Department of Biostatistics University of Copenhagen

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1 Esimaion of average causal effec using he resriced mean residual lifeime as effec measure Zahra Mansourvar Torben Marinussen Research Repor 5/03 Deparmen of Biosaisics Universiy of Copenhagen

2 Esimaion of average causal effec using he resriced mean residual lifeime as effec measure Zahra Mansourvar and Torben Marinussen Deparmen of Biosaisics, Universiy of Copenhagen Absrac Alhough mean residual lifeime is ofen of ineres in biomedical sudies, resriced mean residual lifeime mus be considered in order o accommodae censoring. Differences in he resriced mean residual lifeime can be used as an appropriae quaniy for comparing differen reamen groups wih respec o heir survival imes. In observaional sudies where he facor of ineres is no randomized, covariae adjusmen is needed o ake ino accoun imbalances in confounding facors. In his aricle, we develop an esimaor for he average causal reamen difference in resriced mean residual lifeime as arge parameer. We accoun for confounding facors using he Aalen addiive hazards model. Large sample propery of he proposed esimaor is esablished and simulaion sudies are conduced in order o assess small sample performance of he resuling esimaor. The mehod is also applied o an observaional daa se of paiens afer an acue myocardial infarcion even. Keywords: Aalen model; Average causal effec; Confounding; Resriced mean residual lifeime; Survival daa. Inroducion In medical research and epidemiologic sudies, i is ofen of ineres o compare survival imes beween wo reamen groups. When comparing groups of subjecs, invesigaors may be ineresed in comparing life expecancy. In he saisical lieraure, life expecancy can be characerized by he mean residual life funcion which is he remaining life expecancy of a subjec who has survived up o a cerain ime.

3 Z. Mansourvar and T. Marinussen 2 For a non-negaive survival ime T wih finie expecaion, he mean residual life funcion a ime 0 is defined as m() =E(T T >). However,because of he ineviable righ censoring presen in sudies of survival ime, he crucial upper ail of survival ime disribuion can no be observed. Thereby, he mean residual life funcion of T may no be esimable a some poins. As an alernaive o he mean residual life funcion, i may be more appropriae o use he resriced mean residual life funcion which for fixed L>0 is m L () =E(T L T L >)(0< <L), where T L =min(t,l). This funcion is less sensiive o heavily righskewed survival ime disribuions han he overall mean residual life funcion. For insance, a paien is diagnosed wih prosae cancer may be ineresed in knowing how much longer he is expeced o survive over he nex L years given he is survived so far. The resriced mean residual life funcion of m L () can also be expressed via he survival funcion of S() by m L () = S() S(u)du. () In randomized sudies wih censored survival daa in which he reamens were assigned o subjecs randomly, he resriced mean residual lifeime can be esimaed for each reamen group by () and he difference beween he reamenspecific esimaes can be used as he basis for comparing reamen groups. When reamen is no randomized, as in observaional sudies, he disribuion of prognosic facors is imbalanced beween reamen groups. Therefore a proper regression model needs o be seleced and consruced o adjus for he effecs of prognosic facors. Various auhors proposed mehods o compare resriced mean lifeime beween reamens based on Cox regression. The resriced mean lifeime is he life expecancy resriced o some ime L and can be expressed as he area under he survival curve up o ime poin L. Karrison (987) examined he resriced mean lifeime wih adjusmen for covariaes and presened a mehod o compare wo groups wih respec o survival. To incorporae covariaes ino he analysis, he considered a proporional hazards model in ha separae baseline hazards were posulaed bu he covariaes were assumed o have he same muliplicaive effec on he hazard for boh groups. The baseline hazards were assumed o ake a piece-wise exponenial model. Zucker (998) developed a simplified approach based on he sraified Cox model for implemening Karrison s (987) mehod for a covariae-adjused comparison of wo groups wih respec o he resriced mean lifeime. In his model, reamen effecs can vary over ime and levels of covari-

