Research Artcle Receved 28 October 2009, Accepted 8 June 2010 Publshed onlne 30 August 2010 n Wley Onlne Lbrary (wleyonlnelbrary.com) DOI: 10.1002/sm.4017 The effect of salvage therapy on survval n a longtudnal study wth treatment by ndcaton Edward H. Kennedy, a Jeremy M. G. Taylor, a Douglas E. Schaubel a and Scott Wllams b We consder usng observatonal data to estmate the effect of a treatment on dsease recurrence, when the decson to ntate treatment s based on longtudnal factors assocated wth the rsk of recurrence. The effect of salvage androgen deprvaton therapy (SADT) on the rsk of recurrence of prostate cancer s nadequately descrbed by the exstng lterature. Furthermore, standard Cox regresson yelds based estmates of the effect of SADT, snce t s necessary to adjust for prostate-specfc antgen (PSA), whch s a tme-dependent confounder and an ntermedate varable. In ths paper, we descrbe and compare two methods whch approprately adjust for PSA n estmatng the effect of SADT. The frst method s a two-stage method whch jontly estmates the effect of SADT and the hazard of recurrence n the absence of treatment by SADT. In the frst stage, PSA s predcted n the absence of SADT, and n the second stage, a tme-dependent Cox model s used to estmate the beneft of SADT, adjustng for PSA. The second method, called sequental stratfcaton, reorganzes the data to resemble a sequence of experments n whch treatment s condtonally randomzed gven the tme-dependent covarates. Strata are formed, each consstng of a patent undergong SADT and a set of approprately matched controls, and analyss proceeds va stratfed Cox regresson. Both methods are appled to data from patents ntally treated wth radaton therapy for prostate cancer and gve smlar SADT effect estmates. Copyrght 2010 John Wley & Sons, Ltd. Keywords: treatment by ndcaton; tme-dependent confounder; proportonal hazards model; causal effect; prostate cancer 1. Introducton Prostate cancer s the most commonly dagnosed cancer among Amercan men; however, the ssue of determnng the best course of treatment after ntal dagnoss s relatvely controversal [1]. Often, patents dagnosed wth clncally localzed prostate cancer undergo ether external beam radaton therapy (EBRT) or radcal prostatectomy, sometmes n combnaton wth hormone therapes [2]. After ntal treatment, patents are actvely montored for elevated and/or rsng levels of prostate-specfc antgen (PSA), whch ndcate an ncreased rsk for the clncal recurrence of prostate cancer [3]. In these cases, patents sometmes receve addtonal new treatment (called salvage therapy) n order to prevent or delay recurrence. One such addtonal salvage therapy treatment s androgen deprvaton therapy (ADT). Salvage ADT (SADT) conssts of ether surgcal or medcal castraton, although surgcal castraton (orchectomy) s less prevalent due to the avalablty of safe medcal alternatves, such as gonadotropn-releasng hormone agonsts (GnRH-As). GnRH-As are admnstered as an njecton or mplant, and can last between one and sx months accordng to dosage. GnRH-As produce testosterone levels comparable to those found after surgcal castraton wthn about three weeks [4]. Although SADT s generally thought to be benefcal n delayng the recurrence of prostate cancer, the magntude of the beneft of SADT s not well quantfed. A small number of randomzed trals have been conducted to test the effcacy of early versus deferred androgen suppresson, but these trals took place pror to the use of PSA and yelded nconclusve results [5]. Moreover, lttle attenton has been gven to evaluatng the extent to whch the effect of SADT depends on the current health status of the patent (e.g. on PSA or slope of PSA) or on other patent characterstcs (e.g. age). A better understandng of ths may help doctors to decde when to ntate SADT. a Department of Bostatstcs, Unversty of Mchgan, Ann Arbor, MI, U.S.A. b Peter MacCallum Cancer Centre, Melbourne, Australa Correspondence to: Jeremy M. G. Taylor, Department of Bostatstcs, 1420 Washngton Heghts, Ann Arbor, MI 48109, U.S.A. E-mal: jmgt@umch.edu 2569 Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569 2580
Fgure 1. Typcal log(psa) patterns. 2570 For at least the frst few months after ntaton of SADT, patents experence consderable decreases n PSA levels [6, 7]. Fgure 1 shows the typcal log(psa) patterns for two patents who receved SADT. The frst patent subsequently experenced a clncal recurrence of cancer, whereas the second patent was lost to follow-up. As elevated and/or rsng PSA levels are a rsk factor for recurrence of prostate cancer but are also a predctor of treatment by SADT, PSA and slope of PSA are (tme-dependent) confounders n the relaton between SADT and the prostate cancer recurrence hazard. In general, ths type of a relaton between a tme-dependent confounder and a tme-varyng treatment s typcally present whenever there s treatment by ndcaton [8]. Standard Cox regresson [9] can be used to estmate the effect of a treatment on survval tme n the presence of treatment by ndcaton, as long as the covarates representng the ndcaton are not also ntermedate varables (.e. varables on the causal pathway between treatment and outcome), even f the covarates are tme-dependent. However, snce PSA levels decrease after ntaton of SADT, PSA and slope of PSA are also ntermedate varables n the relaton between SADT and recurrence of prostate cancer. Therefore, usng standard tme-dependent Cox regresson to model the prostate cancer recurrence hazard as a functon of SADT hstory would yeld based estmates of the causal effect of