Alma Mater Studiorum Università di Bologna DOTTORATO DI RICERCA IN METODOLOGIA STATISTICA PER LA RICERCA SCIENTIFICA

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1 Alma Mater Studorum Unverstà d Bologna DOTTORATO DI RICERCA IN METODOLOGIA STATISTICA PER LA RICERCA SCIENTIFICA Cclo XXVII Settore Concorsuale d afferenza: 13/D1 Settore Scentfco dscplnare: SECS-S/02 STATISTICAL METHODS FOR DEALING WITH SELECTIVE CROSSOVER IN RANDOMISED CONTROLLED TRIALS Canddata: Dott.ssa Sara Balduzz Coordnatrce Dottorato Prof.ssa Alessandra Luat Relatrce Prof.ssa Rossella Mglo Correlatore Prof. Roberto D Amco Esame fnale anno 2016

2 A tutt coloro a cu voglo bene

3 Table of contents Table of contents Acknowledgements Introducton Overvew of the thess v Chapter 1 Revew of the lterature Methods Results Adjuvant/Neoadjuvant settng Metastatc settng Revew fndngs 10 Chapter 2 Methods for dealng wth selectve cross over Naïve methods Intenton-to-treat analyss (ITT) Censored analyss Treatment as a tme-varyng covarate More complex methods Inverse Probablty of Censorng Weghtng (IPCW) analyss Loeys and Goetghebeur estmator Rank Preservng Structural Falure Tme Models (RPSFTM) 21 Chapter 3 Smulaton study desgn Scenaros descrpton Performance measures 27 Chapter 4 Results of the smulaton study 29 Chapter 5 Dscusson, conclusons and hnts for future research Dscusson Conclusons and hnts for future research 41 References 42

4 Acknowledgements Frst of all, specal thanks go to Prof. Rossella Mglo and Prof. Roberto D Amco, for havng gven me the opportunty to work on ths project: I hope to keep on workng wth you, your support, gudance, patence and understandng on ths and many other projects n the future. I want to thank Dr. Elsabetta Petracc, who offered me her precous help from the begnnng. Elsabetta, Lnda, Govann, Elena, Aranna: t was an honour (and pretty fun!) to share ths path wth you. My lfe at work, whch I love, would not be the same wthout my beautful colleagues: Cnza, Elena, Nka, Roberto V. Thank you for your support and for lstenng to me every tme I need t. An mportant thank goes to my parents I know you are my bggest fans. Thanks to Alberto, especally for lettng me buy a desk and a shelf: wthout them t would have been much more dffcult to wrte ths thess. Last four years have surely been the most ntense ones of my lfe. I would lke to thank all the people who walked by my sde, at least for a moment, durng ths perod. I am happy because I can understand how lucky I am.

5 Introducton The evdence to support the effcacy and safety of a drug derves from randomsed controlled trals (RCTs) [1]. The ethcal bass of a RCT s provded by the prncple of equpose, statng that the randomzaton s allowed when there s genune uncertanty on the actual beneft of the nterventon under study. An RCT deally should permt an unbased estmaton of the treatment effect (e.g., overall survval (OS) or dsease-free survval (DFS)), through the comparson between an expermental treatment and a comparator, admnstrated n two separate arms. However, sometmes patents may be offered the possblty to cross over from one arm of the tral to the other, a phenomenon usually called selectve crossover (SCO), n order to dstngush t from the studes n whch crossover s planned. Other terms to refer to ths swtch are dropn or cross-n [2]. It can occur as a consequence of the dffuson of (un)favourable results of one treatment that s beng compared, e.g. from nterm analyss or from concurrent studes. In both cases the equpose prncple s challenged. If so, the nvestgators may offer patents the opportunty to cross over to the arm where the more promsng treatment s admnstered. Ths thess consders treatment crossover as the swtchng of patents randomsed to the control group of an RCT on to the expermental treatment, at a certan pont after randomsaton. The occurrence of SCO breaks the randomzaton process and may gve rse to problems n the data analyss and nterpretaton of results. In fact, n presence of SCO, the Intenton to Treat (ITT) analyss, that s the analyss consderng groups as randomsed, wll gve an unbased estmate of the expermental treatment assgnment effect, but that effect wll also

6 nclude the one of the expermental treatment admnstered to patents n the control group who swtched. So the results of ths type of analyss may not reflect the actual effcacy of the expermental nterventon, and adjustments to allow for crossover are needed. My personal nterest n ths topc derves from the work that I conducted for my master degree thess, when I presented the results of a systematc revew on the effcacy and safety of a drug,.e. trastuzumab, for the treatment of early and metastatc breast cancer. The work of my colleagues and me has been publshed n two artcles [3,4]. In both adjuvant/neoadjuvant and metastatc settngs, we observed that more than half of the ncluded studes let patents n the control arm cross over to the trastuzumab arm at a certan pont after randomsaton. Whle n the metastatc settng crossover was permtted generally after progresson, makng t dffcult to estmate the actual beneft of trastuzumab on OS, n the adjuvant/neoadjuvant settng t was permtted before dsease progresson, makng t dffcult to analyse and nterpret both OS and DFS results. The present work ams to: () assess the prevalence of SCO n RCTs assessng the effcacy and safety of bologcal and hormonal therapes for breast cancer and publshed n the scentfc lterature; () dentfy the reported statstcal methods used to handle crossover, n partcular when the effect of the expermental treatment s a tme-to-event outcome; () assess whether dfferent statstcal methods provde dfferent results and nterpretatons. v