4 3 Causal inference for resriced mean residual lifeime aes have consan hazard raios. In addiion, he exended he mehod o achieve robusness agains he siuaion where he covariae effec is no of he underlying sraified Cox model. Chen and Tsiais (200) proposed wo esimaors for he average causal reamen difference in resriced mean lifeime ha accoun for reamen imbalances in prognosic facors assuming a proporional hazards relaionship. Their sraegy can include models wih ineracion erms of reamen and covariaes. Recenly, Zhang and Schaubel (20) developed mehods for esimaing group-specific differences in resriced mean lifeime wih confounding variables when survival imes are subjec o boh dependen and independen censoring. They employed group-specific proporional hazards models o accoun for baseline covariaes. In survival daa analysis, i is crucial o measure survival expecancy progressively, herefore only reporing he resriced mean lifeime is no so conclusive as he resriced mean residual lifeime for a series of ime poins of ineres. This makes resriced mean residual lifeime a more useful quaniy han resriced mean lifeime. In his aricle, we propose a mehod o esimae group-specific differences in resriced mean residual lifeime in seings where covariae adjusmen is needed under righ censoring. Treamen comparison migh be difficul o make in observaional sudies because of confounding. In order o address his issue, we ake he poin of view described by Robins (986) and Pearl (2000) of esimaing he average causal reamen difference in resriced mean residual lifeime. To model he associaion beween he survival ime disribuion and covariaes, he Cox proporional hazards model is he mos widely used model. However, he proporional hazards assumpion of his model may fail very ofen due o ime-varying covariae effecs. In handling lack of proporionaliy assumpion, he Cox model wih ime-varying regression effecs has been considered and nonparameric esimaors for he parameers of his model have been suggesed by some researchers bu hey all uilize smoohness assumpions. See, for example, Zucker and Karr (990), Murphy and Sen (99), Marinussen e al. (2002) and Tian e al. (2005). As a useful alernaive in such a case, we consider he Aalen addiive hazards model (Aalen, 980) since ime-varying covariae effecs can be easily esimaed applying his model wihou needing any kind of smoohing. The aricle is organized as follows. We se up he noaion and sae he required assumpions needed o formalize he problem of ineres in Secion 2. The proposed mehod is described in Secion 3 along wih asympoic properies of he resuling esimaor. In Secion 4, finie sample properies are invesigaed via simulaion sudies, and hen he proposed mehod is applied o an observaional

5 Z. Mansourvar and T. Marinussen 4 sudy of paiens afer an acue myocardial infarcion even. The aricle concludes wih a discussion in Secion 5. Technical proofs are given in he Appendix. 2 Noaion and assumpions Suppose we are ineresed in comparing survival imes of wo reamen groups (A =0, ) in erms of resriced mean residual lifeime up o ime L. Weimagine a se-up where here is a non-randomized binary exposure A and a p-dimensional vecor of addiional prognosic facors X. LeC denoe he censoring ime and assume ha survival ime T and C are condiionally independen given (A, X). Le (T i,c i,a i,x i ), where i =,...,n be independen replicaes of (T, C, A, X). In he presence of censoring, we only observe U =min(t,c), and wheher i is he failure or he censoring ha has occurred, recorded by he indicaor variable =I(T apple C). When righ censoring is presen, he observed daa se hus consiss of n independen copies of {U i, i,a i,x i } and we observe survival imes only in he inerval [0, ] where <is he endpoin of he sudy. To define he average causal reamen effec, we use he noaion of counerfacual (or poenial) oucomes described by Rubin (974, 978). Le T 0 denoe he lifeime of a randomly seleced individual from he populaion under sudy ha would have been received reamen 0, andsimilarlyt he lifeime of he same individual ha would have been received reamen. The variables T 0 and T are referred o as counerfacual (or poenial) oucomes, bu only one of hose oucomes is acually observed for an individual. Neverheless, he average causal exposure effec is defined as () =E{min(T,L) min(t,l) >} E{min(T 0,L) min(t 0,L) >} = P (T >) P (T >u)du P (T 0 >) P (T 0 >u)du. In realiy, each individual will receive only one reamen, 0 or, and he observed survival ime T corresponding o he wo-dimensional counerfacual oucomes (T 0,T ) is equal o T = T 0 I(A =0)+T I(A =). The disribuion of he counerfacual random variable of P (T a >) for a =0, is equivalen o expressions of he form P (T > se(a = a)) or P (T > do(a = a)) used by Pearl (995) which denoe he survival funcion of T a ime if reamen condiion A = a was enforced uniformly over he populaion. Therefore, he average causal