SADT on recurrence, whether or not adjustments were made for past confounder hstory [10, 11]. An analyss whch adjusted for the observed tme-dependent PSA values after SADT would estmate only the beneft of SADT beyond that due to the decrease n PSA at the tme of ntaton of SADT, rather than the recurrence-free survval beneft tself. The methodologcal ssues concernng adjustment n the case of treatment by ndcaton are well-descrbed n the causal nference lterature. Snce Rosenbaum [10] examned the possble bas resultng from adjustment for posttreatment varables n observatonal studes [10], a number of possble approaches have been proposed. Robns developed the g-computaton algorthm estmator [12], structural nested models (SNMs) [11], and margnal structural models (MSMs) [13] to address the problem of adjustment for tme-dependent confounders whch are also ntermedate varables. However, the g-computaton algorthm does not nclude parameters that represent the treatment havng no effect, thereby complcatng the nterpretaton of correspondng confdence ntervals for the estmated treatment effect [12]. SNMs and MSMs do nclude such parameters; however, SNMs do not estmate the effect of treatment on dchotomous outcomes (e.g. recurrence-free survval). Extensons of these methods (for example, dynamc treatment MSMs and hstory-adjusted MSMs) have also been developed [14--16]. More recently, propensty score and other related methods have been adapted for longtudnal observatonal studes [17--20]. In ths paper our goal s to nvestgate alternatve approaches to estmatng the treatment effect n a longtudnal study wth treatment by ndcaton. We wll develop and compare two methods whch approprately adjust for tme-dependent PSA and slope of PSA n estmatng the effect of SADT on the rsk of recurrence of prostate cancer. The frst s a two-stage method whch uses a lnear mxed model to predct PSA and slope of PSA n the absence of SADT [21], and then uses a tme-dependent Cox model to estmate the recurrence-free survval beneft of SADT, adjustng for predcted PSA and slope of PSA. Ths approach condtons on the latent SADT-free PSA process whch, by constructon, s unaffected by SADT. The observed PSA process, however, s of course affected by the recept of SADT. Hence, ths method elmnates the ntermedate varable status of the tme-dependent PSA covarates, thereby allowng the use of a standard Cox regresson analyss n the second stage. The second proposed method has been termed sequental stratfcaton [19, 20] and s related to the approaches suggested n [17, 18]. Ths method reorganzes the observed data to resemble a sequence of randomzed experments occurrng at the ordered SADT ntaton tmes. Estmaton s mplemented va a stratfed Cox model, wth each stratum consstng of a SADT patent and a set of controls, matched on PSA-related covarates at the tme of SADT. Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569--2580
Table I. Descrpton of data. Patents (#) 2781 PSA measures (#) 25 688 Age (years) 71.0 (57.5, 80.5) Pretherapy PSA (ng/ml) 8.0 (2.5, 33.8) Clncal T -stage 1 1016 (36.5 per cent) 2 1571 (56.5 per cent) 3 4 194 (7.0 per cent) Gleason score 2 6 1846 (66.4 per cent) 7 735 (26.4 per cent) 8 10 184 (6.6 per cent) PSA measures/patent 9 (3, 18) SADT 305 (11.0 per cent) Tme to salvage ADT (years) 4.0 (1.5, 8.2) Clncal recurrence Wth pror ADT 58 (2.1 per cent) Wthout pror ADT 280 (10.1 per cent) Total 338 (12.2 per cent) Tme to clncal Recurrence (years) 4.0 (1.4, 9.2) Tme to last Contact (years) 5.2 (1.6, 10.6) For contnuous data: medan (5th, 95th percentles). For categorcal data: number (percentage). Both methods can be extended to allow for the estmaton of nteracton effects between SADT and other fxed or tme-dependent covarates. From a clncal perspectve, the estmaton of these nteracton effects s very useful. Although SADT s thought to be benefcal, t also has potentally serous sde effects [22]; thus, the decson about when to ntate the therapy s dffcult. Any nformaton about when SADT s lkely to be most benefcal would ad n that decson. The remander of ths artcle s organzed as follows. In Sectons 2 and 3, we descrbe the motvatng data set and the basc model of nterest, respectvely. In Sectons 4 and 5, we detal parameter estmaton for the two-stage method, and then for the sequental stratfcaton method. Secton 6 s devoted to the estmaton of nteractons between SADT and fxed or tme-dependent covarates. The prostate cancer data are analyzed n Secton 7, and we compare and contrast the two methods and provde some concludng remarks n Secton 8. 2. Prostate cancer data The data consst of 2781 patents wth clncally localzed prostate cancer, all of whom were ntally treated wth EBRT. Patents came from four cohorts: Unversty of Mchgan (Mchgan, U.S.A.), Radaton Therapy Oncology Group, Peter MacCallum Cancer Centre (Melbourne, Australa), and Wllam Beaumont Hosptal (Mchgan, U.S.A.). PSA (ng/ml), T -stage, and Gleason score were recorded pror to ntal EBRT, wth PSA montored at perodc vsts throughout follow-up. PSA, T -stage, and Gleason score are the three commonly measured varables n prostate cancer, wth hgher values of all three assocated wth worse prognoss. Table I descrbes the pooled data, a more complete descrpton of whch s gven n Proust-Lma et al. [21]. Note that only a small fracton of the patents receved SADT (11.0 per cent), and only a slghtly larger fracton experenced a recurrence of prostate cancer (12.2 per cent). In addton, there were 280 recurrences among the 2476 patents who dd not receve SADT (11.3 per cent), and 58 recurrences among the 305 people who dd receve SADT (19.0 per cent). Therefore, the data are capable of provdng nformaton about the hazard of recurrence for both those who dd and those who dd not receve SADT. 3. Basc model For the th subject ( =1,...,n), let T be the possbly unobserved tme to prostate cancer recurrence, and let C be the censorng tme due to end of study observaton perod, loss to follow-up or death from other cases. We then let X = T C be the observaton tme, and Δ = I (T <C ) be the recurrence ndcator. Let S be the (possbly unobserved) 2571 Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569--2580
