress ceffcets 2 Chse a referece value fr the cvarate Z say z such that there exst may bservats ts eghbrhd say ( z 3 Estmate the survval fuct f fr Z = z usg ( z 4 Use the estmatr 3 t estmate the margal survval fuct f 5 Cmpare the estmatr f the margal survval fuct 4 wth the KME f S The paper s rgazed as fllws I Sect 2 we prpse the MD plt ad ther supplemetary dagstc plts I Sect 3 we preset smulat results the perfrmace f the plt We als cmpare the MD plt t the curret resdual plts I Sect 4 we apply the ew dagstc plt t the lg-term breast cacer fllw-up data aalyzed Wg et al [] Sect 5 s a ccludg remark where we expla that the MD plt ca als be served as a ave hypthess test fr the PH mdels 2 The Ma Results The assumpt ad tats are gve 2 The dea f the margal apprach s trduced 22 The methd s explaed 23 ad 24 2 Assumpts ad Ntats Let Θ ph be the cllect f all PH mdels specfed by where β ad Z β zt ( h tz t = e h t ( z t are w pssble vectrs f fucts f t [2] Fr mdel checkg uder the sem-parametrc set-up the ull hypthess s = ( ( ( H : the data are frm Mdel wth Z t U g t ad gve ad g (2 where U s a p -dmesal radm cvarate vectr ad the basele hazard fuct h ( s ukw Let Θ be the cllect f all pssble jt dstrbut fucts F U f ( U Ntce that F U des t eed t belg t Θ ph Abusg tat by z = ug we mea that ug = ( ug upgp = D( g gp ( u u p where D g gp s a p p dagal matrx wth dagal elemets 94

4 Q Q u et al g t s p Our methd vlves the mde say a vectr c f the dstrbut f the p radm vectr U That s > ad η pr ( U c < pr ( U η < where s a rm eg U = max U where U = ( U U p p Prpst If ( U satsfes Mdel ( the fr each c h ( twt = h exp W { β w } where h = h exp{ β c} ad W = Z c I vew f Prpst hereafter WLOG we ca assume that AS The zer vectr s a mde f U ad t satsfes that ( g( t = Otherwse let U be a mde f U ad defe W = Z c where c = Z( U The by Prpst Mdel ( s equvalet t ather PH mdel where h takes place f the rle f the basele hazard fuct h ad h = h ( t W s als ukw Sce S U may t satsfy the PH mdel the ull hypthess H (see (2 e ca defe a ew cdtal survval fuct f a ew respse varable say fr gve Z such that S satsfes the PH mdel the ull Z hypthess H Crrespdgly e ca defe the ew margal survval fuct f say S That s ( where exp{ d } t β x S ugx t = E S t U S t u e h x x = U U (3 Ntce that h ad S are equvalet f h exsts Abusg tat wrte S ( tu = S( tus β Ntce that S U s a fuct f the ukw parameter β Gve the dstrbut fuct F U f β s t a fuct f t the β s the almst sure lmt f the maxmum partal lkelhd estmatr (MPLE f β uder H (see Example belw therwse t s ccevable β t [2] that β s sme lmtg pt f the estmatr f Example Assume that F Z s a ufrm dstrbut the reg A A2 where A s the set buded by the fur straght les y = y = x y = ad x y = ad A 2 s the set buded by y = y = x = 3 ad x = 4 The the famly f dstrbuts { S ( z : z ( ( 3 4 Z } des t satsfy the PH mdel ad S = S = t fr t [ ] If e fts the TIPH mdel H : h ( tz = h exp Z t ( β z wth data frm F Z wthut β Z e kwg F Z the S = E ( t by (3 where β 45 whch s F uder H the lmt f the MPLE based the radm sample frm Z ad F are uquely determed by F Z Z ad H Lemma If S U des t fllw the mdel defed (a S =/ S ad (b S U U = S ( t = S ( t U Otherwse (a U S = S (b S U U = S ( t = S ( t U ad (c S = S U The prf f Lemma s trval ad s skpped 22 The Margal Dstrbut Plt S H (see (2 the Mtvated by Lemma the ew plttg methd we prpse here s t plt a estmatr f S say S agast S (the KME f S tgether wth the 95