6 5 Causal inference for resriced mean residual lifeime exposure effec can be wrien as () =m L (  =) m L(  =0) = S(  =) S(u  =)du S(  =0) S(u  =0)du, (2) where  = a is shor for do(a = a) which is he do-operaor of Pearl, see Pearl (2000, p. 70). In paricular, m L (  = a) denoes he resriced mean residual life funcion a ime subjec o A was uniformly se o a in he populaion. To calculae he average causal exposure effec from he observable daa, we make use of he G-compuaion formula (Robins, 986; Pearl, 2000) for he disribuion, P (T  = a), hawouldhavebeenobservedunderaninervenion, seing he exposure o a. Inourseing,heG-compuaionformulareads Z P (T  = a) = P (T A = a, X = x) df X (x), where F X denoes he marginal disribuion funcion of X. Therefore, he corresponding survival funcion is given by Z S(  = a) = P (T > A = a, X = x) df X (x), and S(  = a) is called he causal survival funcion. In order o esimae () under a righ-censored survival daa, we need o model he condiional survival funcion of T given reamen and covariaes and derive esimaes for he parameers of his model. We se S a () =S(  = a) and S a ( X) = P (T > A = a, X = x) = exp{ ( a, x)} where ( a, x) = R (u a, x)du is he cumulaive condiional hazard funcion. I 0 is naural o esimae S a () by Ŝa() =n P n i= Ŝa( X i ) where Ŝa( X i ) is an esimaor for S a ( X i ) and he averaging is over all covariae vecors X i, i =,...,n, across boh reamen groups. Then an esimaor for () is given by ˆ() = Ŝ () Ŝ (u)du Ŝ 0 () Ŝ 0 (u)du. (3)

7 Z. Mansourvar and T. Marinussen 6 We now urn o he esimaion of he effec measure () based on daa where here are independen replicaes from he Aalen addiive hazards model. 3 Esimaion and large sample properies We suppose ha he Aalen addiive hazards model (Aalen, 980) ( a, x) = 0 ()+ A () a + x T X(), (4) holds, where () = { 0 (), A(), X()} T is a locally inegrable funcion wih 0() is he baseline hazard funcion, A() is he excess hazard due o reamen and X () denoes he effec of he addiional prognosic facors. We could easily exend his model o incorporae ineracion beween A and X. However,weproceed using model (4) o keep he exposiion simple. Under model (4), he causal survival funcion is S( Â = a) = Z exp{ B 0 () B A () a x T B X ()} df X (x), where B 0 () = R 0 0(u) du, B A () = R 0 A(u) du, andb X () = R 0 X(u) du. Thus, in view of (2), i is seen ha he average causal exposure effec () is a funcion of {B 0 (),B A (),B X ()}. In he couning process framework of Andersen e al. (993), he observed daa are replaced wih N i () and Y i () of funcions of ime, where N i () = i I(U i apple ) is he couning process and Y i () =I(U i ) is a risk process for he ih individual. Wih his noaion, well-esablished mehods of Marinussen and Scheike (2006) leads o he esimaor of B() ={B 0 (),B A (),B X ()} T as ˆB() = Z 0 Z (u) dn(u), where Z () is he generalized inverse of Z() wih he laer being he n (p +2)- marix wih ih row Y i ()(,A i,xi T ). Therefore, he esimaor for he average causal exposure effec, ˆ(), can be obained by equaion (3), where nx Ŝ a () =exp{ ˆB0 () ˆBA ()a} n exp{ i= X T i ˆB X ()}.