tme to treatment by SADT, so that the SADT status ndcator at tme t for subject s I (S t). Tme t s measured from the date of end of EBRT. Further, let Z be the th subject s value of a vector of varous fxed baselne covarates (such as ntal PSA level, T -stage, Gleason score, etc.), and PSA (t) the PSA level at tme t. PSA (t) s observed at dscrete tmes t 1,t 2,...,t n. Cohort membershp s denoted by L, and s a categorcal varable takng four values. Note that death s a competng rsk for cancer recurrence, snce death precludes observng cancer recurrence [9]. As the hazard functon of nterest s the cause-specfc hazard of recurrence (.e. the hazard of recurrence among subjects whle alve), pre-recurrence death can be aggregated wth the other censorng mechansms to compute a lkelhood (n the context of the two-stage method) or to wrte down generalzed estmatng equatons (n the context of sequental stratfcaton). For these data, competng rsks mostly serve as dstractons from the man deas presented n the paper, and further dscusson s postponed untl Secton 8. Let λ (0) (t) be the unknown hazard of recurrence for subject n the absence of treatment by SADT. Consder λ (0) (t) a fxed but unknown functon of tme that defnes the natural dsease progresson ( natural meanng free from nterventon va SADT). We assume that SADT acts multplcatvely on ths natural hazard functon, wth the model then gven by: λ (t)=λ (0) (t)exp[γi(t S )] (1) where γ serves as the parameter of nterest. In both the two-stage and the sequental stratfcaton methods, γ can be generalzed to depend on t, Z, PSA (S ), or other factors. Equaton (1) defnes the parameter γ of nterest. Note that t s defned condtonally and at the ndvdual level,.e. t s the change n the person s log-hazard effectve mmedately upon admnstraton of SADT condtonal on the functon λ (0) (t). Thus t has a mechanstc nterpretaton. Ths contrasts wth the defntons of causal effects mpled by MSM methodology, whch are based on average populaton or margnal effects of nterventons. As the parameter γ of nterest n ths paper s a subject-specfc quantty, we do not necessarly expect the MSM methods to be estmatng γ. In the two-stage approach we estmate γ by utlzng parametrc models for λ (0) (t) and thus the approprateness of the estmates for γ wll be contngent on how well the models approxmate the true λ (0) (t). In the sequental stratfcaton approach we use a matchng and adjustment strategy so that, wthn each strata, λ (0) (t) are smlar and thus dfferences n outcome between those who dd and dd not receve SADT can be used to estmate γ. The approprateness of ths approach for estmatng γ wll rely on the qualty of the matchng and adjustment strateges that we use. MSM approaches to estmatng the effect of nterventons use weghted estmatng equatons, wth weghts determned by models for the probablty of the nterventon. For our example, the use of such weghts would requre buldng models for the probablty of SADT. A nave approach to estmatng γ would be to adjust for baselne covarates Z only, wth the model gven by λ (t)=λ 0 (t)exp[β T 0 Z +γi (t S )]. However, the relatonshp between rsk of recurrence and SADT s confounded by tme-dependent PSA (snce elevated/rsng PSA levels are a rsk factor for recurrence of prostate cancer but are also a predctor of treatment by SADT). Ths suggests that a better approach would be to use a tme-dependent Cox model whch adjusted for baselne covarates and tme-dependent PSA, such as: λ (t)=λ 0 (t)exp[β T 0 Z +βpsa (t)+γi (t S )] (2) However, snce PSA s both a tme-dependent confounder and an ntermedate varable (because PSA levels decrease after ntaton of SADT), model (2) yelds based estmates of γ, the effect of SADT [11]. In model (2), γ only represents the beneft of SADT beyond that due to the decrease n PSA at the tme of ntaton of SADT, rather than the recurrencefree survval beneft tself. Usng the data from all 2781 prostate cancer patents, the estmated hazard rato from model (2) s exp(γ)=1.40, wth a 95 per cent confdence nterval (CI) of (1.03, 1.89). Ths suggests that SADT s harmful wth respect to prostate cancer recurrence, contrary to common belef. In contrast, both the two-stage method and sequental stratfcaton overcome the problem of bas n the estmaton of γ. 4. Two-stage method 2572 In ths approach to estmaton of γ, t s necessary to specfy a parametrc form for the natural hazard λ (0) (t) nthe model λ (t)=λ (0) (t)exp[γi(t S )], and then jontly estmate γ and λ (0) (t). A nave form for the natural hazard, n whch λ (0) (t)=λ 0 (t)exp[β T 0 Z ], assumes that the shape of the natural hazard λ (0) (t) s the same for subjects wth the same baselne covarates, and thus does not allow for much heterogenety among patents. However, snce tme-dependent PSA and slope of PSA are strongly assocated wth recurrence, a better approach (and one that allows for heterogenety n the shapes of the natural hazard curve among patents) s to let λ (0) (t) be lnked to the tme-course of PSA n Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569--2580
