Adaptive dose-response studies to establish proof-of-concept in learning-phase clinical trials
AAdaptive dose-response studies to establishproof-of-concept in learning-phase clinical trials
Shiyang Ma , ∗ , Michael P. McDermott Department of Biostatistics, Columbia University, New York, NY, 10032, USA Department of Biostatistics and Computational Biology, University of Rochester, Rochester,NY, 14642, USA ∗ e-mail: [email protected] Abstract
In learning-phase clinical trials in drug development, adaptive designs can be e ffi cientand highly informative when used appropriately. In this article, we extend the multiplecomparison procedures with modeling techniques (MCP-Mod) procedure with generalizedmultiple contrast tests (GMCTs) to two-stage adaptive designs for establishing proof-of-concept. The results of an interim analysis of first-stage data are used to adapt the candidatedose-response models in the second stage. GMCTs are used in both stages to obtain stage-wise p -values, which are then combined to determine an overall p -value. An alternativeapproach is also considered that combines the t -statistics across stages, employing theconditional rejection probability (CRP) principle to preserve the Type I error probability.Simulation studies demonstrate that the adaptive designs are advantageous compared tothe corresponding tests in a non-adaptive design if the selection of the candidate set ofdose-response models is not well informed by evidence from preclinical and early-phasestudies. KEY WORDS: Adaptive designs; Conditional rejection probability principle; Generalizedmultiple contrast tests; MCP-Mod; Proof-of-concept.
Motivated by the desire for greater e ffi ciency in drug development and the low success ratesin confirmatory (Phase 3) studies, methodological research on adaptive designs and interestin their application has grown tremendously over the last 30 years. In an adaptive design,accumulating data can be used to modify the course of the trial. Several possible adaptationscan be considered in interim analyses, for example, adaptive randomization for dose finding,1 a r X i v : . [ s t a t . M E ] F e b ropping and / or adding treatment arms, sample size re-estimation, and early stopping for safety,futility or e ffi cacy, to name a few.Validity and integrity are two major considerations in adaptive designs (Dragalin, 2006).Because data from one stage of the trial can inform the design of future stages of the trial,careful steps need to be taken to maintain the validity of the trial, i.e., control of the TypeI error probability and minimization of bias. To maintain trial integrity, it is important thatall adaptations be pre-planned, prior to the unblinded examination of data, and that all trialpersonnel other than those responsible for making the adaptations are blind to the results of anyinterim analysis. It is also important to ensure consistency in trial conduct among the di ff erentstages.A general method for hypothesis testing in experiments with adaptive interim analysesbased on combining stage-wise p -values was proposed by Bauer and K ¨ohne (1994). The basicidea behind the construction of a combination test in a two-stage adaptive design is to transformthe stage-wise test statistics to p -values, with independence of the p -values following from theconditional invariance principle (Brannath et al ., 2007, 2012; Wassmer and Brannath, 2016),regardless of the adaptation performed after the first stage. The principle holds as long as thenull distribution of the first-stage p -value ( p ) as well as the conditional distribution of thesecond-stage p -value ( p ) given p are stochastically larger than the U (0 ,
1) distribution (theso-called “p-clud” property). A specified combination function is used to combine the p -valuesobtained before and after the preplanned adaptation of the design into a single global test statis-tic. An extension of combination tests to allow more flexibility regarding the number of stagesand the choice of decision boundaries was provided by Brannath et al . (2002).In dose-response studies, a component of the MCP-Mod procedure (Bretz et al ., 2005) hasgained popularity for the purpose of detecting a proof-of-concept (PoC) signal in learning-phase trials. The procedure consists of specifying a set of candidate dose-response models,determining the optimal contrast statistic for each candidate model, and using the maximumcontrast as the