7 Overvew of the thess Chapter 1 presents the revew of the medcal scentfc lterature carred out n order to assess the prevalence of the SCO n the feld of breast cancer, and the approaches adopted to handle t. Detals on the methods adopted to conduct the revew are gven, along wth the presentaton of the results, dstngushed for adjuvant/neoadjuvant and metastatc settng. The Chapter closes wth a dscusson regardng the fndngs of the revew and the needs for further research. Chapter 2 attempts to collect the more relevant approaches that have been consdered n lterature to address treatment crossover. Along wth ITT analyss, other naïve methods nclude censorng patents at the tme they cross over, or excludng them from the analyss. These methods wll lead to based results f the swtchng process s not random, because of selecton bas. If patents who swtch are pcked completely at random, ther excluson from the analyss would not result n bas, but n the loss of power and ncrease of the uncertanty n the results. Instead, a swtchng process dependng on patents characterstcs mples nformatve censorng, and censorng or excludng from the analyss patents who cross over wll lead to based results. Another method descrbed n lterature s to consder the treatment as a tme-varyng covarate. Ths approach may be subject to selecton bas, as groups may no longer be balanced after a patent s censored or excluded, and bas s lkely f a patent s probablty of swtchng treatment s related to ther underlyng prognoss. Morden et al. [5] studed some statstcal approaches for dealng wth SCO. They consdered naïve methods, as well as more complex ones, such as Robns and Tsats s Rank Preservng Structural Falure Tme Model (RPSFTM) [6]. The key assumpton, though not always reasonable, of the RPSFTM method s the so-called common treatment v

8 effect assumpton: t s assumed that swtchng patents experence the same treatment effect, from the tme they start takng the expermental treatment, as patents randomsed to the expermental group from the begnnng. Another approach consdered n the paper by Morden et al., and evaluated n ths thess, s the one descrbed n Loeys and Goetghebeur [8], who present a method for calculatng the actual treatment effect n stuatons where all patents take ther allocated treatment n one group, and complance s assumed as all-or-nothng n the other. So, f a patent n ths arm cross over, the swtch s assumed to have happened rght after the randomsaton, and the patent s assumed to have only receved the treatment he/she swtched onto and not the treatment he/she was randomsed to. The nverse probablty of censorng weghts (IPCW) method, ntroduced by Robns and Fnkelsten [9], not evaluated n the paper by Morden et al., s also consdered n the present work. The IPCW method do not assume the common treatment effect, but ts fundamental assumpton s the no unmeasured confounders, that s the requrement of data on all covarates that mght nfluence the crossover. All these methods are presented n Chapter 2 of ths thess. In order to assess the potental bas assocated wth the methods descrbed n Chapter 2, the actual effect of an nterventon under study needs to be known, so a smulaton study was performed. The dea was to reproduce a two-arm RCT, smlar to one of the trals emerged from the revew of the lterature of Chapter 1: ths work focuses the attenton on the adjuvant/neoadjuvant settng, where the crossover s usually permtted before dsease recurrence. Chapter 3 detals the smulaton study desgn and Chapter 4 presents the results. The man ssues rased from the present work are dscussed n Chapter 5, along wth potental future developments of the research n ths feld. v

9 Chapter 1 Revew of the lterature In ths chapter t s presented the revew of the scentfc lterature conducted n order to evaluate the prevalence of the SCO among trals evaluatng the effcacy and safety of the bologcal drugs and hormonal therapes for breast cancer, and the statstcal methods used to handle t. 1.1.Methods RCTs (phase III only) publshed between January 2000 and June 2015 n the Annals of Oncology (AoO), Journal of Clncal Oncology (JCO), Journal of the Natonal Cancer Insttute (JNCI), Lancet (L), Lancet Oncology (LO), New England Journal of Medcne (NEJM), assessng the effcacy and safety of bologcal drugs and hormonal therapy n breast cancer patents, were searched. We excluded chemotherapy agents because, n those years we selected as study tme perod, there were few new agents grantng marketng authorsaton: last nnovatve agents doxorubcn pegylated and pacltaxel were approved by the European Medcne Agency n 2000 [9]. The keywords used for the research were random*, breast and cancer wth the restrcton that all words had to be present n the ttle or n the abstract. Trals were dstngushed by the therapy settng they consdered,.e. adjuvant/neoadjuvant or metastatc. The prevalence was calculated as the percentage of trals n whch crossover has occurred out of the total number of trals reflectng the ncluson crtera. For trals n whch crossover has occurred, the followng characterstcs were recorded: prmary end pont, reason for crossover (.e. nterm analyss), number of patents totally 1

10 randomzed, number and type of patents allowed to cross over, statstcal methods used to analyse results after the crossover. When reported, characterstcs of patents who crossed over n respect to patents who remaned n the arm of orgnal allocaton was also recorded. If a tral reported the results for the same outcome (.e. for overall survval (OS) or dsease/progresson-free survval (DFS/PFS)) n terms of hazard ratos (HR) from more than one type of analyss, the rato of HRs (RHR), along wth ts 95% confdence nterval (95%CI) was calculated n order to compare them Results Fgure 1. Flowchart of the bblographc research 1706 references From 6 journals: NEJM, JNCI, Lancet, Lancet Oncology, JCO, Annals of Oncology Perod lmts: January 2000-June references excluded because they consdered non-pertnent topcs or because they consdered phase II studes 351 references reportng the results of phase III or uncertan phase RCTs full text retreved 202 references excluded because the nterventons were nether bologcal nor hormonal 128 references reportng the results of 85 phase III RCTs: - 42 n a metastatc settng; - 43 n an adjuvant/neoadjuvant settng Crossover was present n 14 studes: - 10 out of 42 (24%) RCTs n a metastatc settng; - 10 out of 43 (23%) RCTs n an adjuvant/neoadjuvant settng 2

11 The flowchart of the lterature research s presented n Fgure 1. Out of 1706 references totally retreved (AoO 434; JCO 553; JNCI 192; L or LO 104; NEJM 423), 1355 were excluded because they dd not consder treatments for breast cancer or because they consdered phase II studes. The full text of the remanng 351 references was retreved: 202 were then excluded because the nterventons were therapes nether bologcal nor hormonal and 21 because the study phase was unclear. The remanng 128 references reported the results of 85 RCTs, 43 of whch enrolled women n early breast cancer and 42 metastatc. Crossover was present n 20 RCTs (23.5%) equally dstrbuted across settngs: ten n adjuvant/neoadjuvant settngs (23.3%) and ten n metastatc settng (23.8%). When SCO occurred, the methods used to analyse data were: 1. Intenton To Treat (ITT) analyss; 2. Censored analyss; 3. Inverse Probablty of Censorng Weghtng (IPCW) analyss. In the ITT analyss, controls that crossed over are analysed as belongng to the control arm, despte the fact that they crossed over to the treatment nterventon. In the censored analyss, follow-ups of controls that swtched to the nterventon arm are censored at the tme when the crossover occurred. The IPCW analyss allows the estmaton of the mssng follow-ups of those controls who swtched arm by usng the nformaton comprsed n follow-ups of those controls who nstead decded aganst crossng over and who were smlar n terms of prognostc factors to ther counterparts [8]. 3