the absence of SADT for each person, as f that tme-course were subject-specfc,.e. determned at tme zero by a fnte number of subject-specfc latent varables. Specfcally, we could assume a model for λ (0) (t) of the form λ (0) (t)=λ 0 (t)exp[β T 0 Z +β 1 logpsa (t)+β 2 logpsa (t)], usng the observed PSA and slope of PSA data for subject at tme t n the absence of treatment by SADT. However, there are a number of complcatons n assumng ths form for λ (0) (t) to ft model (1). Frst, PSA s not measured contnuously n tme. Second, PSA s measured wth error, and thus t s not realstc to assume that t s determned by a fnte number of subject-specfc latent varables. And fnally, even f PSA was measured contnuously and wthout error, t would be mpossble to observe PSA n the absence of treatment by SADT for subjects who dd n fact receve SADT. Ths suggests a two-stage approach n whch we frst obtan a smooth contnuous path for PSA n the absence of treatment by SADT, and then use ths contnuous counterfactual PSA n fttng the hazard model. Followng Proust-Lma et al. [21], log(psa) n the absence of SADT for subject was descrbed by a model wth three phases (0: post-therapy, 1: short-term evoluton, 2: long-term evoluton) usng the followng lnear mxed model [21]: logpsa (t) = log P (t)+ε (3) log P (t) = (μ 0 +u 0 +α T 0 Z 0)+(μ 1 +u 1 +α T 1 Z 1) f 1 (t)+(μ 2 +u 2 +α T 2 Z 2) f 2 (t) (4) where P (t) represents true PSA, whch when added to measurement error ε gves the observed data. In ths model, (μ 0,μ 1,μ 2 ), (u 0,u 1,u 2 ), (Z 0,Z 1,Z 2 ), and (α 0,α 1,α 2 ) are phase-specfc ntercepts, random effects, baselne covarates, and parameter coeffcents, respectvely. The functons f 1 (t)=(1+t) 1.5 1and f 2 (t)=t capture the short-term and long-term evoluton, respectvely, and were determned usng a profle lkelhood method. Note that the form f 2 (t)=t corresponds to exponental growth of PSA, whch s well-justfed n the context of tumor growth. Throughout the text we wrte log(psa), although the actual transformaton of PSA used s log(psa+0.1). Fgures 2(a) and (b) show the log(psa) patterns from Fgures 1(a) and (b), respectvely (for two subjects who receved SADT), along wth the subject-specfc estmated log P (t) patterns gven by the lnear mxed model (4). Note that, snce only data pror to nstances of SADT are used n fttng model (4), the correspondng estmated log P (t) patterns are not affected by ntaton of SADT. For values of t whch are greater than the tme of ntaton of SADT, S,these estmated patterns represent log P (t) f SADT was not gven. Usng the estmates of μ, u, andα, the BLUP estmates for log P (t) and slope of log P (t) (gven by log ˆP (t) and log ˆP (t), respectvely) are obtaned. We then obtan the estmates of β and γ n the model: λ l (t)=λ 0l (t)exp[β T 0 Z +β 1 log ˆP (t)+β 2 log ˆP (t)+γi (t S )] (5) where γ s the parameter of nterest, and l references the cohort. Note that model (5) s stratfed by cohort; ths allows for non-proportonal baselne hazards across the four dfferent cohorts. Parameter and covarance estmates are gven by the usual maxmum partal lkelhood estmates and the correspondng nverse nformaton matrx, respectvely. Note that the two-stage method descrbed n ths secton can be ft usng the standard software (e.g. n SAS, PROC MIXED for model (4) and PROC PHREG for model (5), or n R/S-PLUS, lmer() from the lme4 package and coxph() from the survval package). Fgure 2. Typcal PSA data wth BLUP estmates log ˆP (t). 2573 Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569--2580
5. Sequental stratfcaton method 5.1. Estmaton The sequental stratfcaton method reorganzes the observed data set n an attempt to mmc a sequence of condtonally randomzed SADT assgnments (.e. assgned randomly, gven the covarate nformaton). At the tme of each nstance of SADT ntaton, a stratum s created whch ncludes the patent undergong SADT (the ndex case ) and matched patents at rsk who are smlar to the ndex case, but who have not yet undergone SADT. Note that f the matchng s set up such that smlar subjects have smlar natural hazards λ (0) (t) n equaton (1), then ths method wll be estmatng the parameter γ n equaton (1). After strata are defned, a stratfed Cox proportonal hazards model (whch allows for dfferent baselne hazards across strata) can be used to estmate SADT beneft wthout adjustng for tme-dependent varables. Let S ( j) be the jth ordered tme of SADT ntaton, wth j =1,...,n S and S (1) <S (2) < <S (ns ), where n S s the total number of patents undergong SADT. Wth respect to the ( j)th patent to ntate SADT (ndex patent ( j)), the stratum-ncluson ndcator for patent s gven by: e j = I (S X S ( j), L = L ( j), A ( j) =1) (6) where A ( j) equals 1 f patent meets the crtera to warrant matchng to patent ( j) and0otherwse,andl denotes cohort membershp. For the sequental stratfcaton method we chose to restrct matches to those from the same cohort, although ths may not be necessary. Therefore, the ( j)th stratum wll nclude the ndex case undergong SADT at tme S ( j), as well as all matched patents { :e j =1}. For all patents wth e j =1, we ft a model gven by: λ ( j) (t)=λ 0( j) (t)exp[θ T Z +γi ( =( j))], (7) for ( j)=1,...,n S, where I ( =( j)) s an ndcator for patent beng the ndex case. Usng the survval analyss analog of generalzed estmatng equatons [23], γ from model (7) can be estmated wth the estmatng equatons from a stratfed Cox model, where all n S ndex cases defne a total of n S strata. A robust varance estmator s