8 7 Causal inference for resriced mean residual lifeime We now give he asympoic properies of he proposed esimaor for () which can be obained hrough he asympoic properies of he influence funcion of n /2 {ˆ() ()}. To derive he influence funcion for n /2 {ˆ() ()}, wefirs derive he influence funcion for n /2 {Ŝa() S a ()}. WeshowinAppendixha n /2 {Ŝa() S a ()} = n /2 nx i= Sa i ()+o p (), where Sa i (), i =,...,n,areindependenandidenicallydisribuedzero-mean random variables and o p () corresponds o a erm ha converges in probabiliy o zero as n ends o infiniy. The expression for Sa i () is given in Appendix. Hence, n /2 {Ŝa() S a ()} converges weakly o a zero-mean Gaussian process wih covariance funcion E{ Sa i by n P n (s)ˆ Sa i (s) Sa i ()} where ˆ Sa i ()} a (s, ), which can be consisenly esimaed () is obained by plugging in empirical quan- (). This resul can hen be used o show ha i= {ˆ Sa i iies for he unknowns in Sa i nx n /2 {ˆ() ()} = n /2 i ()+o p (), where i (), i =,...,n,areindependenandidenicallydisribuedzero-mean random variables. Accordingly, n /2 {ˆ() ()} converges weakly o a zero-mean Gaussian process whose covariance funcion E{ i (s) i ()} can be consisenly esimaed by n P n i= {ˆ i (s)ˆ i ()} where ˆ i () is defined analogously o i (), wih unknown quaniies subsiued by heir empirical counerpars. The proof of his along wih an expression of i () is given in Appendix 2. i= 4 Numerical resuls 4. Simulaion sudy Asimulaionsudywascarriedouoassesshefiniesampleproperiesofhe proposed esimaor wih respec o various degrees of confounding and effec sizes. In addiion, he proposed esimaor ˆ() was compared wih he unadjused esimaor for confounding denoed by ˆKM(), which is obained by replacing Ŝ() and Ŝ0() in (3) wih he Kaplan-Meier esimaors of he corresponding reamen groups. The survival imes T were generaed from an Aalen addiive hazards model wih hazard funcion equal o 0 + A A + X X, where a single confounding variable X

9 Z. Mansourvar and T. Marinussen 8 was generaed as an exponenial disribuion wih hazard, andgroupindicaora as Bernoulli wih parameer P (A = X) =exp( AX X)/{+exp( AX X)}. The rue parameers were aken o be 0 =0.2, A =0.4, X =0, 0. and AX =0, in order o vary he effec size of X and he magniude of confounding. For each of hese scenarios, independen censoring imes were generaed from he uniform disribuion on (0,c), andhereswerecensoreda = c, correspondingohe sudy being closed a his ime poin. The consan c was seleced o ensure approximaely 20% censoring, and L was chosen o be 3. Each configuraion was based on 000 daa ses wih sample sizes n =200and 400. The simulaion resuls for he proposed esimaor ˆ() a ime poins = 0.5,,.5, and 2 for each of he above configuraions are summarized in columns 4-7 of Table. The resuls indicae ha ˆ() is unbiased and is esimaed sandard error is close o he empirical sandard error. Besides, he 95% coverage probabiliies based on normal approximaion are reasonable and hey become more accurae when he sample size is increased. Similarly, he simulaion resuls for ˆ() when ( X, AX) =(0, 0) perform well (no shown here). Furhermore, in order o evaluae he impac of no accouning for confounding, he sampling bias for ˆKM() is repored in he las column of Table, denoed by Bias. In he case ha ( X, AX) = (0., ), X is a confounding variable since i is relaed o boh survival ime and reamen assignmen. Thus, he esimaor ˆKM() is biased due o confounding. Bu when ( X, AX) =(0., 0) or ( X, AX) =(0, ), hevariablex is no relaed o eiher reamen assignmen or survival ime, respecively. Hence X is no a confounding variable and ˆKM() is unbiased as i is seen in he las column of Table. 4.2 Applicaion We applied he proposed mehod o he daa from an observaional sudy of survival of paiens afer acue myocardial infarcion (AMI) from Universiy Clinical Cenre in Ljubljana, where 040 paiens were followed for up o 4 years. In his sudy, 547 ou of he 040 paiens died during follow-up from any cause, as i had been impossible o gaher cause-specific deah informaion and he res were alive, i.e, censored a he las follow-up. This daa se has also been considered in Sare e al. (2005), and involves several variables recorded a he ime of admission. We concenrae on he effec of aspirin ( = yes, 0=no). There is missing informaion on aspirin for 20 paiens.

