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The Direct Assignment Option as a Modular Design Component: An Example for the Setting of Two Predefined Subgroups

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The Direct Assignment Option as a Modular Design Component: An Example for the Setting of Two Predefined Subgroups
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  Research Article The Direct Assignment Option as a Modular Design Component: An Example for the Setting of Two Predefined Subgroups Ming-Wen An, 1 Xin Lu, 2 Daniel J. Sargent, 3 and Sumithra J. Mandrekar 3 󰀱 Department of Mathematics, Vassar College, 󰀱󰀲󰀴 Raymond Avenue, Poughkeepsie, NY 󰀱󰀲󰀶󰀰󰀴, USA 󰀲 Emory University, Atlanta, GA 󰀳󰀰󰀳󰀲󰀲, USA 󰀳  Mayo Clinic, Rochester, MN 󰀵󰀵󰀹󰀰󰀵, USA Correspondence should be addressed to Ming-Wen An; mian@vassar.eduReceived 󰀲󰀵 November 󰀲󰀰󰀱󰀴; Revised 󰀲󰀹 December 󰀲󰀰󰀱󰀴; Accepted 󰀲󰀹 December 󰀲󰀰󰀱󰀴Academic Editor: Maria N. D. S. CordeiroCopyright © 󰀲󰀰󰀱󰀵 Ming-Wen An et al. Tis is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal work is properly cited. Background  . A phase II design with an option or direct assignment (stop randomization and assign all patients to experimentaltreatment based on interim analysis, IA) or a prede󿬁ned subgroup was previously proposed. Here, we illustrate the modularity o the direct assignment option by applying it to the setting o two prede󿬁ned subgroups and testing or separate subgroup maineffects.  Methods . We power the 󰀲-subgroup direct assignment option design with 󰀱 IA (DAD-󰀱) to test or separate subgroup maineffects, with assessment o power to detect an interaction in a post-hoc test. Simulations assessed the statistical properties o thisdesign compared to the 󰀲-subgroup balanced randomized design with 󰀱 IA, BRD-󰀱. Different response rates or treatment/controlin subgroup 󰀱 (󰀰.󰀴/󰀰.󰀲) and in subgroup 󰀲 (󰀰.󰀱/󰀰.󰀲, 󰀰.󰀴/󰀰.󰀲) were considered.  Results . Te 󰀲-subgroup DAD-󰀱 preserves powerand type I error rate compared to the 󰀲-subgroup BRD-󰀱, while exhibiting reasonable power in a post-hoc test or interaction. Conclusion . Te direct assignment option is a 󿬂exible design component that can be incorporated into broader design rameworks,while maintaining desirable statistical properties, clinical appeal, and logistical simplicity. 1. Introduction Te primary goal o phase II clinical trials is to betterunderstand a treatment’s efficacy and saety pro󿬁le to inorma phase III go/no-go decision. Te phase II design withoption or direct assignment (i.e., stop randomization andassign all patients to the experimental arm based on one ortwo interim analyses (IA)) or a single prede󿬁ned subgroupwas previously proposed [󰀱]. In theory, such a design canbe readily incorporated into existing and broader designrameworks.Speci󿬁cally,theoptionordirectassignmentcanbe integrated into any design with an IA where a decisionmust be made or how to allocate treatment to patients(typically the decision is between continuing to randomizepatients to one o the treatments and stopping the trial dueto either efficacy or utility). In this paper, we present thedirect assignment option as a modular design component by applying it to the setting o two prede󿬁ned subgroups.In some therapeutic settings, we may expect treatmentheterogeneity across subpopulations identi󿬁ed by some ac-tor, or example, biomarker status. Speci󿬁cally, the treatmentmay be effective in one subgroup but ineffective in another(qualitative treatment-subgroup interaction), or the treat-ment may be effective in both subgroups but with differentmagnitude (quantitative treatment-subgroup interaction). Ineither case, primary interest may be in both subgroups, andthe design should enroll patients in both subgroups intoa single trial. Such a design could allow or prospectiveplanning o a design to identiy predictive markers. Forexample, KRAS mutations were identi󿬁ed in a retrospectiveanalysis to be predictive or overall survival response tocetuximab in colon cancer [󰀲]. We could imagine havinginstead proposed a prospective phase II direct assignmentdesignenrollingtwosubgroupstoenabletheKRASdiscovery in phase II. Alternatively in some settings, primary interestmay only be in one o the subgroups. However, there may  Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2015, Article ID 210817, 6 pageshttp://dx.doi.org/10.1155/2015/210817  󰀲 Computational and Mathematical Methods in Medicinebesecondaryinterestinthesecondsubgroup,anddependingon available resources and the clinical setting, one may wishto enroll the second subgroup as well. One design option orthis setting is a strati󿬁ed balanced randomized design. Tisdesign could be readily modi󿬁ed to incorporate and enjoy thebene󿬁tsothedirectassignmentoptionintroducedinAnet al. [󰀱].In this paper thereore, we consider a design with directassignment option or the setting o two prede󿬁ned sub-groups o patients. Te proposed design is a 󰀲-subgroupdirect assignment design with 󰀱 IA (DAD-󰀱), enrolling thesubgroupsinparallel,eachusingaDAD-󰀱withtheoptionstostop early or utility, continue with randomization, continuewith direct assignment, or stop early or efficacy at IA(able 󰀱). We power the design to test or separate subgrouptreatment effects, where the direction o treatment effect isprespeci󿬁ed,andalsoincludeapost-hocassessmentopowerto detect a treatment-subgroup interaction. We compare the󰀲-subgroup DAD-󰀱 with a 󰀲-subgroup balanced randomizeddesign with 󰀱 IA (BRD-󰀱), with options to stop early or util-ity, continue with randomization, or stop early or efficacy atIA. We perorm a simulation study to examine the statisticalproperties o the designs, under a variety o response ratesettings. Finally, we discuss a planning exercise or enrollinga second subgroup, when primary interest is only in onesubgroup. 2. Methods 󰀲.󰀱. When Both Subgroups Are of Primary Interest.  Weconsider a binary outcome, such as response, as the trialendpoint, and two patient subgroups, which we call M+ andM − . Since we are interested in separate treatment effects inthe two subgroups, we consider two independent primary hypotheses. Let  Δ +  and  Δ −  denote the ratio o response ratesor treated versus control groups (i.e., treatment effect) inthe M+ and M −  subgroups, respectively. We assume that Δ +  is positive (i.e., treatment is bene󿬁cial) and assume that Δ −  is speci󿬁ed a priori as either positive or negative by theinvestigator based on expert knowledge. Tus in the M+subgroup, we are interested in the one-sided test H 0+ :  Δ +  =1  versus H 1+ :  Δ +  > 1 ; in the M −  subgroup, we are alsointerestedinaone-sidedtestH 0− : Δ −  = 1  versusH 1− : Δ −  > 1 or H 0− :  Δ −  = 1  versus H 1− :  Δ −  < 1 , depending on whether Δ −  isspeci󿬁edaprioritobegreaterorlessthan󰀱.Wemakenoassumption on the relationship o treatment effect betweenthe two subgroups. Speci󿬁cally, we do not assume that thetreatment will be bene󿬁cial in the M −  subgroup only i it is󿬁rst bene󿬁cial in the M+ subgroup. Since this is the phaseII setting, a treatment-subgroup interaction is typically noto primary interest. However, we include an assessment o power to detect a treatment-subgroup