used n order to account for the ncluson of the same patents n multple strata. Note that t n ths model represents tme from the end of radaton therapy, the same tme axs used for the two-stage approach. However, f tme t was measured from S ( j) for each strata n model (7), the relatve orderng of the falure tmes wthn strata (and hence the resultng estmator of γ) would be unchanged. As wth the two-stage method, the sequental stratfcaton method can be ft usng the standard software (e.g. by usng the start/stop, or countng process, nput fle structure, wth PROC PHREG n SAS or coxph() n R/S-PLUS). In addton, the sequental stratfcaton method s ntended for observatonal data, wth the absence of randomzaton requrng that unbased contrasts between treatment groups be obtaned through covarate adjustment. Accurate covarate adjustment s acheved through the combnaton of factors used to determne A ( j) and factors ncorporated nto Z. In practce, t would be desrable to adjust for all factors that affect the hazard functon, whether or not these factors were assocated wth the treatment assgnment. Owng to the non-lnearty of the hazard model, substantal bas n the Cox model could result f an mportant hazard predctor were omtted, regardless of whether or not such a predctor was ndependent of treatment. The practtoner needs to make decsons regardng whch adjustment factors should be ncluded n the covarate vector, and whch should be ncluded as matchng crtera. Factors that are very strong predctors of treatment and/or the hazard functon are prme canddates for matchng, as are factors that would be dffcult to accurately model (e.g. a categorcal covarate wth 50 levels). Fnally, non-ndex-case patents who later undergo SADT are censored at the tme of ther SADT. As outlned n detal n Schaubel et al. [20], the sequental stratfcaton method assumes that treatment (SADT, n ths case) s assgned randomly gven the matchng varate and the tme-dependent covarates. Dependng on the applcaton, nverse probablty of censorng weghtng (IPCW) may be requred [24]; however, n ths case, nverse weghtng s not requred snce (as mentoned prevously) only a small fracton of patents receved SADT. 2574 5.2. Analyss of prostate cancer data We now descrbe the applcaton of the sequental stratfcaton method to the prostate cancer data set. The most mportant factors wth respect to generatng comparable sets of patents are tme-dependent PSA and slope of PSA, factors for whch we adjust usng the matchng ndcator, A ( j). Three dfferent sets of PSA-related covarates were used wth the ndcator A ( j) for ths data set. For each, t was requred that the locaton for patent be equal to that of patent ( j) (.e. L = L ( j) ). Method (1) uses only estmated log(psa), method (2) uses estmated log(psa) and slope of log(psa), and method (3) uses estmated log(psa), estmated slope of log(psa), and the long-term phase (LTP) coeffcent α 2,whch s the BLUP estmate for the (μ 2 +u 2 +α T 2 Z 2) term n model (4). In addton, both thresholdng and nearest-neghbor Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569--2580
methods were used wth the ndcator A ( j) to defne whch patents could be consdered suffcently smlar wth respect to the three dfferent sets of PSA-related covarates. Method (1), whch does not match on slope of PSA, s probably not suffcent to ensure that the matched set s smlar enough wth respect to varables that are assocated wth the decson to ntate SADT. However, method (2), whch matches on slope of PSA n addton to PSA, should provde a sutably smlar set of matches to each ndex case. Method (3) adds the crteron of matchng on expected future slope of PSA, whch s a determnant of dsease progresson. Ths could provde mproved effcency for the sequental stratfcaton method. 5.3. Matchng crtera Note that, as we have mplemented them, all three matchng methods requre longtudnal modelng of PSA to obtan estmates of log(psa), slope of PSA, and LTP. Smlar methods based only on the observed PSA data could be devsed, thus avodng the need to specfy a longtudnal model. Let log ˆP (t) be the standardzed estmated log(psa): log ˆP (t)= log ˆP (t) log ˆP(t) (8) sd(log ˆP(t)) where log ˆP(t)=(1/n S ) log ˆP (t) and sd(log ˆP(t))= (1/n S ) [log ˆP (t) log ˆP(t)]. Smlarly, let the standardzed estmated slope of log(psa) log ˆP (t) and LTP coeffcent α 2 be log ˆP (t) and α 2, respectvely, obtaned from the longtudnal model. The threshold-based A ( j) ndcator functons used were: A( T 1 j) = I ( log ˆP (t) log ˆP ( j) (t) c) (9) A T 2 ( j) = I ( log ˆP (t) log ˆP ( j) (t) c, log ˆP A( T 3 j) = I ( log ˆP (t) log ˆP for threshold values c =0.2, 0.5, and 1.0, and for tme t = S ( j). ( j) (t) c, log ˆP (t) log ˆP ( j) (t) c) (10) (t) log ˆP ( j) (t) c, α 2 α 2( j) c) (11) The nearest-neghbor ndcator functons A( N1 j), AN2 ( j),andan3 ( j) equal one for the 10 patents nearest to the ndex case (wth respect to Eucldean dstance) n 1-dmensonal log(psa) space, 2-dmensonal log(psa) and slope of log(psa) space, and 3-dmensonal log(psa), slope of log(psa), and LTP coeffcent space, respectvely. For example, for patent, A( N2 j) =1f [log ˆP (t) log ˆP ( j) (t)]2 +[log ˆP (t) log ˆP ( j) (t)] 2 (12) s among the 10 smallest values across all patents such that S X S ( j) and L = L ( j). Matched patents who subsequently receved SADT were censored, and such censorng was treated as ndependent. Ths should not be a gross volaton of the ndependent censorng assumpton snce, wthn strata (for matchng methods 2 and 3), all subjects have smlar PSA and slope of PSA. In a separate analyss of tme to SADT, t was determned that the only factors that were strongly and consstently assocated wth ntaton of SADT were PSA and slope of PSA. Thus, wthn each stratum, there s approxmately ndependent censorng. 