10 9 Causal inference for resriced mean residual lifeime Table : Summary of he simulaion resuls (n, X, AX) True Bias SE SEE CP Bias (200, 0., ) (400, 0., ) (200, 0., 0) (400, 0., 0) (200, 0, ) (400, 0, ) True, he rue average causal exposure effec; Bias, sample mean of he esimaed () minus he rue value; SE, he sample sandard error of he esimaes; SEE, he mean of he esimaed sandard error; CP, he coverage probabiliy of he nominal 95% poin-wise confidence inervals; Bias,biasforˆKM()

11 Z. Mansourvar and T. Marinussen 0 (a) (b) Causal exposure effec Causal exposure effec Time (years) Time (years) Figure : (a) Esimae of average causal exposure effec (solid line) wih 95% poinwise confidence inervals (dashed lines) and (b) Esimae of average causal exposure effec (solid line) wih esimae of average causal exposure effec based on Kaplan-Meier esimaor (dashed line) The objecive of he analysis is o compare 5-year mean residual lifeime beween reaed versus unreaed paiens wih aspirin. A marginal analysis of he aspirin effec alone shows ha paiens reaed wih aspirin have significanly higher survival han hose unreaed. For some reason aspirin was more likely o be given o younger paiens as specifically he median ages of reaed and unreaed paiens were 59 and 66 years, respecively. Since age is a highly significan predicor of deah, i is of ineres o esimae he difference beween resriced mean residual lifeime of reaed and unreaed paiens adjusing for age as a confounder by using model (4). The esimae of reaed and unreaed difference in mean residual lifeime resriced o 5 years, ogeher wih 95% poin-wise confidence inervals are shown in Fig. (a). The paern displays ha he average causal exposure effec ˆ()

12 Causal inference for resriced mean residual lifeime is esimaed o be significanly posiive wih a decrease over 5 years, suggesing he resriced mean residual lifeime for he reaed paien wih aspirin is higher han he unreaed paien wih aspirin. As i was poined ou, he average causal reamen difference in resriced mean residual lifeime is more informaive han he average causal reamen difference in resriced mean lifeime since he former quaniy depends on a series of ime poins. For insance a 2 years of follow-up, ˆ() is esimaed o 0.4 wih 95% confidence inervals ranging from 0.02 o Therefore, in a comparison beween a reaed and an unreaed paien, he resriced mean lifeime lef a 2 years of follow-up for he aspirin-reaed paien is esimaed o be 0.4 years higher han he aspirin-unreaed paien over 5 years wih a sandard error of 0.06 years. In Fig. (b), he esimaor for (), ˆ(), is conrased wih he unadjused esimaor for age, ˆKM(). As i is seen, here is a subsanial difference beween he unadjused esimaor ˆKM() and ˆ(). For example, a 2 years of follow-up, ˆKM () is 0.2 years, which is differen han ˆ(2) = Discussion In his aricle, we proposed an esimaor for he average causal reamen difference in resriced mean residual lifeime where he reamen is no randomized. To accoun for reamen imbalances in confounding facors, we considered he Aalen addiive hazards model. We also derived he large sample propery of he proposed esimaor. In Secion 3, one could have applied he Cox model as he working associaion model and have developed esimaors of sandard errors in a similar way. We, however, prefer o use he Aalen addiive hazards model as i is our experience ha i ofen provides a good fi in pracice due o is grea flexibiliy. Acknowledgemen This work was suppored by he Minisry of Science, Research and Technology of Iran.