12 Adjuvant/Neoadjuvant settng Characterstcs of the ten RCTs (IMPACT, HERA, MA17, NSABP-B-33, BIG 1-98, NOAH, BCIRG-006, TEAM, B31, N9831) assessng effcacy and safety of treatments n the adjuvant/neoadjuvant settng n whch SCO occurred are shown n Table 1.a. The studes B31 and N9831 were analysed and presented jontly, because they evaluate the same nterventon, admnstered on a smlar schedule. Fve trals evaluated bologcal drugs and fve hormonal therapes. Nne trals had DFS as prmary outcome, one (IMPACT) had clncal tumour overall response. OS was always a secondary outcome. Two trals (IMPACT, NOAH) consdered neoadjuvant strateges, and these are the trals wth the lowest number of randomzed patents (330 and 334 respectvely). In fve bg trals (BIG 1-98, HERA, MA17, B31+N9831, wth 8010, 3401, 5170, and 4390 randomzed patents respectvely), crossover was allowed after postve results obtaned at a pre-planned nterm analyses. HERA and MA17 were the trals wth the hghest percentage of patents crossng over, 52% and 61% respectvely. In three trals (IMPACT, NSABP-B-33, TEAM) patents were permtted to cross over after results from other smlar studes were publshed leadng to a protocol amendment. For two trals (BCIRG- 006, NOAH) the motvaton for crossover was not reported. These trals had the lowest percentage of patents crossed over, 2.1% and 16% respectvely. Other two trals (IMPACT, TEAM) dd not clearly report the percentage of patents who crossed over. In two trals (BIG 1-98, HERA) the crossover was allowed only to patents who dd not experence recurrence yet (HERA also requred an adequate left ventrcular ejecton fracton); n the other studes, the crossover seemed to be offered to all patents n the control arm. All the studes presented ITT analyses; two studes (BIG 1-98, HERA) conducted censored analyss; two studes (BIG 1-98, MA17) conducted the IPCW analyss. The only study whch conducted all three analyses s the BIG 1-98: n 2009 the results of the ITT and 4

13 censored analyses were publshed and n 2011 another paper reported an update of the ITT analyss and the IPCW analyss. Only two trals (HERA, MA17) reported the characterstcs n terms of age, prevous therapy, menopausal status, hormone-receptor status, and lymph nodal status of patents of the control arms who crossed over to the treatment arms, along wth the characterstcs of patents who dd not. In both cases, patents n the SCO cohort compared wth patents remanng n the control arm were more lkely to be younger and have hormone-receptorpostve dsease. In table 2 are reported the results, expressed n HR (95%CI), for the three trals reportng censored or IPCW analyss, for DFS and OS respectvely. The ITT analyss always seemed to be the more conservatve one, although for the BIG 1-98 tral the RHRs both for OS and DFS were not statstcally sgnfcant. OS results dervng from censored or IPCW analyses were more dstant from the ones obtaned wth the ITT analyss n respect to DFS results Metastatc settng Characterstcs of the ten RCTs (Slamon 2001, Mourdsen 2003, TANDEM, AVADO, EGF104900, RIBBON-1, NCT , NCT , NCT , CONFIRM) assessng effcacy and safety of treatments for metastatc breast cancer n whch SCO occurred are shown n Table 1.b. All the trals but three (Mourdsen 2003, NCT , CONFIRM) evaluated bologcal drugs. All the trals consdered PFS as prmary outcome and OS as secondary outcome. All the trals protocols permtted crossover to the expermental treatment arm for a patent n the control arm who experenced progresson. RIBBON-1 was a four-arms tral, two controls and two expermental arms; after progresson, patents n both the control arms were permtted to swtch to the respectve expermental arm. A partcular case s represented by the Mourdsen 2003 tral, where two dfferent hormonal therapes were 5

14 compared and n whch patents at progresson were permtted to swtch to the other arm, rrespectve of the arm n whch they were ntally allocated. The percentages of swtched patents were over the 40% n all the studes but three (AVADO 36%, NCT %, CONFIRM 2%). All the studes reported the ITT analyss and two (EGF104900, Mourdsen 2003) conducted censored analyss. Only one study (EGF104900) reported the man characterstcs age, performance status, pror therapes, hormone-receptor status of the control arm by crossover status (crossover versus non-crossover), wthout statstcally evaluatng the dfferences between the two groups. In table 3 are reported the OS results, expressed n HR, for the two trals reportng censored analyss. Mourdsen 2003 calculated the medan tme to death n each group, from whch t was possble to estmate the HR; too less nformaton was provded to calculate the RHR. The HRs from the ITT and the censored analyses reported by the EGF tral do not seem to dffer sgnfcantly. 6

15 Table 1.a Characterstcs of the trals n whch SCO occurred Trals assessng effcacy of treatments n an adjuvant/neoadjuvant settng Study BD/ HT Journal Year of publcaton* Year of reported SCO 1 IMPACT HT JCO Prmary End Pont Clncal OR tumour Motvaton for SCO After ATAC results Pts totally randomzed/ to be enrolled Pts crossed over ITT Cens IPCW 330/330 Not reported 2 HERA BD NEJM,L,LO DFS Interm analyss 5102/ /1698 (52%) 3 MA17 HT JCO,AoO,NEJ M DFS 4 NSABP-B-33 HT JCO DFS 5 BIG 1-98 HT NEJM,JCO,LO DFS Interm analyss, unblndng After MA17 (nterm analyss) results Planned nterm analyss 5187/ /2587 (61%) 1598/ /779 (44%) 8010/ /2459 (25%) 6 NOAH BD L DFS (EFS) Not reported 235/232 19/118 (16%) 7 BCIRG-006 BD NEJM DFS Not reported 3222/ /1073 (2.1%) 8 TEAM HT L DFS After IES results 9779/9300 Not reported B31+N9831 BD NEJM,JCO DFS Planned nterm analyss 4390/ /2018 (20%) * If more than one publcaton refers to the same study, the year of frst publcaton and the year of last publcaton are reported Pts=patents NEJM=New England Journal of Medcne; JNCI=Journal of the Natonal Cancer Insttute; L=Lancet; LO=Lancet Oncology; JCO=Journal of Clncal Oncology; AoO=Annals of Oncology DFS=Dsease Free Survval; OR=Objectve Response HT=Hormonal therapy; BD=Bologcal Drug Neoadjuvant trals 7