6. Estmaton of treatment-by-covarate nteractons Both the two-stage method and the sequental stratfcaton method allow for easy generalzaton of γ to depend on covarates (such as tme of SADT, age at tme of SADT, estmated PSA or slope of PSA at tme of SADT, T-stage, tme, etc.). Ths allows for the estmaton of nteracton SADT effects. For example, t may be of nterest to determne whether the effect of SADT s constant wth respect to age. As androgens n men tend to decrease after the age of approxmately 40, SADT mght be expected to have less beneft for older men, who have less androgens present to modfy [25]. We consder a number of dfferent extensons of equaton (1) to represent dfferent types of nteractons. In the case of non-tme-varyng covarates, such as age, we extend equaton (1) to be λ (0) (t)exp[γ(age )I (t S )]. In the case of tme snce SADT extend equaton (1) to be λ (0) (t)exp[γ(t S )I (t S )]. In the case where the nteractng varable of nterest s based on PSA, we extend equaton (1) to be λ (0) (t)exp[γ(p (S ))I (t S )], where P (S ) represents the true value of PSA for subject at tme S. For estmaton purposes we would replace P (S )by ˆP (S ). 2575 Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569--2580
For the two-stage method, the approach for estmatng the nteracton effect of SADT wth categorcal age would be to assume the model (n place of model (5)): λ l (t) = λ 0l (t)exp[θ T Z +β 1 log ˆP (t)+β 2 log ˆP (t)+γ 1I(t S,a (S ) 71)+γ 2 I (t S,71<a (S ) 76) +γ 3 I (t S,76<a (S ) 80)+γ 4 I (t S,a (S )>80)] (13) where γ s the parameter of nterest, l references the cohort, and a (S ) s the age of subject at the tme of SADT. For the sequental stratfcaton method, the correspondng model (n place of model (7)) for estmatng the nteracton effect of SADT wth categorcal age would be: λ m (t) = λ 0m (t)exp[β T 0 Z +γ 1 I ( =( j),a (S ) 71)+γ 2 I ( =( j),71<a (S ) 76) +γ 3 I ( =( j),76<a (S ) 80)+γ 4 I ( =( j),a (S )>80)] (14) where I ( =( j)) s an ndcator for whether patent s the ndex case, and a (S ) s the age of subject at the tme of SADT. Smlar generalzatons of γ can be made for other, possbly tme-dependent, covarate nteractons, for both the two-stage and sequental stratfcaton methods. 7. Results Table II shows the estmates (and correspondng 95 per cent CIs) of the hazard rato of recurrence for the two-stage and sequental stratfcaton methods. For the sequental stratfcaton method, results are gven for each of the three sets of matchng varables, and for each of the three threshold values along wth the 10 nearest-neghbor method; n addton, strata sze statstcs (medan, and 5th and 95th percentles) are gven for each combnaton of matchng varables and matchng technque. Usng the two-stage method, SADT s assocated wth an estmated 76 per cent decrease n the hazard of recurrence of prostate cancer (HR=0.24, 95 per cent CI: (0.17, 0.33)). Usng the sequental stratfcaton method, the estmated beneft ranges from a 56 per cent decrease, matchng on locaton and log(psa), and usng a threshold of c =1.0 (HR=0.44, 95 per cent CI: (0.32, 0.61)), to a 76 per cent decrease, matchng on locaton, log(psa), slope of log(psa), and LTP coeffcent, and usng a threshold of c =0.5 (HR=0.24, 95 per cent CI: (0.15, 0.38)). Each estmate from the sequental stratfcaton method gves wder confdence ntervals than that from the two-stage method. Note that for the sequental stratfcaton method usng the thresholdng technque, strata sze decreases as the number of matchng varables ncreases. Ths s expected snce the matchng crtera are more restrctve wth addtonal matchng varables (as long as the threshold value c s constant), hence fewer patents are defned as matches to the ndex Table II. Estmated effect of SADT. Method and matchng varables Strata sze* Hazard rato 95 per cent CI 2576 Two-stage: NA NA 0.24 (0.17, 0.33) Sequental stratfcaton: PSA, locaton 0.2 Threshold 11.0(1.0,112.2) 0.29 (0.19, 0.43) 0.5 Threshold 28.0(3.0,281.4) 0.33 (0.23, 0.47) 1.0 Threshold 73.0(7.2,508.4) 0.44 (0.32, 0.61) 10-NN 11.0(11.0,11.0) 0.28 (0.20, 0.40) PSA, spsa, locaton 0.2 Threshold 3.0(1.0,30.0) 0.26 (0.15, 0.45) 0.5 Threshold 13.0(1.0,159.2) 0.25 (0.17, 0.38) 1.0 Threshold 42.0(4.0,370.8) 0.31 (0.22, 0.45) 10-NN 11.0(11.0,11.0) 0.29 (0.21, 0.41) PSA, spsa, LTP, locaton 0.2 Threshold 1.0(1.0, 6.0) 0.37 (0.16, 0.87) 0.5 Threshold 5.0(1.0,71.8) 0.24 (0.15, 0.38) 1.0 Threshold 28.0(2.0,297.8) 0.31 (0.21, 0.44) 10-NN 11.0(11.0,11.0) 0.27 (0.19, 0.39) Abbrevatons: PSA ndcates matchng on standardzed log(psa+0.1), spsa ndcates matchng on standardzed slope of log(psa+0.1), and LTP ndcates matchng on standardzed long-term phase coeffcent. Strata sze s gven by: medan (5th, 95th percentles). Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569--2580