5 Q Q u et al 95% cfdece bad f S ad t check whether the graphs f y = S x ad y = S ( x are clse ( y = S ( x S eg whether s wth r utsde the 95% cfdece bad f where ( S t S tu = S ( tu S( tus β = U = (see (3 β s a U csstet estmatr f β uder H ad S s a csstet estmatr f S = S ( U uder the assumpt that FU Θ eve f the data d t satsfy the pre-assumed PH mdel H (see Remark I the latter case t s ccevable that S gves a clse mage f S If the graphs f S ad S are clse the t suggests that H (2 s true We thus call ths plt the margal dstrbut plt Sce the ma ssue f the MD plt s the estmatr S ad t s t trval t fd S due t h ( (3 the ma fcus f the paper s t trduce hw t cstruct S We shall expla detals hw t bta S thrugh g t We als gve g t r varus ( g S fr the geeral pecewse ctuus Fr smplcty we shall frst expla ur methd whe U r Z s a uvarate cvarate ad z = ( u g = ug We trduce the geeralzat t the case f a cvarate vectr (r matrx Remark 3 ad t the tme-depedet mdel fr geeral ( ugt 234 Suppse that ( U C ( U C vectr ( are d cpes frm a radm UC where may be subject t rght cesrg by cesrg varable C The success f the MD plt reles the prper estmatrs f S S ( tu ad S U Dete them by S S ( tu ad S U respectvely Sce s a mde f the cvarate U by assumpt AS a csstet estmatr f S s the KME deted by S based the data satsfyg U < where k = r (3 p wth k > (eg = r ad r s a gve pstve umber (eg the ter-quartle-rage r the stadard devat f U s WLOG let the frst bservats be all the bservats satsfyg U < If s are rght cesred the the KME S s based ( U C δ = δ = C Fr ease f explaat we ly csder the case f where cmplete data ths sect The extes t the rght cesred data s straghtfrward ad we preset smulat results wth rght-cesred data Sect 3 Fr the cmplete data the KME f servats s S based the frst (3 p ( > t ( U < r (3 j= ( j < = = ( > = p = U r S t t S ( tu U ( wll be trduced detals 23 ad S t = S tu = U Remark Sce S certa regularty cdts S ( t cverges t S b- (4 t s a kerel estmatr t s well kw that uder t prbablty ad 96

6 Q Q u et al ( β S tu S cverges t S = t prbablty fr each gve β ad F U Θ If the gve PH mdel hlds the S = S ad we expect that the graphs f S ( t ad S are clse as S ad S are csstet estmatrs f S ad S respectvely Otherwse t s lkely that S S ad tw curves y = S x ad y = S ( x are apart Remark 2 Oe may wder whether S (4 ca be replaced by the exstg estmatrs f the basele survval fuct uder the PH mdel deted by S Fr stace uder the TIPH mdel several csstet estmatrs f S say S ca be btaed frm the stadard statstcal packages Fr example the Breslw estmatr f basele survval fuct ca be btaed by applyg survftcxph R Hwever f the gve TIPH mdel des t satsfy the data S s csstet whereas S gve (4 s stll csstet I fact gve a jt dstrbut f a radm vectr ( U f t des t satsfy the TIPH mdel there exsts at least e par f survval fuct S ad exp vectr b satsfyg { } ( bu E S t = S eg ( S b = S It meas that the estmatr S usg S wll always suggest that the gve TIPH mdel fts the regress data eve thugh the data set may t satsfy the TIPH mdel ad hece S s t a prper chce Smulat study 3 suggests that uder the assumpt Example S cverges t S ( t prbablty ad usg S fals t detfy the wrg mdel assumpt 23 Estmat f S( tzs β We shall frst llustrate the ma dea thrugh three typcal cases whe β s a cstat: g( t = e the TIPH mdel 2 g = ( ( t < a ( t a e g t = t a t a e the LDPH mdel We als the PWPH mdel 3 dscuss the geeral case f g( t Case f g t = e the TIPH mdel Sce { } exp( βu S tus β = S t by (3 defe S exp( exp( t u S t u U u S S t β ad S t S t β = = = (5 U ( β ( = where β s a csstet estmatr f β uder the selected PH mdel eg the MPLE Remark 3 Eve thugh U (4 ad (5 s a radm varable t s easy t exted t the case that U s a vectr Assume U = ( U U p s a p - dmesal radm vectr s much larger tha p ad U s buded p 2 We ca defe U = max p U r U = I the smulat study we kw the mde f U I applcats the sample mde s t well defed We p chse a sample mde f U s say c as fllws: Select a prper radus r ad chse pts q such that a eghbrhd f q wth rasus r r ( q there are mre tha 2 (r 3 p bservats Of curse we d t wat r t be t large 2 Amg these pts q chse the e say c wth the largest umber f bservats r ( c amg all r ( q 3 The set U = U c = ( vew f Prpst 97