overall test statistic. Other authors have considered extensions of this pro-cedure to adaptive dose-response designs. Miller (2010) investigated a two-stage adaptive2ose-response design for PoC testing incorporating adaptation of the dosages, and possiblythe contrast vectors. He developed an adaptive multiple contrast test (AMCT) that combinesthe multiple contrast test statistics across two stages under the assumption that the varianceis known. Franchetti et al . (2013) extended the MCP-Mod procedure to a two-stage dose-response design with a pre-specified rule of adding and / or dropping dosage groups in Stage 2based on the Stage 1 results. The PoC test uses Fisher’s (1932) combination method to combinethe two stage-wise p -values, each obtained by applying the MCP-Mod procedure to the datafrom each stage. This method includes a restrictive requirement of equal total sample sizes foreach stage. Also, the authors claimed that the independence of the two stage-wise p -values ispotentially compromised if the number of dosages used in Stage 2 is not the same as that usedin Stage 1 and proposed a method for assigning weights to the di ff erent dosage groups to dealwith this problem. We do not believe that such weighting is necessary as long as the statisticused to combine the stage-wise p -values (Fisher’s, in this case) does not include weights thatdepend on the Stage 1 data.Early work related to adaptive designs for dose-response testing includes a general proce-dure with multi-stage designs proposed by Bauer and R¨ohmel (1995), in which dosage adapta-tions were performed at interim analyses. Bornkamp et al . (2011) proposed a response-adaptivedose-finding design under model uncertainty, which uses a Bayesian approach to update the pa-rameters of the candidate dose-response models and model probabilities at each interim analy-sis. In this article, we propose new methods for adaptive dose-response studies with normally-distributed outcomes. We extend the MCP-Mod procedure to include generalized multiplecontrast tests (GMCTs; Ma and McDermott, 2020) and apply them to adaptive designs; werefer to these as adaptive generalized multiple contrast tests (AGMCTs). These tests are in-troduced in Section 2. In Section 3 we extend the AMCT of Miller (2010) to accommodatemore flexible adaptations and to the important case where the variance is unknown using theconditional rejection probability (CRP) principle (M¨uller and Sch¨afer, 2001, 2004). Numericalexamples are provided in Section 4 to illustrate the application of the AGMCTs and AMCT. In3ection 5, we conduct simulation studies to evaluate the operating characteristics of the variousmethods as well as the corresponding tests for non-adaptive designs. The conclusions are givenin Section 6. In this section, we propose a two-stage adaptive design in which we use data from Stage 1 toget a better sense of the true dose-response model and make adaptations to the design for Stage2. We then use data from both Stage 1 and Stage 2 to perform an overall test to detect thePoC signal. The rationale is to overcome the problem of potential model misspecification atthe design stage.
We consider the case of a normally distributed outcome variable. Suppose that there are n i subjects in dosage group i in Stage 1, i = , . . . , k . Denote the first stage data as YYY = ( Y , . . . , Y n , . . . , Y k , . . . , Y k n k ) (cid:48) . The statistical model is Y i j = µ i + (cid:15) i j , (cid:15) i j iid ∼ N (0 , σ ) , i = , . . . , k , j = , . . . , n i . The true mean configuration is postulated to follow some dose-response model µ i = f ( d i , θθθ ),where d i is the dosage in the i th group, i = , . . . , k . The dose-response model is restrictedto be of the form f ( · ; θθθ ) = θ + θ f ( · ; θθθ ), where f ( · ; θθθ ) is a standardized dose-responsemodel indexed by a parameter vector θθθ (Thomas, 2017). A candidate set of M dose-responsemodels f m ( · , θθθ ), m = , . . . , M , including values for θθθ , is pre-specified. For each candidatemodel, an optimal contrast is determined to maximize the power to detect di ff erences amongthe mean responses; the contrast coe ffi cients are chosen to be perfectly correlated with the meanresponses if that model is correct (Bretz et al ., 2005; Pinheiro et al ., 2014).For each candidate model, the following hypothesis is tested: H m : k (cid:88) i = c mi µ i = , vs. H m : k (cid:88) i = c mi µ i > , m = , . . . , M , c m , . . . , c mk are the optimal contrast coe ffi cients associated with the m th candidatemodel in Stage 1. The multiple contrast test statistics are T m = k (cid:88) i = c mi ¯ Y i (cid:44) S (cid:118)(cid:116) k (cid:88) i = c mi n i , m = , . . . , M , where ¯ Y i = (cid:80) n i j = Y i j / n i and the pooled variance estimator is S = (cid:80) k i = (cid:80) n i j = ( Y i j − ¯ Y i ) /ν ,where ν = (cid:80) k i = n i − k . The joint null distribution of ( T , . . . , T M ) (cid:48) is multivariate t (with ν degrees of freedom) with common denominator and correlation matrix having elements ρ mm (cid:48) = k (cid:88) i = c mi c m (cid:48) i n i (cid:44) (cid:118)(cid:116) k (cid:88) i = c mi n i k (cid:88) i = c m (cid:48) i n i , m , m (cid:48) = , . . . , M . Let p m = − T ν ( T m ) be the p -values derived from T m , m = , . . . , M , where T ν ( · ) is thecumulative distribution function of the t distribution with ν degrees of freedom. We considerthree combination statistics to combine the M dependent one-sided p -values in Stage 1 (Maand McDermott, 2020):(i) Tippett’s (1931) combination statistic, Ψ T = min ≤ m ≤ M p m ;(ii) Fisher’s (1932) combination statistic, Ψ F = − M (cid:88) m = log( p m );(iii) Inverse normal combination statistic (Stou ff er, 1949), Ψ N = M (cid:88) m = Φ − (1 − p m ) . Note that the use of Tippett’s combination statistic is equivalent to the original MCP-Modprocedure; the use of di ff erent combination statistics results in a generalization of the MCP-Mod procedure, yielding GMCTs (Ma and McDermott, 2020). When the p -values are inde-pendent, these statistics have simple null distributions. In our case the p -values are dependent,but the correlations among T , . . . , T M are known. For Tippett’s combination method, one5an obtain multiplicity-adjusted p -values from T m , m = , . . . , M , given the correlation struc-ture using the mvtnorm package in R . A PoC signal is established in Stage 1 if the minimumadjusted p -value p min, adj1 < α (Bretz et al ., 2005). For Fisher’s and the inverse normal combi-nation methods, excellent approximations to the null distributions of Ψ F and Ψ N have beendeveloped (Kost and McDermott, 2002), enabling computation of the overall p -value p forStage 1 using a GMCT (Ma and McDermott, 2020).After obtaining the Stage 1 data, we make design adaptations and determine the optimalcontrasts for the updated models in Stage 2 (see Section 2.2 below). We then conduct a GMCTin Stage 2 and obtain the second-stage p -value p . Under the overall null hypothesis H : µ = · · · = µ k ∗ , where k ∗ is the total number of unique dosage groups in Stages 1 and 2 combined,the independence of the stage-wise p -values p and p can be established using the conditionalinvariance principle (Brannath et al ., 2007). To perform the overall PoC test in the two-stageadaptive design, we combine p and p using one of the above combination statistics.A procedure that ignores the adaptation, i.e., that simply pools the data from Stage 1 andStage 2 and applies a GMCT to the pooled data as if no adaptation had been performed, wouldsubstantially increase the Type I error probability. Here we consider adaptations for the second stage that are arguably most relevant for PoCtesting, namely those of the candidate dose-response models and the dosages to be studied. Thechoice of the candidate dose-response models and dosages for Stage 1 would depend on priorknowledge from pre-clinical or early-stage clinical experience with the investigative agent.If there is great uncertainty concerning the nature of the dose-response relationship, it wouldseem sensible to select a more diverse set of candidate dose-response models with pre-specifiedparameters when the trial begins.After collecting the Stage 1 data, these data can be used to estimate θθθ for each of the M candidate dose-response models and adapt each of the models by substituting ˆ θ ˆ θ ˆ θ for the originalspecification (guess) of θθθ . The optimal contrast vectors can be constructed for each of the6pdated models f m ( · , ˆ θ ˆ θ ˆ θ ), m = , . . . , M , for use in Stage 2. A potential problem occurs when thetrue dose-response model di ff ers markedly from some of the specified candidate models and ifthose candidate models are nonlinear models with several parameters. In such cases there canbe a failure to fit the model using the Stage 1 data. To handle this problem, one can