Table III. Estmaton of nteracton SADT effects. Two-stage method Seq. Strat. (10-NN) Interacton Hazard rato 95 per cent CI p Hazard rato 95 per cent CI p Tme of SADT (after radaton therapy) 0 2.5 years 0.26 (0.16, 0.42) 0.41 0.30 (0.18, 0.50) 0.76 2.5 4 years 0.16 (0.09, 0.26) 0.24 (0.13, 0.46) 4 6 years 0.27 (0.15, 0.47) 0.28 (0.14, 0.54) 6+ years 0.33 (0.15, 0.75) 0.47 (0.17, 1.30) Age (at tme of SADT) 71 years 0.22 (0.13, 0.37) 0.33 0.30 (0.16, 0.55) 0.31 71 76 years 0.23 (0.13, 0.43) 0.19 (0.09, 0.39) 76 80 years 0.18 (0.10, 0.33) 0.25 (0.13, 0.48) 80+ years 0.36 (0.21, 0.62) 0.44 (0.24, 0.79) Predcted PSA (at tme of SADT) 1.5 0.20 (0.09, 0.46) 0.36 0.37 (0.16, 0.86) 0.51 1.5 2 0.14 (0.06, 0.31) 0.22 (0.09, 0.53) 2 3 0.31 (0.19, 0.48) 0.36 (0.22, 0.61) 3+ 0.24 (0.15, 0.40) 0.23 (0.13, 0.39) Predcted slope of PSA (at tme of SADT) 0.4 0.65 (0.31, 1.35) 0.04 0.94 (0.41, 2.15) 0.02 0.4 0.7 0.27 (0.15, 0.49) 0.35 (0.17, 0.70) 0.7 1 0.21 (0.12, 0.37) 0.17 (0.08, 0.36) 1+ 0.19 (0.12, 0.30) 0.26 (0.16, 0.41) Predcted PSA, slope of PSA (at tme of SADT) PSA 2, Slope 0.7 0.30 (0.15, 0.60) 0.06 0.49 (0.22, 1.07) 0.10 PSA > 2, Slope 0.7 0.41 (0.22, 0.78) 0.47 (0.22, 0.98) PSA 2, Slope > 0.7 0.07 (0.02, 0.22) 0.13 (0.05, 0.34) PSA > 2, Slope > 0.7 0.24 (0.16, 0.37) 0.25 (0.16, 0.39) Tme (after SADT) 0 1.5 years 0.31 (0.20, 0.47) 0.14 0.37 (0.24, 0.55) 0.18 1.5 3 years 0.20 (0.11, 0.35) 0.19 (0.11, 0.33) 3 5 years 0.12 (0.06, 0.27) 0.24 (0.07, 0.81) 5+ years 0.28 (0.13, 0.59) 0.66 (0.11, 4.12) P-value for Wald test of equal nteracton effects. Predcted PSA and slope of PSA are based on log(psa+0.1). cases undergong SADT. Although precson decreases, the estmated beneft tself ncreases as the number of matchng varables ncreases (agan keepng the threshold value constant), except for the analyses usng the 0.2 threshold. Conversely, the estmated beneft decreases as the threshold value ncreases (keepng the number of matchng varables constant), except n the case of matchng on all of PSA, spsa, LTP, and locaton. Ths s reasonable snce the matchng crtera are less restrctve wth ncreasng threshold values, therefore less smlar patents are matched to ndex cases. In an unadjusted analyss (smlar to a sequental stratfcaton method whch matched ndex cases to all patents at rsk), SADT would be assocated wth an ncrease n the rsk of recurrence of prostate cancer [21], snce treatment assgnment s not randomzed, and patents who are at rsk for recurrence are more lkely to receve treatment by SADT than patents who are not at rsk. For the nearest-neghbor technque, the estmated beneft and the correspondng precson are roughly constant across varyng sets of matchng varables. Table III shows the SADT nteracton effect estmates (hazard ratos), along wth 95 per cent CIs. Results are presented for the two-stage method and the sequental stratfcaton method usng 10 nearest-neghbors, wth matchng based on locaton, PSA, and slope of PSA. Cutponts for nteracton covarates were chosen so as to ensure roughly equal subgroup szes, whle also allowng for easy nterpretaton. In general, the hazard rato estmates from both the two-stage method and the sequental stratfcaton method are smlar. However, as was the case for the results dsplayed n Table II, the sequental stratfcaton method gves slghtly wder confdence ntervals than the two-stage method. 2577 Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569--2580
For both the two-stage method and the sequental stratfcaton method, a Wald test of the null hypothess of no nteracton effect suggests that there s nsuffcent evdence for sgnfcant nteracton effects between SADT and the tme at whch SADT s gven, the age at whch SADT gven, predcted PSA at the tme of SADT, and tme (p-values range from 0.14 to 0.76). However, both methods suggest sgnfcant nteracton effects between SADT and slope of PSA at the tme of SADT (p =0.04 and p =0.02 for the two-stage and sequental stratfcaton methods, respectvely), and margnally sgnfcant nteracton effects between SADT and PSA and slope of PSA jontly (p =0.06 and p =0.10, respectvely). SADT appears to be most benefcal for patents wth hgher slopes of PSA (at the tme of SADT), and least benefcal for patents wth lower slopes of PSA. When nteracton effects for PSA and slope of PSA are explored jontly, patents wth log(psa) less than 2 and wth slope of log(psa) greater than 0.7 receve the most beneft from SADT, wth an estmated 93 per cent decrease n the hazard of recurrence usng the two-stage method (HR=0.07, 95 per cent CI: (0.02, 0.22)), and an estmated 87 per cent decrease usng the sequental stratfcaton method (HR=0.13, 95 per cent CI: (0.05, 0.34)). 8. Dscusson 2578 We have proposed two methods whch approprately adjust for tme-dependent PSA and slope of PSA n estmatng the effect of SADT on the rsk of recurrence of prostate cancer. The frst s a two-stage method whch uses a lnear mxed model to predct PSA and slope of PSA after EBRT, and then uses a tme-dependent Cox model to estmate the recurrence-free survval beneft of SADT, adjustng for predcted PSA and slope of PSA. Ths method elmnates the ntermedate varable status of the tme-dependent PSA and slope of PSA covarates, thereby allowng the use of a standard Cox regresson analyss. The second method, sequental stratfcaton, attempts to mmc a sequence of randomzed experments, occurrng at the ntaton tmes of SADT. After strata are defned by SADT patents who are matched wth approprate controls, a stratfed Cox model can be used to estmate the beneft of SADT. Although the methods developed n ths paper were targeted at prostate cancer applcatons, they have broader applcablty to other stuatons where the goal s to estmate a treatment effect whch s gven by ndcaton n an observatonal data set. For the two-stage method, the key ngredent s a tme-dependent marker of a dsease whch s assocated wth the outcome of nterest, and whch lends tself to longtudnal modelng. In most stuatons ths marker wll also be a strong determnant of when treatment s ntated. The sequental stratfcaton method s applcable whenever there s a marker that determnes the ntaton of treatment. Whle longtudnal modelng of the marker