7 Q Q u et al 2 Case f g ( ( t a ( t a pt Nw β = ( β β ad = < e the PWPH mdel wth e cut 2 ( β = ( { exp( βu exp( β2v } ( t a exp{ βu( t< a + β2v( t a } S tuvs S a S t The cvarate z( t (2 s ( ( z t = D t < a t a u v where D s S tuv = S tuvs β ad a 2 2 dagal matrx The U V ( S ( t { } ( S ( a ( S ( t where exp exp 2 exp( { 2 β U β V t a β U t < a + β V t a } = (6 = a may deped the -th bservat Ad = ( ( < ( g t t a t a crrespds t a specal case f the PWPH mdel wth e cut-pt Fr the PWPH mdel wth mre tha e cut-pt the dervat f S s smlar Typcally fr the PWPH mdel wth tw cut-pts ( exp{ β } exp{ β β2 ( [ β3 } h t u r v = z t h t = u t < a + r t a b + v t b h t where = ( 2 3 the cvarate ( ( β β β β z t = D t < a a t < b t b u r v ad D s a 3 3 dagal matrx The (3 yelds exp( βu ( S f t a exp( β2r exp( βu S S( turvs β = ( S( a f t [ ab S ( a exp( β2r exp( β3v exp( βu S( b S ( S ( a f t [ b S( a S( b = ( β S t S tu R V S where β s a estmatr f β = 3 Case f g ( t a( t a S( tus β s t a smple frm terms f S ( t Let (4 ad let a = b < b < < bk be the dsctuus pts f S t > a we prpse t estmate S( tus β ad S by S f t < b b auβ e S ( tu = U j S( b S( a f t b j b = j+ S ( b ad ( = U = e the LDPH mdel I ths stuat S be defed as t fr S t = S tu The reas s as fllws Ntce that S (7 defed (4 s a step fuct It has the same csstecy prperty as S t f t a S = t S( a exp ( h d x x f t > a S where h t = t a b h h = l fr t b b ( ( ] ( b ( b ( ] S (8 98

8 a = b 2 k S t = S t b b j k ad S t Sce = = m { j j : = } at the dsctuus pts f f ( β = t ( x a βu Q Q u et al S t u = r t a S tus ( by( 3 S a exp ( h x e dx f u t > a a S tus β ( S f t a r u = exp{ j a a uβ} S exp exp{ a h uβ b a } = = u β f u t bj aj+ j k exp{ ( a a uβ j } S exp exp{ a h uβ t a } = u β f u t aj bj j k S tu = S tus U β At the dsctuus pts t f S j exp{ ( a a uβ} S { tu = S a exp h exp uβ ( b a } U uβ = j exp exp{ S a h b a uβ}( b a = exp{ ( b auβ } j S( b = S( a f u t = bj j k( by ( 8 = S( b Wrte Defe exp b S ( tu = U j S( b = S( b leads t (7 S ( tu s smpler tha S ( tu S t f t < b {( au β} S a f t bj bj+ mplemetat ad they U U have the same asympttc prpertes Smulat results sect 3 suggest that S s csstet uder the selected PH mdel g t It ca be shw that the 4 The case f ther frms f g( t r ( estmatr S ( tu the prevus three cases f g t are all the frm U that t s a step fuct wth dsctuus pts b b whch are as the same as thse f S ad S f t < b exp ( { ugb β S } tu = U j S( b S( a f t b j bj = + S( b where m b s are defed the case f g ( t a ( t a j It (9 ( = It ca be shw 99