13 Z. Mansourvar and T. Marinussen 2 Appendix. Weak convergence of n /2 {Ŝa() S a ()} According o Marinussen and Scheike (2006, p. 8), n /2 { ˆB() wrien as B()} can be where n /2 { ˆB() B()} = n /2 nx Z i= 0 {n Z T (u)z(u)} Z i (u)dm i (u), (5) M i () =N i () Z 0 Z T i (u)db(u), is he maringale process. When n is large, represenaion (5) is equivalen o a sum of independen and idenically disribued zero-mean maringales where n /2 { ˆB() nx B()} = n /2 B i ()+o p (), (6) B i () = Z 0 i= w(u)z i (u)dm i (u), and w() is he limi in probabiliy of {n Z T ()Z()}. We now urn o he proof for he weak convergence of n /2 {Ŝa() S a ()}. Define Z S a () =exp{ B 0 () B A () a}, S a2 () = exp{ x T B X ()} df X (x), so ha S a () =S( Â = a) =S a() S a2 (), and likewise, define nx Ŝ a () =exp{ ˆB0 () ˆBA () a}, Ŝ a2 () =n exp{ where Ŝa() =Ŝa() Ŝa2(). Then i easily follows ha i= X T i ˆB X ()}, n /2 {Ŝa() S a ()} = n /2 {Ŝa() S a ()}Ŝa2()+n /2 {Ŝa2() S a2 ()}S a (). (7)

14 3 Causal inference for resriced mean residual lifeime Taking he Taylor series expansion of Ŝa(), ogeher wih he consisency of ˆB() yields ha n /2 {Ŝa() S a ()} = n h{ /2 ˆB 0 () B 0 ()} + { ˆB i A () B A ()} a S a ()+o p () where = n /2 nx i= Sa i ()+o p (), (8) Sa i () = S a (){ B0 i ()+a B A i ()}. In he laer display, B0 i () and B A i () are defined as he influence funcion of n /2 { ˆB 0 () B 0 ()} and n /2 { ˆB A () B A ()}, respecively,givenin(6). By a Taylor series expansion of Ŝa2() along wih he consisency of ˆB() and he uniform srong law of large numbers, i furher follows ha where n /2 {Ŝa2() S a2 ()} = n /2 nx i= Sa2 i () =exp{ X T i B X ()} µ() B X i () S a2 (), Sa2 i ()+o p (), (9) wih µ() being he limi in probabiliy of n P n i= exp{ XT i B X ()}Xi T. In he laer display, B X i () is defined as he influence funcion of n /2 { ˆB X () B X ()}, given in (6). Therefore, combining he resuls from equaions (7), (8) and (9) follows ha n /2 {Ŝa() S a ()} can be decomposed as a sum of independen and idenically disribued erms as where n /2 {Ŝa() S a ()} = n /2 nx i= Sa i ()+o p (), (0) Sa i () =S a2 () Sa i ()+S a () Sa2 i ().

15 Z. Mansourvar and T. Marinussen 4 2. Weak convergence of n /2 {ˆ() ()} To find he influence funcion for n /2 {ˆ() ()}, i follows ha ( n /2 {ˆ() ()} = n /2 Ŝ () ( n /2 Ŝ 0 () Ŝ (u)du Ŝ 0 (u)du S () S 0 () S (u)du S 0 (u)du ) ). I can also be wrien ha, for a =0,, is equal o Ŝ a () Ŝ a () Ŝ a (u)du nŝa (u) S a (u)o du S a () S a (u)du, nŝa () S a ()o S a ()Ŝa() S a (u)du. ()} can be dis- Using expression (0), asympoic approximaion for n /2 {ˆ() played by where i () = i () i 0 () wih nx n /2 {ˆ() ()} = n /2 i ()+o p (), i= a i () = S a () Sa i (u)du Sa i () S a () 2 S a (u)du, for a =0,.

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