interaction in a post-hoc test: H 0, Int :  Δ + /Δ −  = 1  versus H 0, Int :  Δ + /Δ −  ̸= 1 .We speciy a desired treatment effect size  Δ  , (  = + or  − ) or each subgroup, an acceptable type I error rate( 󽠵 ; the probability o rejecting the subgroup-speci󿬁c null,when the null is true), and the desired power to detect thetreatmenteffect size Δ   (i.e.,  1−􍠵 ,theprobabilityorejectingthe subgroup-speci󿬁c null, when the alternative  Δ   is true).We assume common  󽠵  and  􍠵 , but possibly different  Δ  , orthe two subgroups. Sample sizes are calculated separately ineachsubgroup.Speci󿬁cally,wecalculatesamplesizebasedonaone-sided󰀲-sampletestorproportionsora󰀲-stagedesignwith 󰀱:󰀱 randomization, 󰀱 interim analysis, and O’Brien-Fleming stopping rules using EAS sofware, as in theoriginaldesignwithdirectassignmentoption(ordetails,see[󰀱]). Simulation Study  . o understand the statistical properties o the 󰀲-subgroup direct assignment option design with 󰀱-IA(DAD-󰀱), we conducted a simulation study. For testing thesubgroup main effects, we speci󿬁ed  1 − 􍠵 = 0.80  and  󽠵 =0.20 , corresponding to widely recommended and acceptedstandards in the phase II setting [󰀳]. In the M+ subgroup,we considered a control arm response rate o 󰀰.󰀲 versusa treatment arm response rate o 󰀰.󰀴. Tat is, we assumethat the treatment is effective in the M+ subgroup ( Δ +  =2.0 ), based on preliminary studies. In the M −  subgroup, weconsider a control arm response rate o 󰀰.󰀲 versus treatmentarm response rates o 󰀰.󰀱, 󰀰.󰀲, and 󰀰.󰀴, re󿬂ecting 󰀳 possiblescenarios:areversetreatmenteffect (Δ −  = 0.5 ),notreatmenteffect ( Δ −  = 1 ), and a treatment bene󿬁t ( Δ −  = 2.0 ),respectively. Note that, here, a treatment arm response ratein the M −  subgroup o either 󰀰.󰀱 or 󰀰.󰀲 corresponds to atreatment-subgroup interaction. Te resulting sample sizecalculations are summarized in able 󰀲.Wesimulated󰀵󰀰󰀰trialsoreachothetwocases(able 󰀲).In Case I (no interaction), we consider a treatment effect inboth subgroups (i.e., treatment versus control response rates:󰀰.󰀴 versus 󰀰.󰀲, in both M+ and M − ); in Case II (interaction),we consider a treatment effect in M+ (treatment versuscontrolresponserates:󰀰.󰀴versus󰀰.󰀲)andareversetreatmenteffect in M −  (treatment versus control response rates: 󰀰.󰀱 versus 󰀰.󰀲). For each trial, we tested main effects separately in each subgroup and an interaction effect and recorded theresults or testing each o the three independent hypotheses:one-sided tests or main effects—H 0+ :  Δ +  = 1  versus H 1+ : Δ +  > 1 ; H 0− :  Δ −  = 1  versus H 1− :  Δ −  > 1  or H 1− : Δ −  < 1  (depending on the a priori hypothesized treatmenteffect)andthetwo-sidedtestoraninteractioneffect—H 0, Int : Δ + /Δ −  = 1  versusH 0, Int : Δ + /Δ −  ̸= 1 .Forthepost-hoctestorinteraction,weusedaconservative 󽠵 -levelo󰀰.󰀱󰀰(two-sided)or rejecting the null hypothesis o no interaction. Averagingthese outcomes over the 󰀵󰀰󰀰 simulated trials, we obtainedestimates o type I error rate and power or each hypothesis.For comparison, we also studied the outcomes under a 󰀲-subgroupbalancedrandomized(󰀱:󰀱)designwith󰀱IAandnooption or direct assignment (BRD-󰀱), based on the O’Brien-Fleming stopping rules. 󰀲.󰀲. When One Subgroup Is of Primary Interest: ProspectivePlanning Exercise for Enrolling a Second Subgroup.  