9 Q Q u et al that f ( ugt sstet estmatr f S s pece-wse ctuus t the t als leads t a c- 24 Supplemetary Dagstc Plts The MD plt eeds t kw g( t Oe pssble way t cjecture the frm f the fuct g( t fr a tme-depedet cvarate s t exted the PWPH plttg methd Wg et al [] as fllws: ( S ( S t > ( Plt l l fr ad check whether t s pecewse lear where S (3 p ( > t Z z < r p ( Z z < r = = (3 = r s a pstve cstat ad z belgs t the supprt f Z Fr stace fr a PWPH mdel defed (6 l S ( β exp β u ( exp u l S t f t < a = ( exp( 2 β v l S a + exp ( β2v l S f t a whch crrespds t tw les: y= bx ad y= a2 + bx 2 ad the cut-pts ca be determed frm the PWPH plt (see Fgure 3 Sect 3 (2 Let b < < bm be all the dstct exact bservats I vew f the exg t = t a t a f b a the press (7 uder the PH mdel wth S ( b U + z S( b+ l exp{ ( b + a zβ} l S ( b z S( b U S b l exp{ ( S b + b a zβ } l + + S b S b f S b b a + S b+ = < G = l l l S b b a z f b a S b β + where Ŝ ad z are gve ( ad (2 Thus ather dagstc plttg methd s ( b G { m} t plt fr (3 ad check whether t appears as a tw-pecewse-lear curve: e s y = ad ather e s y ( x a b = If s t s lkely that g ( t a ( t a = where a s the tersect f the tw le segmets We shall call the plttg methd g t = t a t a crrespdg t the LDPH mdel the LDPH plt as The advatage f the PWPH plt ad the LDPH plt s that they prvde clues the cut-pts whch are eeded the MD plt uless the cut-pt s gve Remark 4 If the cut-pts the PWPH r TDPH mdel vary frm bservat t bservat the the PWPH plt as ( ad LDPH plt as (3 d t wrk Hwever the cut-pts a are als bservats such U s ( the case f cmplete data Thus we d t cases addt t

10 Q Q u et al eed t guess the cut-pts ad e ca replace S j (2 by 3 (3 p ( > t U zj < r a a < r 2 3 (3 ( Ul zj < r a l l a < r = 2 = Sj = p where a s a predetermed referece cut-pt ad r ad r 2 are pstve cstats 3 Smulat Study I rder t cmpare the MD plt t the ther plts we preset three sets f smulat results 3 32 ad 33 respectvely The mde s as assumed AS S ca be ufrm r expetal dstrbuts The cvarate ca be dscrete r ctuus The data d t eed t be frm a PH mdel There s ut tme t due t the smulat 3 Smulat Study uder the Assumpts Example Tw radm samples f = 3 ad = 3 pseud radm umbers are geerated frm U( dstrbut Fr each = y geerate Z frm U( y y wth prbablty 5 ad frm U ( 3 4 wth prbablty 5 These ( Z satsfy the assumpts gve Example Let W = ( Z 3 The the famly f dstrbuts { S ( z : z ( ( 3 4 } des t satsfy Z the ull hypthess H : htz = h exp t ( β z but t ca be shw that { S ( z : z { W }} des The sample f sze = 3 s ly used fr the MD plts paels (3 ad W s respectvely (33 f Fgure wth data ( Z s ad data 3 The MD Plts Successfully Idetfy the Vlat f the TIPH Mdel Assumpt Whe Data Are ( Z s Sce S Z des t satsfy the PH mdel a prper estmate f S s expected S It s see frm the tw MD plts wth data ( Z t devate frm s paels (2 ad (3 f Fgure that (a whe = 3 S les mst f the tme utsde r arud the edge f the 95% cfdece bad f S ; (b whe = 3 S s ttally utsde the cfdece bad f S The MD plts S appears qute dfferet frm S eve whe suggest that ur estmatr 3 TIPH Mdel as expected = ad they suggest that the data ( Z = 2 are t frm the 32 The MD Plts Crrectly Supprt the TIPH Mdel fr ( W s Sce S W satsfes the PH mdel a prper estmate f S shuld be clse t S It s see frm bth f the MD plt wth data ( W s paels (32 ad (33 f Fgure that the curves f S crrespdg t = 3 ad = 3 bth le wth the cfdece bads f S It s a strg evdece that the data W s are frm the TIPH mdel as expected 33 Abut S Defed Remark 2 T shw that the csstet estmatr f S s the key the MD apprach we