s not strctly necessary for sequental stratfcaton, t does facltate matchng. The sequental stratfcaton method does requre choces to be made regardng the composton and sze of strata. We recommend matchng on varables that are strongly assocated wth the ntaton of treatment and wth the hazard of the event. In the prostate cancer applcaton, the frst matchng crtera dd not nclude the slope of PSA. As the slope of PSA s probably the sngle most mportant determnant of recurrence and of the decson to start SADT, ths s not deal and may explan the larger varablty n the estmated hazard rato as the strata sze changes. If multple factors defne the strata then there are many possble ways of combnng the factors, some of whch could be desgned to gve more weght to the factors that are consdered more mportant. In the prostate cancer applcaton, we used Eucldean dstance on normalzed covarates, but that could potentally be optmzed. The longtudnal model plays a crucal role, partcularly for the two-stage method. Msspecfcaton of that model could certanly cause bas n the estmate of the treatment effect. The model s explctly used for extrapolaton of marker values, whch makes t even more mportant to have a model that can be trusted. In the prostate cancer applcaton, the large longtudnal data set dd allow for a consderable amount of model buldng before arrvng at a fnal form that fts the data well. In ths applcaton, extrapolaton s somewhat justfed snce, once PSA starts to ncrease, the pattern s qute determnstcally drven and lnear (on a logarthmc scale) for typcal patents. The sequental stratfcaton approach also uses the ft of the longtudnal model, but n a far weaker way. Specfcally, ths model helps defne smlar subjects for each stratum; thus, mnor or even moderate msspecfcatons of the longtudnal model wll not be crucal n the sequental stratfcaton settng. In applyng the two-stage and sequental stratfcaton methods to the prostate cancer data, estmates were qute smlar; ths suggests that msspecfcaton of the longtudnal model s not a major concern. In other applcatons wth less longtudnal data and more heterogeneous patterns, the longtudnal model may be more crtcal. In practce, assumng no unmeasured confounders, the decson to ntate the treatment can only be based on measured varables; however, there are some subtle devatons from ths prncple n the approaches descrbed n ths paper. Frst, the longtudnal model smoothes or nterpolates the observed data, and t s mplctly assumed that t s ths smoothed ft whch determnes the treatment ntaton, rather than the observed data tself. In practce, the person makng the decson to ntate treatment may be mplementng ther own form of smoothng, hence the longtudnal model smoothng and nterpolaton could be smlar to what happens n a clncal settng. A second subtle devaton s that the longtudnal Copyrght 2010 John Wley & Sons, Ltd. Statst. Med. 2010, 29 2569--2580
model s ft to all the data (not just past data), and the predcton ˆP (t) s based on ths ft. Ths can be vewed as an advanced form of smoothng [26], but t could also lead to bas. For the sequental stratfcaton method, a choce must be made regardng the sze of each stratum. Smaller szes gve strata for whch matched patents are more smlar, but ths could also result n a loss of effcency. In ths paper, we utlzed two approaches n determnng the strata, one based on a dstance measure and the other usng a fxed stratum sze. Whle the results dd vary, the dfferences between the two approaches were not substantal. Other methods of choosng strata could certanly be developed. One lmtaton of the work presented n ths paper s related to covarance estmaton for the two-stage model. Covarance estmates for the parameters n model (5) are gven by the usual nverse nformaton matrx values; however, such an approach does not take nto account the varance of the estmated log(psa) and slope of log(psa) quanttes from model (4). Thus the standard errors from the two-stage method may be underestmates. One soluton to ths problem would be to use the bootstrap for nference (.e. by teratvely fttng models (4) and (5) to bootstrap samples of the data). Another soluton would be to use a jont longtudnal-survval modelng approach [27, 28], whch s flexble but consderably more computatonally complex. Jont modelng could also elmnate some of the possble bas n the parameter estmates when the longtudnal and survval analyses are performed separately. In ths paper, we have focused on the relatve hazard as the summary measure of the effectveness of the SADT treatment. In practce, SADT s often thought of as delayng recurrence by a year or more, or by stretchng the tme to recurrence by a factor of two or more. Both methods presented n ths paper could be adapted to provde these knds of summary measures of the effect of SADT. For example, model (7) n the sequental stratfcaton method could be replaced by an accelerated falure tme model. The correspondng modfcaton could potentally be more complex for the two-stage method, but a tme-dependent accelerated falure tme model could be used n place of equaton (5). Acknowledgements Ths research was partally supported by NIH grants CA083654 and CA110518. References 1. Shpley WU, Thames HD, Sandler HM, Hanks GE, Zetman AL, Perez CA, Kuban DA, Hancock SL, Smth CD. Radaton therapy for clncally localzed prostate cancer: a mult-nsttutonal pooled analyss. Journal of the Amercan Medcal Assocaton 1999; 281(17): 1598--1604. 2. Agarwal PK, Sadetsky N, Konety BR, Resnck MI, Carroll PR. 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