16 Table 1.b Characterstcs of the trals n whch SCO occurred Trals assessng effcacy of treatments n a metastatc settng Study BD/ HT Journal Year of publcaton* Year of reported SCO Prmary End Pont Motvaton for SCO Pts totally randomzed/ to be enrolled Pts crossed over ITT Cens IPCW 1 Slamon 2001 BD NEJM PFS After progresson 469/ /234 (66%) 2 Mourdsen 2003 HT JCO PFS After progresson 907/ /458 (51%) + 226/458 (49%) 3 TANDEM BD JCO PFS After progresson 208/208 73/104 (70%) 4 AVADO BD JCO PFS After progresson 736/705 83/231 (36%) 5 EGF BD JCO PFS After progresson 296/296 77/145 (53.1%) 6 RIBBON-1 BD JCO PFS After progresson 1237/ /206 (54.4%) + 105/207 (50.7) 7 NCT BD JCO PFS After progresson 432/430 77/215 (36%) 8 NCT BD JCO PFS After progresson 519/ /258 (62%) 9 NCT HT NEJM PFS After progresson 695/ /345 (41%) 10 CONFIRM HT JNCI PFS After progresson 736/834 8/374 (2%) * If more than one publcaton refers to the same study, the year of frst publcaton and the year of last publcaton are reported Pts=patents NEJM=New England Journal of Medcne; JNCI=Journal of the Natonal Cancer Insttute; L=Lancet; LO=Lancet Oncology; JCO=Journal of Clncal Oncology; AoO=Annals of Oncology PFS=Progresson Free Survval; EFS=Event Free Survval HT=Hormonal therapy; BD=Bologcal Drug Neoadjuvant trals 8

17 Table 2 DFS and OS results, expressed n HR (95%CI), for the three trals assessng effcacy and safety of treatments, n an adjuvant/neoadjuvant settng, reportng censored or IPCW analyss results from the most recent publcaton for each tral are reported. RHRs (95%CI) were calculated n order to compare the results from dfferent type of analyses (ITT as reference analyss). Study DFS Type of analyss ITT Censored IPCW RHR HERA (2011) 0.76 (0.66;0.87) 0.69 (0.59;0.79) 0.91 (0.66; 0.87) BIG 1-98 (2011) 0.86 (0.78;0.96) 0.82 (0.74;0.92) 0.95 (0.82; 1.11) MA17 (2012) 0.68 (0.56;0.83) 0.52 (0.45;0.61) 0.76 (0.60; 0.98) OS HERA (2011) 0.85 (0.70;1.04) 0.53 (0.44;0.65) 0.62 (0.47; 0.82) BIG 1-98 (2011) 0.87 (0.77;0.99) 0.79 (0.69; 0.90) 0.91 (0.76; 1.09) MA17 (2012) 0.99 (0.79;1.24) 0.61 (0.52;0.71) 0.62 (0.47; 0.81) Table 3 OS results, expressed n HR (95%CI), for the two trals assessng effcacy and safety of treatments, n a metastatc settng, reportng censored or IPCW analyss results from the most recent publcaton for each tral are reported. RHRs (95%CI) were calculated n order to compare the results from dfferent type of analyses (ITT as reference analyss). Study Type of analyss ITT Censored IPCW EGF (2012) 0.74 (0.57;0.96) 0.80 (0.56;1.12) 1.08 (0.70; 1.67) Mourdsen 2003 (2003) 0.88 (p=0.53) 0.71 (nr) - nr = not reported RHR 9

18 1.3. Revew fndngs Between January 2000 and June 2015, one out of fve RCTs assessng effcacy of nnovatve bologcal drugs and hormonal therapy for breast cancer permtted patents to cross over at a certan pont durng the course of the study. From a clncal standpont, early dscontnuaton of RCT due to unequvocal observed beneft, harm or futlty are always justfed by ethcal ssues. An early nterrupton for beneft leads to stoppng further recrutment n a potental nferor arm, and patents randomzed to the control arm can opt to cross over to receve the expermental treatment. However, from a methodologcal standpont, ths approach leads to uncertantes surroundng the true magntude of the actual effect. The scenaro mght be dfferent when consderng the early or the advanced dsease settng. Indeed, t s well accepted that patents wth metastatc dsease, at the tme of dsease progresson, are gven the opportunty to swtch to the arm wth the more promsng therapy. Ths approach has no effect on the earler measures of treatment effect such as PFS, whch represents the prmary outcome n all the found studes. On the other hand, the crossover can defntvely preclude the possblty to demonstrate an OS beneft, whch s often consdered the ultmate test of effcacy. The stuaton s even more crtcal n the adjuvant settng, where SCO s generally offered to patents stll free of dsease recurrence, thus affectng the clear nterpretaton of both DFS and OS. The scentfc communty has to deal wth the ethcal mperatve to offer the best treatment to those patents who decde to enter a clncal tral, and wth the need of obtanng the most clean evdence to be appled n the whole populaton. Indeed, all the nterventons whch rase uncertantes n data nterpretatons can delay the full acceptance of a clncally relevant nterventon. 10