11 Q Q u et al Fgure Dagstc plts based Example 2

12 Q Q u et al preset paels ( ad (3 f Fgure the plts f the curves f ph est ( S ad the curves f S tgether wth ts cfdece bad wth data ( Z s ad ( W s respectvely It s see that the curves f ph est ad S almst ccde fr bth data sets eve whe = 3 These plts suggest that the curve f ph est s always clse t the curve f the KME regardless f whether the pre-assumed mdel fts the data Thus S s t a prper chce fr the dagstc plt 34 Abut Resdual Plts I paels (2 ad (4 f Fgure the resdual plts usg the scre prcess by the cumulatve martgale resduals methd are pltted wth data ( Z s ad ( W s respectvely There s bg dfferece patter betwee these tw plts Ntce that S Z des t satsfy the TIPH mdel but S W des Thus ths resdual plt ca t tell whether the data satsfy the TIPH mdel ths example I paels (22 ad (42 f Fgure the devace resduals are pltted wth data ( Z s ad ( W s respectvely I paels (23 ad (43 f Fgure the scaled Schefeld resduals are pltted wth data ( Z s ad ( W s respectvely There s bg dfferece patter betwee these tw pars f plts Thus these tw types f resdual plts ca t tell whether the data satsfy the TIPH mdel ths example 35 Abut Resdual Tests We als carred ut smulat study the testg H : htz = h exp t ( β z wth data ( Z s ad usg the resdual test the exstg R package Our smulat study suggests that fr = 5 r 2 ad wth a replcat f 5 the resdual test des t reject the crrect H fr mre tha 93% f the tme Thus t s t surprsed that the resdual plts d t wrk well 32 Smulat Based Data frm the TIPH Mdel A sample f cmplete data wth = 3 s geerated frm the TIPH mdel: 2 htz = exp ( β z h where β = h ( t = ad Z Nrm ( Paels ( (2 (3 ad (3 Fgure 2 presets fur dfferet plts by fttg the data set t the TIPH mdel htz = exp ( β z h Thus the cvarate s ms-specfed The secd sample f cmplete data wth sze = 3 s geerated frm the mdel htz = exp ( β z h where β = Z Nrm ( ad h ( t = t > Paels (2 (22 (23 ad (4 Fgure 2 presets 4 plts by fttg the data set t the TIPH mdel htz = exp ( β z h Thus the cvarate s crrectly specfed 32 Resduals Plts Paels ( ad (2 Fgure 2 dsplay the resdual plts usg the scre prcess by the cumulatve martgale resduals methd by fttg data frm the htz = exp β z h t It s frst ad the secd TIPH mdel respectvely t 3

13 Q Q u et al z Fgure 2 Dagstc plts based data frm Mdel: htz = h e β r 2 z htz = h t e β 4

14 Q Q u et al t easy t dstgush the ms-specfed mdel frm the crrect e by ths par f resdual plts Smlarly Fgure 2 Paels (3 ad (4 dsplay the scaled Schefeld resdual plts fr fttg the frst ad secd samples respectvely t the TIPH mdel htz = exp ( β z h Mrever Paels (2 ad (22 preset the lg-lg plts a smlar patter Aga thse methds d t prvde eugh frmat abut the verall fttg f the mdels 322 Resduals Tests I fact wth the data frm the frst ms-specfed mdel ad wth a mderate sample szes 5 ur smulat study wth a replcat f 5 suggests that the resdual test the exstg R package (eg cxzph wuld t reject the ms-specfed TIPH mdel fr mre tha 7% f the tme Thus t s t surprsed that the resdual plts cat detect the ms-specfed TIPH mdel 323 MD Plts The MD plt pael (3 fts the ms-specfed TIPH mdel wth data frm the frst sample It successfully detfes that the fuctal frm f the cvarates Z s ms-specfed fr the frst data set as S s almst ttally utsde the 95% cfdece bad f S I ther wrds the MD apprach suggests that the frst data set des t fllw the PH mdel htz = exp ( β z h O the ther had the MD plt pael (23 successfully detfes that the fuctal frm f the cvarates Z s crrect fr the secd data set as S s ttally sde the 95% cfdece bad f S Based the secd sample the mdfed PWPH plt ad the LDPH plt are dsplayed paels (32 ad (42; the MD plts uder the PWPH ad LDPH Mdels are dsplayed paels (33 ad (43 The PWPH plt pael (32 suggests that the data are ether frm a TIPH Mdel r frm a PWPH mdel wth e cut-pt at a 7 The MPLE f the regress ceffcet uder the TIPH Mdel s β = 26 ad the MPLE f the regress ceffcets uder the PWPH mdel wth k = s ( β β 2 = ( 32 wth SE = ( 7 36 Bth β ad 2 β are t sgfcatly dfferet frm β as ther dffereces frm β are wth tw SEs Bth MD plts paels (23 ad (33 suggest that the PWPH mdel wth at mst e cut-pt fts the data as expected as bth curves f S are ttally sde the 95% cfdece bad f S The LDPH plt pael (42 suggests that the LDPH Mdel may ft the data but t s see frm pael (43 that eve wth the terval [] ly less tha 3% f the curve f S les sde the cfdece bad f S Thus the MD plt suggests that the data are t frm the LDPH Mdel as expected It s see that the LDPH plt perfrms t as gd as the MD plt 33 Smulat the LDPH Mdel A sample f = 3 rght-cesred data s geerated uder the LDPH Mdel h t z exp βug t h t g t = t a t a a = 2 ad where = ( 5