In somesettings, interest may be only in one subgroup, say M+.However, instead o altogether excluding the other subgroup,M − , i resources are available, a design could include the M − subgroup (as long as there are no saety or efficacy concerns)by accruing to the M −  subgroup while the M+ subgroup isaccruing. An exploratory analysis in the M −  subgroup could  Computational and Mathematical Methods in Medicine 󰀳 󰁡󰁢󰁬󰁥 󰀱: Options available at interim analysis (IA) in a 󰀲-subgroup direct assignment design with 󰀱 IA (DAD-󰀱) versus a 󰀲-subgroup balancedrandomized design with 󰀱 IA (BRD-󰀱). Te decisions are independently made in each subgroup at the time o IA. Some cells in the BRD-󰀱table are intentionally lef blank, to highlight the missing option o direct assignment in this design. Te options in bold are those that areavailable only in the design with direct assignment option.Options at interim analysis (IA)󰀲-subgroup direct assignment design with 󰀱 IA (DAD-󰀱) 󰀲-subgroup balanced randomized design with 󰀱 IA (BRD-󰀱)M −  M+ M −  M+Stop, utility Stop, utility Stop, utility Stop, utility Continue, randomize Continue, randomize Continue, direct Stop, efficacy Stop, efficacy Continue, randomizeStop, utility Continue, randomizeStop, utility Continue, randomize Continue, randomize Continue, direct Stop, efficacy Stop, efficacy  Continue, directStop, futility Continue, randomizeContinue, directStop, efficacy  Stop, efficacy Stop, utility Stop, efficacy Stop, utility Continue, randomize Continue, randomize Continue, direct Stop, efficacy Stop, efficacy 󰁡󰁢󰁬󰁥 󰀲: Sample size calculations or 󰀱-sided  󽠵 = 0.20  and  1 − 􍠵 = 0.80 , or different treatment effects in the two subgroups. RR  trt  is theresponse rate in the treatment group, RR  control  is the response rate in the control group, and RRR  trt : control  is the ratio o response rates in thetreatment versus control groups (i.e., treatment effect).Case M+ subgroup M −  subgroupreatment effect, RRR  trt : control  RR  trt /RR  control  󝠵  reatment effect, RRR  trt : control  RR  trt /RR  control  󝠵 I (no interaction) 󰀲 󰀰.󰀴/󰀰.󰀲 󰀶󰀵 󰀲 󰀰.󰀴/󰀰.󰀲 󰀶󰀵II (interaction) 󰀲 󰀰.󰀴/󰀰.󰀲 󰀶󰀵 󰀰.󰀵 (i.e., reverse bene󿬁t) 󰀰.󰀱/󰀰.󰀲 󰀱󰀶󰀱 then yield preliminary indication o treatment effect in theM −  subgroup.o decide between enrolling M+ patients only andadditionally enrolling the M −  subgroup as an exploratory companiongroup,aprospectiveplanningsimulationexercisecould be conducted. Since the M −  subgroup is not o primary interest, it is likely that there is no precise prelimi-nary inormation about the treatment effect in the M −  sub-group. Instead, one might speciy in the M −  subgroup theresponse rate in the control arm (RR  control ) to be uniormly distributed over some interval [RR  󽠵, control , RR  􍠵, control ] andthe response rate ratio comparing treated versus controlarms (RRR  trt : control ) to be uniormly distributed over anotherinterval [RRR  󽠵, trt : control , RRR  􍠵, trt : control ]. I the interval or theRRR  trt : control  includes󰀰,thensuchaspeci󿬁cationwouldallow or the treatment to have a negative (RRR  trt : control  < 0 ),neutral (RRR  trt : control  = 0 ), or positive (RRR  trt : control  > 0 )effect in the M −  subgroup, thus re󿬂ecting vague knowledgeaboutthetreatmentactivityintheM − subgroup.Itispossiblethen to simulate a trial and record the observed differencein response rates. Averaging across the simulated trials, onecan obtain a probability o observing a difference in responserates that exceeds some threshold, say    , that is o clinicalinterest. O course this probability will depend on the samplesize. Since accrual to M −  will occur while accrual to M+ isopen,thesamplesizeorM − willdependontheaccrualratesto both M+ and M −  and the prevalence o the subgroupsand may not be known in advance. Probabilities that thedifference exceeds    can be generated, under a variety o plausible