19 All the studes ncluded n the present analyss were always analyzed followng the ITT approach, whch s the tradtonal analyss. Accordng to the ITT prncple, patents are analysed n ther assgned treatment arm regardless of the actual treatment receved. Therefore, when a substantal fracton of the patents from the less actve treatment cross to the more effectve treatment, the net beneft of the latter tends to be reduced. Censored analyss can be used to account for dsruptons n treatment allocaton. Ths approach censures patents after crossover, and can be more nformatve on the real performance of the expermental arm. Only 2/10 adjuvant and 2/10 metastatc studes ncluded n the present analyss reported the censored analyss. The censored analyss was assocated wth an ncreased beneft for the expermental arm as compared to the ITT analyss. However, censored analyss can ntroduce bas tself, n partcular when censored patents are more lkely or less lkely to experence the event than uncensored patents (nformatve censorng). In both HERA and MA17 trals, patents n the SCO cohort were more lkely to be younger and have hormone-receptor-postve dsease, as compared to patents remanng n the control arm. One of the most recent type of analyss, IPCW [8], whch accounts for prognostc factors, was rarely used (2/10 adjuvant trals). Smlarly to the censored analyss, IPCW analyss led to results whch favour the expermental arm n respect to the ITT analyss, whch nstead tends to dlute the treatment effect. By the way, the adjustment made by the IPCW analyss s vald only f the varables whch determne crossover are known and measurable, as ponted out by Rmaw et al. [10]. Ths s not always possble, leavng the choce to mply ths type of analyss doubtful. The man lmtaton of the research presented n ths chapter s the fact that only artcles that appeared wthn a 15-year perod were searched n sx selected medcal journals, reasonng that these ones publshed most of the RCTs n breast cancer. It would be helpful to further nvestgate the phenomenon wth a more comprehensve lterature research. The 11

20 prevalence and mpact of SCO n felds other than breast cancer should also be studed. The magntude and drecton of the potental bas ntroduced by the SCO needs to be clearly evaluated, as well as the mpact on the results for dfferent effect szes when results concern safety and when the reason for swtchng depends on a combnaton of prognostc factors. However, the lack of an approprate reportng s not a trval concern. In 2005, Montor et al. publshed a systematc revew of RCTs stopped early for beneft [11], whch mght lead to SCO, n whch they hghlghted the lack of adequately reported such an mportant nformaton as the motvaton to stop the tral. More attenton should be pad by the authors also n the reportng of the characterstcs of patents to whom the crossover s offered. Consderng the ncreasng frequency of the phenomenon, the Consoldated Standards of Reportng Trals (CONSORT) statement [1] could be modfed n order to recommend the specfcaton of that nformaton, fundamental for a better understandng and nterpretaton of the results of the tral. Ths chapter clearly ponts out that the treatment crossover phenomenon s qute common n breast cancer trals. Dfferent methods may lead to dfferent results and nterpretatons, but there s stll no consensus on the approprate approach to deal wth SCO. 12

21 Chapter 2 Methods for dealng wth selectve cross over Chapter 1 hghlghted the need to fnd approprate strateges to deal wth treatment crossover. The present chapter descrbes the most relevant exstng statstcal approaches to handle SCO. These methods wll be then assessed through smulaton, as explaned n Chapter 3, and the results of the smulatons wll be presented n Chapter Naïve methods Intenton-to-treat (ITT) analyss As emerged from the medcal lterature revew presented n Chapter 1, all authors use an ITT analyss. Accordng to ths approach, patents are analysed dependng on whch treatment group they were ntally allocated to, and data from all randomsed patents s used. ITT analyss results should always be reported, as ths method reflects the desgn of the study. By the way, f the expermental treatment s actually superor to the control, and a fracton of patents have crossed over from the control to the expermental arm, the ITT analyss wll tend to dlute the magntude of the expermental treatment effect estmate, makng t appear more smlar to the control effect. A Cox proportonal hazard model s usually ftted n order to estmate the treatment effect. 13

22 Censored analyss A possble approach s to censor patents at the tme of crossover; ths method s used n two out of ten breast cancer trals where SCO has occurred, as seen n Chapter 1. Groups may actually no longer be balanced after patents are censored, so ths type of analyss may be exposed to selecton bas, and ths s partcularly true f patents probablty of crossng over s related to ther underlyng prognoss Treatment as a tme-varyng covarate Although no study from the revew n Chapter 1 has reported t, an approach to deal wth SCO s consderng the treatment as a tme-varyng covarate. It s an extenson of the Cox proportonal hazards model: λ (t) = λ 0 (t) exp(βx (t)) where λ 0 (t) s the baselne hazard functon and X (t) takes a value of zero when a patent s on control and 1 when on expermental treatment. Ths approach also may be subject to selecton bas f SCO s related to prognoss More complex methods Inverse Probablty of Censorng Weghtng (IPCW) analyss Ths method s rarely used n RCTs assessng the effcacy of bologcal drugs and hormonal therapes for breast cancer, and n partcular only n the adjuvant settng, as descrbed n Chapter 1. Although n another clncal feld, Robns and Fnkelsten [8] frstly used the IPCW approach to try to overcome noncomplance ssue. It can be thought as a method to mprove the censorng strategy descrbed n paragraph 2.2. Instead of smply censorng 14

23 patents, the covarates of censored patents are consdered n order to try to remove selecton bas. IPCW method creates a scenaro of mssng follow up data by censorng the follow up of each subject at the tme of crossover. The weght n the analyss for tme perods after crossover s then equal to 0. For subjects n the control wth smlar characterstcs that do not cross over, IPCW method assgns bgger weghts to re create the populaton that would have been observed wthout crossover. So, a patent n the control arm who remans n the control arm wll be assgned a weght > 1 f other patents wth smlar characterstcs crossed over. Weghts are based on factors affectng a patent s decson to cross over. The method reles on the assumpton usually called no unmeasured confounders n the censorng process. Condtonal on the treatment arm R and on the recorded hstory V (t) of the tme-dependent covarates V(t), the cause-specfc hazard of censorng C at tme t does not further depend on the possbly unobserved falure tme T: λ C (t V (t), R, T, t < T) = λ C (t V (t), R, t < T) where λ C R T V (t) = cause-specfc hazard of censorng C = treatment arm = possbly unobserved falure tme = { V (x); 0 x < t } s the recorded hstory up to tme t V(x) s a vector of all measured tme-dependent factors for falure tme recorded at tme x. 15