15 Q Q u et al t h e subject t rght cesrg ad the rght cesrg varable C U( 2 = U has a Pss dstrbut wth mea ad β = It s The mdfed PWPH plts ad the LDPH plt are gve paels ( (2 ad (3 Fgure 3 The MD plts ad the qqplts fr the TIPH PWPH ad LDPH Mdels are gve the secd ad the thrd clums f Fgure 3 Bth the PWPH plt ad the MD plts wth crrespdg qqplt paels ( (2 ad (3 suggest that the data are t frm the TIPH Mdel We Fgure 3 Dagstc plts based data frm the LDPH Mdel 6

16 Q Q u et al eed the frmat frm the PWPH plt t decde the cut-pt eeded the MD plts The PWPH plt pael (2 suggests that the data may be frm a PWPH mdel wth a cut-pt a satsfyg l S( a 5 that s a Hwever the MD plt pael (22 suggests that the data are t frm the PWPH mdel Ths aga dcates that the MD plt perfrms better tha the mdfed PWPH plt The LDPH plt pael (3 Fgure 3 suggests that the data are frm the LDPH Mdel wth the cut-pt a ( 24 The MD plt ad the crrespdg qqplt paels (32 ad (33 suggest that the data are frm the LDPH Mdel as expected Thus the LDPH plt has the advatage that t ca suggest the cut-pt a the case that a s t gve the LDPH Mdel Sce U s dscrete ths case the PWPH plt ad the LDPH plt perfrm better tha thse the prevus smulat studes whe U s ctuus 4 Data Aalyss A cmm stuat that wll vlve the use f the PH mdel s a lg-term clcal fllw-up study I such a study the mpact f a prgstc varable may chage at dfferet tme perds Ths s the case the breast cacer data aalyzed Wg et al [] The data are btaed frm a Isttutal Revew Bard apprved lg-term clcal fllw-up study 37 wme wth stages I-III ulateral vasve breast cacer treated by surgery at Memral Sla- Ketterg Cacer Ceter New rk Cty betwee 985 ad 2 The meda fllw-up tme f the study s 74 years (rage s mth-8 mths (48 years whch s the lgest amg publshed studes be marrw mcrmetastass (BMM Oe bjectve f the study s t vestgate whether tumr dameter s sgfcat predctg early r late relapse The the relapse tme s the respse ad the tumr dameter Z s the cvarate Clcal csderat ad survval plts suggest late falure ca be csdered at tme greater tha 5 years frm tal breast cacer surgery Data aalyss based a PWPH mdel wth tw cut-pts at 2 years ad 5 years wth cvarate Z s carred ut Wg et al [] We apply ur dagstc plttg methds t check whether such a PWPH mdel fts ur data the case that Z s ctuus ad Z s dscretzed as a dchtmzed varable ( > 2 cm r 2 cm as s de Wg et al [] I the latter case we try the lg-lg plts as well We apply ur data t the TIPH mdel wth cvarate Z (whch s bascally ctuus the regress ceffcet s β = 35 ad s sgfcat We shall ly preset ur ew methds fr the case that the tumr dameter Z s ctuus pael (2 f Fgure 4 The mdfed lg-lg plts pael ( seems t suggest that the TIPH Mdel fts the data but the MD plt pael (2 suggests that the TIPH Mdel des t ft ur data as the curve f S les almst ttally utsde the 95% cfdece bad f S Thus the partal lkelhd estmate β = 35 s t vald Wg et al [] suggest t d data aalyss by dchtmzg Z accrdg 7