considerpolynomial approximations to the nonlinear candidate models. If there is a failure to fit thenonlinear candidate models using the Stage 1 data, they can be replaced with the correspondingpolynomial models. Specifically, consider the following 5 candidate dose-response models: E max model: f ( d , θθθ ) = E + E max d / ( ED + d )Linear-log model: f ( d , θθθ ) = θ + θ log(5 d + f ( d , θθθ ) = θ + θ d Quadratic model: f ( d , θθθ ) = θ + θ d + θ d Logistic model: f ( d , θθθ ) = E + E max / [1 + exp { ( ED − d ) /δ } ]Among those 5 candidate models, the E max and Logistic models are the ones for whichpolynomial approximations might need to be considered since the others can be expressed aspolynomials in d (or a simple function of d ). The approximation can be at most of order k −
1. Of course, if a polynomial approximation is needed for more than one model, theapproximations would have to be of di ff erent orders. For example, one could fit a 3 rd orderpolynomial approximation θ + θ d + θ d + θ d for the E max model and a 4 th order polynomialapproximation θ + θ d + θ d + θ d + θ d for the Logistic model if they fail to fit the Stage 1data.One could also consider di ff erent numbers of candidate models (or contrast vectors) in Stage1 and Stage 2. One non-model-based option, for example, would be to use a single contrastin Stage 2 based on the sample means of the dosage groups from Stage 1. We found that thisstrategy, while intuitively appealing, yielded tests with reduced power, likely due to the relianceon a single contrast combined with the uncertainty associated with estimation of the means ofeach dosage group in Stage 1. One could also consider a small number of other contrasts based7n values that are within the bounds of uncertainty reflected in the sample means, though howto choose these contrasts is somewhat arbitrary.Adaptation of the dosage groups in Stage 2, including the number of dosage groups, couldalso be considered. One would have to establish principles for adding and / or dropping dosages;for example, dropping active dosages that appear to be less e ffi cacious than placebo or that ap-pear to be less e ffi cacious than other active dosages, or adding a dosage (within a safe range)when there appears to be no indication of a dose-response relationship in Stage 1. Relevant dis-cussion of these issues can be found in Bauer and R ¨ohmel (1995), Miller (2010), and Franchetti et al . (2013). Instead of combining the stage-wise p -values p and p , each based on a GMCT, Miller (2010)suggested combining the test statistics for each candidate dose-response model across the twostages, and then derving an overall p -value from a multiple contrast test applied to those statis-tics, assuming a known variance σ . For each candidate model, we have Z m = k (cid:88) i = c mi ¯ Y i + k (cid:88) i = c mi ¯ Y i (cid:44) σ (cid:118)(cid:116) k (cid:88) i = c mi n i + k (cid:88) i = c mi n i , m = , . . . , M . Since k , c mi , and n i can depend on the interim data (adaptation), the null distribution of Z m isnot standard normal in general.In order to control the Type I error probability of the overall test, Miller (2010) appliesa conditional error approach based on the conditional rejection probability (CRP) principle(M¨uller and Sch¨afer, 2001, 2004). Computation of the conditional Type I error probabilityrequires pre-specification of what Miller (2010) calls a “base test”, i.e., pre-specified valuesfor the contrast coe ffi cients ( c ∗ mi ), number of dosage groups ( k ∗ ), and group sample sizes ( n ∗ i )in Stage 2, i = , . . . , k ∗ , m = , . . . , M . There is not a clear best strategy for choosing thesepre-specified values. Miller (2010) considers an example where all possible Stage 2 designs8an be enumerated and have k = k and n i = n i , i = , . . . , k , and the pre-specified valuesinvolving c ∗ mi , i = , . . . , k , m = , . . . , M , are averaged over the possible Stage 2 designs.More generally one cannot enumerate all possible Stage 2 designs, so in the development belowwe pre-specify c ∗ mi = c mi , k ∗ = k , and n ∗ i = n i , i = , . . . , k , m = , . . . , M . Since the dosagescan also be adapted, we suggest pre-specifying ddd ∗ Stage2 = ddd Stage1 = ( d , . . . , d k ) (cid:48) .The Z -statistics for the base test are Z ∗ m = k (cid:88) i = c mi (cid:16) ¯ Y i + ¯ Y i (cid:17) (cid:44) σ (cid:118)(cid:116) k (cid:88) i = c mi n i , m = , . . . , M . Under