24 Ths assumpton specfes that, wthn a treatment arm, patents censored at tme t have the same dstrbuton of falure tme as those uncensored at tme t wth the same recorded hstory. Gven the no unmeasured confounders assumpton, the IPCW estmators based on the tme-dependent prognostc factors can be constructed as follow. Tme-dependent Cox proportonal hazards models are used to estmate the treatmentspecfc hazards of censorng condtonal on tme-dependent prognostc factors (reason for censorng can dffer between arms). λ C (t V (t), R, t < T) = λ 0R (t) exp (α R V (t)) IPCW Kaplan-Meer (KM) estmator for falure dffers from the ordnary KM estmator for falure by weghtng the contrbuton of a subject at rsk at tme t by the nverse of an estmate of the condtonal probablty of havng remaned uncensored untl tme t, based on the ft of ths model. By denotng α R as Cox partal lkelhood estmate of α R n treatment arm R X = mn(t, C) Y(u) = I (x u) as the at rsk ndcator τ = I (T = x) as the falure ndcator (1 = falure; 0 = censored) an estmate of the condtonal probablty of patent havng remaned uncensored untl tme t s provded by the tme-dependent extenson of KM product lmt estmator of censorng 16

25 17 j j j R R t X j j R j R V X V X t K 0,, ; ' )}] ( )exp{ ˆ ( ˆ [1 ) ( ˆ where )] ( ) ( )} ( exp{ ˆ ) /[ (1 ) ( ˆ 1 ' n j j R j j R R R I X Y X X V s the Cox estmator of the baselne hazard functon for censorng λ 0R n arm R. The subject-specfc weght can be defned ) ( ˆ ) / ( ˆ ) ( ˆ 0 t K t K t W V Where ) ( ˆ 0 t K s the usual treatment arm specfc KM estmator of the probablty of beng uncensored by the tme t n treatment R. So the IPCW KM estmator for falure n treatment arm r, r {0, l}, dffers from the ordnary KM estmator for falure only n that the contrbuton of a subject at rsk at any tme X s weghted by the subject-specfc weght ) ( ˆ X W. The IPCW KM estmate of the treatment arm specfc margnal probablty of remanng alve through tme t s } ; { 1 ) ( ) ( ) ( ) ( ) ( 1 ) ( ˆ t X n k k k k T r R I X W X Y r R I X W r t S where ) ( ) ( r R I X W s the estmate of the number of subjects n arm r who would have been observed to fal at tme X n the absence of any censorng and n k k k k r R I X W X Y 1 ) ( ) ( ) ( s the estmate of the number of subjects n arm z who would have been at rsk at tme X n the absence of any censorng.

26 Sˆ T ( t r) estmates the probablty S T ( t r) of survvng wthout falure untl tme t n the absence of censorng. It s possble to compare the margnal survval n the two arms by usng the Cox proportonal hazards model λ T (t R) = λ 0 (t)e βr The IPCW Cox partal lkelhood score for β s U( ) Wˆ ( X ) [ R n j1 Y ( X n j1 j j Y ( X ) Wˆ j ) Wˆ ( X) Z e j j ( X) e R j R j ] The estmatng equaton U( ) 0 gves a consstent and asymptotcally normal estmator of parameter β. The major concerns on ths method are the no unmeasured confounders assumpton, whch s untestable, and the fact that the IPCW approach cannot work f there are any covarates whch ensure (that s, the probablty equals 1) treatment crossover wll or wll not occur Loeys and Goetghebeur estmator An approach to estmate the real treatment effcacy n stuatons where all patents take ther allocated treatment n one arm of the tral and complance s all-or-nothng n the other arm s reported by Loeys and Goetghebeur [7]. All-or-nothng means that, f a patent allocated to that arm crosses over to the other one, the crossover s assumed to have 18

27 happened at the very begnnng, rght after the randomsaton, and the patent s assumed to have only receved the treatment he/she swtched onto. The authors present the method by consderng that all patents n the control arm comply fully, and patents n the expermental arm may ether comply fully (compler) or not at all (non-compler). Indvduals n the control arm are also classfed as complers and noncomplers accordng to how they would have behaved f they had been randomzed to the expermental arm. Ths thess consders the opposte case n whch all patents n the expermental arm comply fully, and patents n the control arm may ether comply fully or not at all. The proporton of noncomplers, α, s assumed to be the same n both arms due to randomsaton; ths assumpton s often called excluson restrcton assumpton. The probablty of survval to tme t s denoted by S n0 (t) and S c0 (t) for noncomplers and complers randomzed to control, and S n1 (t) and S c1 (t) for noncomplers and complers randomzed to the expermental arm. For each arm j = 0, 1: S j (t) = α*s nj (t)+(1 α)*s cj (t) Let us assume that allocaton to nterventon has no effect on noncomplers and has hazard rato ψ for complers: S n0 (t) = S n1 (t) S c0 (t) = S c1 (t) 1/ψ S c0 (t) ψ = S c1 (t) The complance-adjusted nterventon effect estmate, ψ, s obtaned by usng Kaplan Meer estmates of Sn 0 (t) and Sc 0 (t) to estmate the survvor functon n the expermental arm: S (t 1 ψ) = α *S n0 (t)+(1 α )* S c0 (t) 1/ψ A value of ψ s found at whch ths quantty matches the observed survval n the expermental arm. 19

28 Defnng Λ 1 (t ψ) = logs (t 1 ψ), G 1 (ψ) = [Λ 1 (T j ψ) δ j ] j where the sum s over all ndvduals n the expermental group, T j s the censorng/event tme for the jth ndvdual, and δ j s the falure ndcator for the jth ndvdual. G 1 (ψ) can be thought of as the dfference between observed and expected events n the expermental arm, based on predctons from the control arm f the hypotheszed ψ s correct. The value of ψ that represents the fnal estmate of the complance-adjusted nterventon effect s found by solvng G 1 (ψ) = 0. The authors demonstrate that where s(ψ) 2 = 2 Λ 1 (T j ψ) j. G 1 (ψ)~n{0, s(ψ) 2 } Confdence lmts for ψ are found, as descrbed n Km and Whte [12], by solvng G 1 (ψ) = ±z crt s(ψ), where z crt s the crtcal value for the approprate sgnfcance level. The estmaton of the pont estmate and confdence lmts can be obtaned usng a loop employng nterval bsecton. The target value s frstly set as ether 0, z crt s or + z crt s, dependng whether the pont estmate, lower confdence lmt, or upper confdence lmt s beng estmated. Then, mnmum and maxmum values of ψ are ntalzed. At the start of each run of the loop, ψ s defned as the mdpont of the current mnmum and maxmum values, and G 1 (ψ) s calculated. If G 1 (ψ) s greater than the target value, the mnmum s reset to the value of ψ used n ths run, or f t s less than the target, the maxmum s reset to ψ. The loop s then run agan, applyng these new mnmum and 20