settings. Te investigator can use the probability distributions as a guide to decide whether to enroll the M − subgroup as an exploratory companion group, depending onwhether he or she believes the probability to be sufficiently high to make enrollment into M −  worthwhile.As an example o the prospective planning exercise, weconsidered   = 0.15 . Tat is, an observed difference inresponseratescomparingtreatedversuscontrolo󰀱󰀵%wouldbe considered clinically relevant. We urther speci󿬁ed thecontrol group response rate to be uniormly distributed over [0.1,0.3]  and RRR  trt : control  to be uniormly distributed over [0.5,1.5] . We considered sample sizes o   󝠵 1,−  = 6 , 󰀱󰀶,and 󰀴󰀸 patients or the 󿬁rst stage in the M −  group. Tesecorrespond to M+/M −  prevalence o   󝠵 1,+ /6 ,  󝠵 1,+ /16 , and 󝠵 1,+ /48 , assuming a sample size o   󝠵 1,+  in the 󿬁rst stage or  󰀴 Computational and Mathematical Methods in Medicine 󰁡󰁢󰁬󰁥󰀳:Simulationstudyresultsbasedon󰀵󰀰󰀰simulatedtrials.Samplesizeandtreatmenteffectsasspeci󿬁edin able 󰀲.Statisticalpropertiesothe󰀲-subgroupdirectassignmentoptiondesignwith󰀱IA(DAD-󰀱)versusa󰀲-subgroupbalancedrandomizeddesignwith󰀱interimanalysis(BRD-󰀱). One-sided  󽠵 = 0.20  and  1 − 􍠵 = 0.80  or the subgroup main effects; two-sided  󽠵 int  = 0.10  or the post-hoc test or an interactioneffect. (a) Separate subgroup main effectsCaseM+ subgroup M −  subgroupPower ype I error rate Power ype I error rateDAD-󰀱 BRD-󰀱 DAD-󰀱 BRD-󰀱 DAD-󰀱 BRD-󰀱 DAD-󰀱 BRD-󰀱I (no interaction) 󰀷󰀸.󰀴 󰀷󰀸.󰀸 󰀲󰀳.󰀸 󰀲󰀰.󰀶 󰀷󰀹.󰀸 󰀸󰀱.󰀸 󰀲󰀰.󰀸 󰀱󰀸.󰀶II (interaction) 󰀷󰀸.󰀴 󰀷󰀸.󰀸 󰀲󰀳.󰀸 󰀲󰀰.󰀶 󰀸󰀲.󰀸 󰀸󰀴.󰀴 󰀱󰀹.󰀰 󰀱󰀸.󰀲 (b) Subgroup-treatment interaction effectCase Power ype I error rateDAD-󰀱 BRD-󰀱 DAD-󰀱 BRD-󰀱I (no interaction) — — 󰀱󰀱.󰀳 󰀱󰀱.󰀴II (interaction) 󰀶󰀴.󰀳 󰀶󰀷.󰀶 — — the M+ group. For each sample size, we simulated 󰀵󰀰󰀰 trials.We created histograms o the observed treatment differencesand recorded the proportion o trials where the absoluteobservedtreatmentdifferenceinresponseratesexceeded  =0.15 . 3. Results 󰀳.󰀱. When Both Subgroups Are of Primary Interest.  Tenominal power and type I error rate are preserved in the 󰀲-subgroup DAD-󰀱, relative to a 󰀲-subgroup BRD-󰀱 (able 󰀳).In particular, or the M+ group, the power to detect aRRR  trt : control  o󰀲orDAD-󰀱is󰀷󰀸.󰀴%(versus󰀷󰀸.󰀸%oraBRD-󰀱), and the type I error rate is 󰀲󰀳.󰀸% (versus 󰀲󰀰.󰀶% or a BRD-󰀱).FortheM − group,thepowertodetectaRRR  trt : control  o󰀰.󰀵or the DAD-󰀱 is 󰀸󰀲.󰀸% (versus 󰀸󰀴.󰀴% or a BRD-󰀱), and thetype I error rate is 󰀱󰀹.󰀰% (versus 󰀱󰀸.󰀲% or a BRD-󰀱).We were also interested in the properties o a post-hoctest or an interaction effect. ype I error rate is preservedat the nominal rate, and power decreases slightly relative tothe nominal rate. Speci󿬁cally, or Case I (no interaction), thetype I error rate is 󰀱󰀱.󰀳% or the DAD-󰀱, compared with 󰀱󰀱.󰀴%or a BRD-󰀱. For Case II (interaction), the power to detect aninteractioneffectatatwo-sidedalphalevelo󰀰.󰀱󰀰or󰀲versus󰀰.󰀵is󰀶󰀴.󰀳%ortheDAD-󰀱,comparedwith󰀶󰀷.󰀶%oraBRD-󰀱. 󰀳.󰀲. When One Subgroup Is of Primary Interest: ProspectivePlanning Exercise for Enrolling a Second Subgroup.  Fromthe 󰀵󰀰󰀰 simulated trials, when  [ RR  󽠵, control , RR  􍠵, control ] =[0.1,0.3]  and  [ RRR  󽠵, trt : control , RRR  􍠵, trt : control ] = [0.5,1.5] , theprobability o the absolute observed difference exceeding   =0.15  is 󰀶󰀷% or   = 6  patients per treatment arm in the󿬁rst stage (Figure 󰀱). In contrast, using   = 16  (󰀳󰀲) patientsper arm in the 󿬁rst stage, the probability o the observeddifference exceeding   = 0.15  is 󰀳󰀲.󰀸% (󰀱󰀳%).        