17 Q Q u et al Fgure 4 Dagstc plts fr cacer data wth the tumr dameter measuremets t Z > 2 cm r Z 2 cm The the cvarate s dscrete ad the stadard lg-lg plts as well as the PWPH plt prpsed by Wg et al [] s applcable These tw plts are gve paels (2 ad (3 f Fgure 4 I vew f the graph the lg-lg plts suggest that the TIPH Mdel des t ft the dscretzed data Mrever t des t preset useful frmat the ther pssble PH mdels Hwever the PWPH plt pael (3 des suggest that the PWPH mdel wth tw cut-pts clse t 5 years ad 6 years fts the dscretzed data Wg et al [] suggest t d data aalyss usg a PWPH mdel wth the cut-pts at 2 ad 5 years (whch are clse t 5 ad 6 Thus we further cmpute the partal lkelhd MLE fr mdel wth g = ( t < 2 t [ 25 t 5 The estmates β 2 β ad 3 β are 76 ( 8

18 Q Q u et al 22 ad 34 wth p-value 77 ad 52 respectvely I paels (22 ad (23 f Fgure 4 we prvde the MD plts based the dscrete ad ctuus cvarates respectvely fr fttg the PWPH mdel I partcular sce the curve f S les ttally sde the 95% cfdece bad f S pael (22 the MD plt suggests that the PWPH mdel csdered Wg et al [] fts the dscretzed data Thus ther data aalyss s vald Hwever the MD plt pael (23 dcates that the rgal data set wth the ctuus cvarate Z des t ft a PWPH mdel as the curve f S les ttally e edge f the 95% cfdece bad f S 5 Ccludg Remarks Our smulat results ad the data aalyss suggest that the MD plt has certa advatages ver the exstg resdual plts especally whe the ull hypthess H (2 s ms-specfed r the data are t frm ay PH mdel Our MD plt des t vlve resduals studed the lterature ad ths s the frst dfferece betwee the resdual appraches ad the MD apprach 5 Drawback ad Remedy f the MD Plt H (2 wth ( The MD apprach s clsely related t Z t = U g t where the parameter β s ukw but ( g s gve The MD plt s applcable t all PH mdels wth Z = ( U g ad wth all types f cvarates U prvded that ( g s gve The assumpt f a gve ( g may be vewed as a drawback f the MD plt f e wats t fd a certa PH mdel t ft the data There are several ways t vercme ths drawback Fr the case that ( u g = ug we als prpse a mdfed PWPH plt ad LDPH plt 24 fr ferrg the fuctal frm f g( t 2 Oe ca apply the MD apprach t several pssble typcal mdels Fr stace 32 ad 33 gve a data set the MD apprach fds whch f the 3 sem-parametrc PH mdels fts 3 Of curse e ca als make use f the exstg resdual appraches the lterature fr guessg g( t There s harm t spect all dagstc plts avalable based the data O the ther had the MD plt ca further check the valdty f the fuct frms suggested by the exstg resdual plts As llustrated the paper wth the help f the cfdece bad f the KME S t s mre relable ad mre frmatve tha the resdual plts whether the mdel suggested by the resdual plts s apprprate fr the data Thus the MD plt s a ce cmplemet t the exstg dagstc methds t a replacemet t them 52 A Nave but Vald Mdel Test As see frm the smulat results whe the data are t frm ay PH mdel (see Example 3 r H s ms-specfed (see smulat the TIPH Mdel 32 the MD plts ca successfully reject H prvded that s large eugh ( 3 Thus the MD plt ca play a rle f a mdel checkg 9