29 maxmum values, unless the dfference between them s less than a user-defned value (e.g. 0.01). An mportant lmtaton of ths method s clearly the all-or nothng complance assumpton, as ths type of complance s only lkely to occur n very specfc stuatons. By the way t remans nterestng to evaluate the method n a smulaton. The excluson restrcton assumpton, though untestable, s lkely to hold n the majorty of the cases, due to randomsaton Rank Preservng Structural Falure Tme Model (RPSFTM) Robns and Tsats [6] developed the RPSFTM n order to estmate causal effects n the presence of non-complance n an RCT. Ths method dentfes the treatment effect by usng the randomsaton of the tral, observed survval and observed treatment hstory. An assumpton of ths method s that, gven two patents and j, f faled before j when on one treatment, then would also fal before j f both patents took the same alternatve treatment. That s way ths approach s called rank preservng. Another mportant assumpton s the so called common treatment effect : the treatment effect s assumed to be the same for patents crossng over to a treatment as for those ntally allocated to receve t. The observed event tme T s related to an underlyng event tme U that would have been observed n the absence of treatment, through an accelerated lfe model. The parameter ψ of the model represents the factor by whch lfe s accelerated by treatment and s estmated as the value at whch U s balanced between the treatment groups (on a user-specfed test). The method splts the observed event tme for each patent (T ) nto two: T = T 0 + T 1 21

30 where T 0 and T 1 are the lengths of tme that the patent spent on control and on expermental treatment before the event, respectvely. For patents randomsed to the expermental treatment, who do not cross over onto the control treatment, T 0 s equal to 0. For patents randomsed to the control group who do not swtch onto the expermental treatment, T 1 s equal to 0, whle for patents who cross over both T 0 and T 1 wll be greater than 0. The observed event tme T s related to counterfactual treatment-free event tme U by a causal model U = T 0 + e ψ 0 T 1 where ψ 0 s the true causal parameter. Assumng that U R, where R = 0/1 ndcates the randomzed treatment arm, for any gven value of ψ, the hypothess ψ = ψ 0 s tested by computng U (ψ)= T 0 + e ψ T 1 and calculatng Z(ψ) as the test statstc for the hypothess U(ψ) R. Bascally U s estmated usng the causal model for each value of ψ, and the true value of ψ s that for whch U(ψ) s ndependent of randomsed groups. A log-rank test can be used for testng the hypothess that the baselne survval curves are dentcal n the two treatment groups. Z(ψ) s a step functon, and the pont estmate s the value of ψ for whch Z(ψ) crosses 0. The RPSFTM makes dfferent mportant assumpton: - f a patent fals before another ndvdual on one treatment arm, he/she wll also fal before that other ndvdual on all other treatment regmens; - the tme at whch a patent would experence the outcome f never treated s not related to the allocaton arm (randomsaton assumpton); 22

31 - the treatment effect does not change n relaton to the tme n whch a patent starts recevng the treatment (common treatment effect assumpton). The randomsaton assumpton should be reasonable n the context of an RCT. The common treatment effect, nstead, could be a concern because t s assumed that there s not a dfference n the treatment effect n patents ntally randomsed to the nterventon compared to patents n the control group who cross over. 23

32 Chapter 3 Smulaton study desgn The effort to estmate the real effect of an nterventon s of crucal mportance, and SCO makes t undoubtedly more dffcult. Chapter 1 has descrbed SCO n the feld of therapes for breast cancer, hghlghtng the non-neglgble spread of ths phenomenon. The most relevant methods for dealng wth SCO were presented n Chapter 2. In order to assess the bas that the dfferent methods may lead to, the real effect of an nterventon under study needs to be known. Ths s possble through a smulaton study. The attenton s focused on the adjuvant/neoadjuvant settng, where the crossover s offered before dsease recurrence. So a two-arm RCT smlar to one emerged from the medcal lterature search was smulated. In the frst paragraph of ths chapter, the hypotheszed scenaros are explaned, whle n the second one the descrpton of the performance measures s gven Scenaros descrpton A sample sze of 3000 was chosen, wth 1500 patents allocated each to receve the expermental treatment or the control. As mentoned n prevous chapters, bas s lkely to occur when patents wth dfferent underlyng prognoses have dfferent probabltes of crossng over between treatment arms. In order to explore ths, patents were dvded nto two groups, those called wth a good prognoss, or at low rsk, and those wth a poor prognoss, or at hgh rsk. 24

33 The probablty of beng at low rsk was set at ether 30% or 70%. Survval tmes were generated from an exponental dstrbuton. The rate chosen for patents at low rsk was 0.05, whle the one for patents at hgh rsk was Randomsaton guarantees that the proporton of patents at low rsk and at hgh rsk s balanced between treatment arms. Three dfferent scenaros for the actual treatment effect were hypotheszed: the hazard rato (HR) was chosen to be 0.55 (β = ), to represent a hghly effectve treatment, or 0.80 (β = ), a less effectve treatment, or 1 (β = 0) to represent a treatment wth no effect. All patents were assumed to have entered the tral at the same tme, and an admnstratve censorng at 3 years was consdered, to represent the end of follow-up. The crossover was assumed to be undrectonal, from the control to the treatment arm. Two assumptons on the crossover probabltes were made: n one case, the probablty dd not change between the two prognostc groups and was chosen to be 0.50; n the other case, patents at hgh rsk were consdered more lkely to crossover, wth a probablty of 0.80, whle for patents at low rsk the probablty was set to be These probabltes were then used to generate a bnary varable representng the presence or the absence of crossover for each patent. If present, crossover was assumed to have occurred after 1 year from randomsaton. The summary of all the smulaton scenaros s gven n Table 3.1. For each scenaro, 1000 dfferent datasets were generated as descrbed n ths paragraph, and the varous methods appled to each dataset. Smulatons were made usng the R statstcal software, and the STATA packages stcomply [12] and strbee [13]. 25