F     r     e     q     u     e     n     c     y       F     r     e     q     u     e     n     c     y       F     r     e     q     u     e     n     c     y 0.00.51.00.00.51.00.00.51.0 N 1  = 6 ;mean  = 0.011 ;SD  = 0.5486 ; pr (diff   ≥ 0.15) = 0.67N 1  = 16 ;mean  = 0.0018 ;SD  = 0.5582 ; pr (diff   ≥ 0.15) = 0.328250150500150100500100500P trt  − P control P trt  − P control P trt  − P control −1.0 −0.5−1.0 −0.5−1.0 −0.5N 1  = 48 ;mean  = 8e − 04 ;SD  = 0.5632 ; pr (diff   ≥ 0.15) = 0.13 F󰁩󰁧󰁵󰁲󰁥 󰀱: Distributions o observed treatment differences in theM −  subgroup at interim analysis, across 󰀵󰀰󰀰 simulated trials,using control response rate uniormly distributed over  [0.1,0.3] and response rate ratio or treated versus control arms uniormly distributed over  [0.5,1.5] . Sample sizes in the 󿬁rst stage,  󝠵 1  = 6 ,󰀱󰀶, and 󰀴󰀸. Proportion o trials where observed treatment differenceexceeds   = 0.15  in absolute value is 󰀶󰀷%, 󰀳󰀲.󰀸%, and 󰀱󰀳%,respectively. 4. Discussion Te direct assignment option design was 󿬁rst proposed asa design enrolling a single cohort [󰀱]. We have applied thedesign in the 󰀲-subgroup setting. Te 󰀲-subgroup DAD-󰀱 preserves power and type I error rates at their nominal  Computational and Mathematical Methods in Medicine 󰀵levels.Further,thisdesignhasreasonablepowerorplanningpurposes or a post-hoc test or detecting a treatment-subgroup interaction. Te 󿬁rst 󿬁nding is to be expected sincethe 󰀲-subgroup DAD-󰀱 applies the direct assignment optiondesign in parallel to each subgroup. However, the secondresult that there is reasonable (post-hoc) power to detect atreatment-subgroupinteractionispreviouslyunexploredandis o potential interest.Te assessment o power to detect an interaction in apost-hoc test using the 󰀲-subgroup DAD-󰀱 and an  󽠵 -level o 󰀰.󰀱󰀰 suggests reasonable power (󰀶󰀴.󰀳% to detect a responserate o 󰀲.󰀰 in M+ versus response rate o 󰀰.󰀵 in M − ). Werecognize that this  󽠵 -level is conservative or an interactiontest when the main effects use a one-sided  󽠵 -level o 󰀰.󰀲󰀰.Te sample size that was used to detect the interaction waslarger than that o a typical phase II trial (󰀶󰀵 patients inM+ and 󰀱󰀶󰀱 patients in M − ). However, the reality is that nophase II trial can reliably detect interaction effects o the sizeexplored in this simulation study using small sample sizes. Inact, an alternative strategy o prospectively planning or aninteraction test would have similarly yielded a large samplesize (󰀲󰀱󰀴 total patients, based on a balanced randomizeddesign with no IA). Our strategy thereorehas no sample sizedisadvantage in detecting an interaction effect relative to onethatprospectivelyplansorasimilarlysizedinteractioneffectin a phase II setting.Te 󰀲-subgroup DAD-󰀱 could be applied in any settingwhere there are two subpopulations o interest. A naturalsetting is that o targeted, biomarker-based therapies. Otherso-called integrated biomarker designs have previously beenproposed.Wehighlightacoupleosuchdesigns;comprehen-sive reviews o such designs are available elsewhere (e.g., [󰀴,󰀵]). Te parallel subgroup-speci󿬁c design [󰀶] evaluates treat- menteffectsseparatelyintwosubgroups.Whenthetreatmenteffect is homogeneous across subgroups, this design has lesspowerordetectingatreatmentbene󿬁t,comparedtoadesignthat tests or an overall treatment effect. In an attempt toaddress the lack o power in a homogeneous treatment set-ting, other designs have adopted a sequential testing strategy.Speci󿬁cally, the Marker Sequential est (MaS) design 󿬁rsttests or a treatment effect in the M+ subgroup [󰀷]. I thistestisstatisticallysigni󿬁cant,thentheM − subgroupistested.However, i the test is not statistically signi󿬁cant, then