19 Q Q u et al test φ md that s ( ( reject H f S les sde the cfdece bad case I d t reject H f S les utsde the cfdece bad case II cclusve therwse ( case ( III O the ctrary such case ur smulat results suggest that the resdual plts fte cat detect that H s crrect ad the resdual tests fte d t reject H There must be sme reass As summarzed Thereau ad Grambsch [3] Iterestgly early all f the tests fr prprtal hazards that have bee prpsed the lterature are T( G tests ad dffer ly ther chce f the tme trasfrm g(t The parameter space Θ r fr the resdual apprach ctas all dstrbuts that γzt = β Ut + θ Ut γ= βθ Fr fllw Mdel ( wth 2 where stace the case f Example let γz βu θut = + S a resdual test s t r test : r H θ = vs H : θ =/ stead f testg H (2 vs H: H s t r r true The parameter space Θ r fr testg H vs H s a subset f the parameter space Θ fr testg H vs H where Θ s the cllect f all F U where s ctuus Whe the data are t frm ay PH mdel r Θ s ms-specfed the resdual tests are t vald Mrever that case t s sme- what true that θ = mst f the tme ad thus whe H s t rejected t s lkely t make type II errr I real applcats we d t kw whether the data are deed frm sme PH mdel r whether we select a crrect sub-mdel f the PH mdel Remark 5 I summary f the exstg resdual tests reject H the decs s lkely t be crrect Otherwse we have cfdece t beleve that the test s vald I ths regard t s terestg t pt ut that the parameter space fr the afremeted test duced by the MD plt s actually Θ Thus the MD apprach s vald eve f FU Θ r It ca als detect the crrect mdel assumpt Θ r by testg the ew ull hypthess H : FU Θ r whe s t t small but the resdual apprach cat accmplsh ths task Ths s the secd dfferece betwee the MD apprach ad the resdual apprach I applcat f H s t rejected the resdual tests fte make type II errr ad we strgly suggest t csder the MD apprach the Of curse very fte ether case (I r (II happes the we eed sme mre rgrus mdel checkg tests The dfferece f S ad S s a atural chce fr a test statstc ad t s e f ur -gg research The ther s t exted the MD apprach t ther cmm regress mdels such as the lear regress mdel ad the geeralzed addtve mdel Ackwledgemets We thak the Edtr ad the referee fr ther cmmets Refereces [] Cx DR ad Oakes D (984 Aalyss f Survval Data Chapma ad Hall New r

20 Q Q u et al rk [2] Kalbflesch JD ad Pretce RL (98 The Statstcal Aalyss f Falure Tme Data Wley New rk [3] Zhu M (2 Uderstadg the Cx Regress Mdels wth Tme-Chage Cvarates Amerca Statstca [4] Thereau TM Grambsch PM ad Flemg TR (99 Martgale-Based Resduals fr Survval Mdels Bmetrka [5] Baltazar-Aba I ad Pea E (995 Prpertes f Hazard-Based Resduals ad Implcats Mdel Dagstcs Jural f the Amerca Statstcal Asscat [6] Barlw WE ad Pretce RL (998 Resduals fr Relatve Rsk Regress Bmetrka [7] L D We LJ ad g Z (993 Checkg the Cx Mdel wth Cumulatve Sums f the Martgale-Based Resduals Bmetrka [8] Scheke TH ad Martusse T (24 O Estmat ad Tests f Tme-Varyg Effects the Prprtal Hazards Mdel Scadava Jural f Statstcs [9] Haste TJ ad Tbshra RJ (99 Geeralzed Addtve Mdels Chapma ad Hall New rk [] Grambsch PM ad Thereau TM (994 Prprtal Hazards Tests ad Dagstcs Based Weghted Resduals Bmetrka [] Wg GC Osbre MP Da Q ad u Q (26 Pecewse Cx Mdels Wth Rght-Cesred Data Cmmucat Statstcs: Smulat ad Cmputat Ole [2] Ta L Zucker D ad We LJ (25 O the Cx Mdel wth Tme-Varyg Regress Ceffcets Jural f the Amerca Statstcal Asscat [3] Thereau TM ad Grambsch PM (2 Mdelg Survval Data: Extedg the Cx Mdel Sprger Berl Submt r recmmed ext mauscrpt t SCIRP ad we wll prvde best servce fr yu: Acceptg pre-submss qures thrugh Emal Facebk LkedI Twtter etc A wde select f jurals (clusve f 9 subjects mre tha 2 jurals Prvdg 24-hur hgh-qualty servce User-fredly le submss system Far ad swft peer-revew system Effcet typesettg ad prfreadg prcedure Dsplay f the result f dwlads ad vsts as well as the umber f cted artcles Maxmum dssemat f yur research wrk Submt yur mauscrpt at: Or ctact js@scrprg