34 Table 3.1 Summary of the smulaton scenaros # Scenaro Treatment effect (ln(hr)) % good prognoss Crossover probabltes* = % at low rsk % good prognoss % poor prognoss * Computed on patents who reach 1 year after randomzaton 26

35 3.2. Performance measures Crtera for evaluatng the performance of the obtaned results from the dfferent scenaros and statstcal approaches are summarsed n ths paragraph. Performance measures as descrbed n Burton et al [14] were used n order to compare the smulated results wth the true values used to generate the data. They nclude an assessment of bas, accuracy and coverage. The bas of each method was calculated as δ = β β, B where β s the true ntal treatment effect for the scenaro under study, and β = β /B, β s the estmate of nterest wthn each of the = 1,, B smulatons. =1 The mean square error (MSE) provdes a useful measure of the overall accuracy, as t ncorporates both measures of bas and varablty. Coverage s defned as the proporton of tmes the 100(1 - α)% confdence nterval (.e. 95% confdence nterval) for a partcular method contans the true treatment effect, β. The coverage should be approxmately equal to the nomnal coverage rate, e.g. 95 per cent of samples for 95% confdence ntervals, to approprately control the type I error rate for testng a null hypothess of no effect. Over-coverage, where the coverage rates are above 95%, suggests that the results are too conservatve: more smulatons wll not fnd a sgnfcant result when there s a true effect, leadng to a loss of statstcal power wth too many type II errors. Conversely, under-coverage, where the coverage rates are lower than 95%, ndcates over-confdence n the estmates: more smulatons wll ncorrectly detect a sgnfcant result, whch leads to hgher than expected type I errors. 27

36 In Table 3.2. the consdered performance measures and formulas are gven. Table 3.2 Performance measures Evaluaton crtera Formula BIAS Bas δ = β β Percentage bas Standardsed bas ( β β β ) 100 β β ( SE(β ) ) 100 ACCURACY Mean square error COVERAGE (β β) 2 + (SE(β )) 2 Proporton of tmes the 100(1 - α)% confdence nterval ± Z 1 α β 2 SE(β ) nclude β, for = 1,, B SE(β )= 1 B (B 1) (β =1 β s the emprcal standard error, and t represents an ) 2 assessment of the uncertanty n the estmate of nterest between smulatons. An alternatve measure of uncertanty s the average of the estmated wthn smulaton SE for the estmate of nterest SE(β ) B =1 /B. The emprcal SE should be close to the average of the estmated wthn smulaton SE f the estmates are unbased, so t may be approprate to consder both estmates of uncertanty. 28

37 Chapter 4 Results of the smulaton study The results of the smulatons for the dfferent scenaros are reported n Tables Scenaros from 1 to 4, reproducng a tral n whch the expermental treatment s hghly effectve (β = ), are reported n Tables In the frst scenaro (Table 4.1), the patents are mostly at hgh rsk (only 30% are at good prognoss) and the crossover does not depend on prognoss, but the probablty to swtch s the same,.e. 50%, for patents at low and hgh rsk. It s clear from the smulaton that the ITT analyss gves the most based results. The other methods, both naïve and non-naïve, perform better; the method by Loeys and Goetghebeur gves a low based estmate, but wth a wder confdence nterval, whle the RPSFTM performs partcularly well, wth a very low based and accurate estmate. When adjusted by prognoss, the IPCW method gves an unbased and very accurate estmate. The second scenaro (Table 4.2) reproduces a tral smlar to the one n the frst scenaro, but consders dfferent probabltes to cross over for the two prognoss groups, wth a hgher probablty for patents wth a poor prognoss (20% among patents wth a good prognoss, 80% for patents wth a poor prognoss). ITT analyss stll gves based results, but the other naïve methods also do not perform well. The Loeys and Goetghebeur and the RPSFTM methods gve low based results. Also n ths case, when adjusted by prognoss, the IPCW method gves an estmate whch s the most smlar to the true value of the effect. 29

38 In scenaros 3 and 4 (Tables 4.3 and 4.4), trals smlar to the ones of the frst and second scenaro, respectvely, are reproduced, wth the dfference that the patents are mostly at low rsk (70% are at good prognoss). Results are smlar to the ones observed for the frst two scenaros: when the crossover probablty does not depend on prognoss, all the consdered methods perform well, except the ITT whch underestmates the true effect; when the crossover probablty depends on prognoss, all the naïve methods and the unadjusted IPCW method gve based results, whle the Loeys and Goetghebeur and the RPSFTM approaches perform well. Agan, when adjusted by prognoss, the IPCW method provdes an unbased and accurate estmate. Scenaros from 5 to 8, reproducng a tral n whch the expermental treatment s less effectve than n prevous scenaros (β = ), are reported n Tables In scenaro 5 (Table 4.5), patents are mostly at hgh rsk (30% are at good prognoss) and the crossover probablty s the same,.e. 50%, for patents at low and hgh rsk. The ITT s stll the method that gves the most based results, although the bas s lower f compared to the one observed n the frst scenaro (that s dentcal to scenaro 5, except for the value of the true treatment effect). The other methods, both naïve and non-naïve perform well, wth the Loeys and Goetghebeur and the RPSFTM approaches gvng wder confdence ntervals f compared to the other methods, wth the RPSFTM tendng to slghtly overestmate the treatment effect. In scenaro 6 (Table 4.6), a tral smlar to the one n scenaro 5 s reproduced, but the probablty to cross over s hgher for patents wth a poor prognoss (20% among patents wth a good prognoss, 80% for patents wth a poor prognoss). All the naïve methods do not perform well, but n partcular the model n whch the treatment s consdered as a tme-dependent covarate gves results wth the hgher bas. Also the unadjusted IPCW analyss does not perform well. The method by Loeys and Goetghebeur and the RPSFTM 30

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