anoverall population is tested. MaS increases power to detecttreatment bene󿬁t when the treatment effect is homogeneousacross subgroups and preserves power when the treatment iseffective in the M+ subgroup but not the M −  subgroup, ascompared to a parallel subgroup-speci󿬁c design.At 󿬁rst glance, the proposed design resembles a parallelsubgroup-speci󿬁c design in spirit. As noted in Freidlin etal. [󰀷], under the case o homogeneous treatment effect, theparallelsubgroup-speci󿬁cdesignlackspowerrelativetoatestor overall treatment effect and the MaS design. However,the proposed design differs rom the setting consideredby Freidlin et al. [󰀷] in two important ways. First, theproposed design is or the phase II setting and not the phaseIII setting. At this early exploratory stage, it is importantto 󿬁rst understand the treatment effect in each subgroupseparately without necessarily examining overall treatmenteffects.Tatis,althoughatestoroveralltreatmenteffectmay be more powerul than separate tests or subgroup-speci󿬁ctreatment bene󿬁t when the treatment effect is homogeneousacross subgroups, an overall treatment effect would not be o primary interest in the phase II setting. Second, the MaSdesign’s property o improving power in the homogeneoustreatment setting rests on a key assumption: i the treatmentdoes not work in the M+ subgroup, then it cannot work in the M −  subgroup. We reer to this as the  treatment monotonicity   assumption and do not invoke this assumptionor the proposed design setting. Rather, we allow or thesubgroup-speci󿬁ctreatmenteffectstobeindependentoeachother.It is not always the case that primary interest is in twosubgroups. However, even i primary interest is only in onesubgroup, it may be inormative to enroll a second subgroupas an exploratory companion group (i ethical) in early phase trials. Te decision to enroll a second subgroup willdependonthetradeoffamongavailableresources,theclinicalimportance o the difference in response rates   , and theestimated probability o the observed difference exceeding   .Teresultsromtheprospectiveplanningexercise, thereore,could aid the study design team in this decision. 5. Conclusion In summary, we have illustrated at a practical level a 󰀲-subgroup design with direct assignment option or the phaseII setting where primary interest is in separate subgrouptreatment effects. Such a design not only preserves thenominal type I error rate and power or testing or separatesubgroup treatment effects but also enjoys reasonable post-hoc power to detect a treatment-subgroup interaction, i oneexists, as well as clinical appeal and logistical simplicity. Inthe case where primary interest is only in one subgroup butresources may be available to enroll the second subgroup, wehaveproposedaplanningexerciseoraidingthestudydesignteam in deciding to enroll the second subgroup, using theramework o the direct assignment option design. Conflict of Interests Te authors declare that there is no con󿬂ict o interestsregarding the publication o this paper.  Acknowledgment Sumithra J. Mandrekar and Daniel J. Sargent are supportedin part by the National Cancer Institute Grant no. CA-󰀱󰀵󰀰󰀸󰀳(Mayo Clinic Cancer Center). References [󰀱] M.-W. An, S. J. Mandrekar, and D. J. Sargent, “A 󰀲-stage phaseII design with direct assignment option in stage II or initialmarker validation,”  Clinical Cancer Research , vol. 󰀱󰀸, no. 󰀱󰀶, pp.󰀴󰀲󰀲󰀵–󰀴󰀲󰀳󰀳, 󰀲󰀰󰀱󰀲.[󰀲] A. Li`evre, J.-B. Bachet, V. Boige et al., “KRAS mutations asan independent prognostic actor in patients with advanced
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