H , the joint distribution of ZZZ ∗ = ( Z ∗ , . . . , Z ∗ M ) (cid:48) is multivariate normal with mean 000 andcovariance matrix RRR ∗ = ( ρ mm (cid:48) ), m , m (cid:48) = , . . . , M . One can then obtain the non-adaptive α -level critical value u ∗ − α based on the null distribution of Z ∗ max = max { ZZZ ∗ } using the R -package mvtnorm .In order to obtain the conditional Type I error probability A = P H ( Z ∗ max ≥ u ∗ − α | YYY ), where YYY are the Stage 1 data, it can be seen that the conditional distribution of ZZZ ∗ given YYY = yyy ismultivariate normal with mean vector k (cid:88) i = c i ¯ y i (cid:44) σ (cid:118)(cid:116) k (cid:88) i = c i n i , . . . , k (cid:88) i = c Mi ¯ y i (cid:44) σ (cid:118)(cid:116) k (cid:88) i = c Mi n i (cid:48) and covariance matrix R R R ∗ = RRR ∗ /
2, where ¯ y i = (cid:80) n i j = y i j / n i , i = , . . . , k . Hence, the condi-tional Type I error probability is A = P H ( Z ∗ max ≥ u ∗ − α | YYY ) = − P H ( ZZZ ∗ ≤ ( u ∗ − α , . . . , u ∗ − α ) (cid:48) | YYY ) , which can be obtained using the pmvnorm function in the R -package mvtnorm .In general, the interim analysis at the end of Stage 1 could yield adapted values of c mi , k , and n i for Stage 2 and, hence, the adapted Z -statistics Z m , m = , . . . , M . Denote ZZZ = ( Z , . . . , Z M ) (cid:48) and Z max = max { ZZZ } . The adaptive critical value ˜ u − α can be obtained by solvingthe equation ˜ A = P H ( Z max ≥ ˜ u − α | YYY ) = − P H ( ZZZ ≤ ( ˜ u − α , . . . , ˜ u − α ) (cid:48) | YYY ) = A , ZZZ given
YYY is multivariate normal with mean vector k (cid:88) i = c i ¯ y i (cid:44) σ (cid:118)(cid:116) k (cid:88) i = c i n i + k (cid:88) i = c i n i , . . . , k (cid:88) i = c Mi ¯ y i (cid:44) σ (cid:118)(cid:116) k (cid:88) i = c Mi n i + k (cid:88) i = c Mi n i (cid:48) and covariance matrix ˜ R ˜ R ˜ R = (cov( Z m , Z m (cid:48) | YYY )), m , m (cid:48) = , . . . , M , wherecov( Z m , Z m (cid:48) | YYY ) = k (cid:88) i = c mi c m (cid:48) i n i (cid:44) (cid:118)(cid:116) k (cid:88) i = c mi n i + k (cid:88) i = c mi n i k (cid:88) i = c m (cid:48) i n i + k (cid:88) i = c m (cid:48) i n i . Use of ˜ u − α as the critical value for the AMCT controls the Type I error probability at level α (M¨uller and Sch¨afer, 2001, 2004; Miller, 2010). Miller (2010) briefly discusses the possibility of extending the AMCT to accommodate esti-mation of the variance σ , the complication being that the conditional Type I error probabilitydepends on the unknown variance. Posch et al . (2004) developed methods to calculate the con-ditional Type I error probability for the one sample t -test given the interim data, but the authorsonly consider the univariate case and the approach does not directly apply to either the singlecontrast test or the multiple contrast test.In this subsection, we extend the AMCT to the unknown variance case by considering thecombined T -statistics T m = k (cid:88) i = c mi ¯ Y i + k (cid:88) i = c mi ¯ Y i S (cid:118)(cid:116) k (cid:88) i = c mi n i + k (cid:88) i = c mi n i = σ Z m S , m = , . . . , M , where the pooled variance estimator is S = k (cid:88) i = n i (cid:88) j = ( Y i j − ¯ Y i ) + k (cid:88) i = n i (cid:88) j = ( Y i j − ¯ Y i ) (cid:44) k (cid:88) i = n i − k + k (cid:88) i = n i − k . As in Section 3.1, we pre-specify c ∗ mi = c mi , k ∗ = k , n ∗ i = n i , and ddd ∗ Stage2 = ddd Stage1 , i = , . . . , k ∗ , m = , . . . , M . The T -statistics for the base test are T ∗ m = k (cid:88) i = c mi ( ¯ Y i + ¯ Y i ) (cid:44) S ∗ (cid:118)(cid:116) k (cid:88) i = c mi n i = σ Z ∗ m S ∗ , m = , . . . , M , where S ∗ = k (cid:88) i = n i (cid:88) j = (cid:104) ( Y i j − ¯ Y i ) + ( Y i j − ¯ Y i ) (cid:105) (cid:44) (2 ν ) , where ν = k (cid:88) i = n i − k . Since S ∗ is independent of Z ∗ m and 2 ν S ∗ /σ ∼ χ ν , the null joint distribution of TTT ∗ = ( T ∗ , . . . , T ∗ M ) (cid:48) is multivariate t with 2 ν degrees of freedom and correlation matrix RRR ∗ . Thenon-adaptive α -level critical value c ∗ − α can then be obtained using the qmvt function in the R -package mvtnorm .The main di ffi culty in the unknown variance case is that the approach outlined in Section3.1 cannot be employed because the conditional distribution of T ∗ m given YYY is not central t under H . We develop the conditional Type I error probability as follows. Denote T ∗ m | YYY = k (cid:88) i = c mi (¯ y i + ¯ Y i ) (cid:118)(cid:116) k (cid:88) i = c mi n i (cid:118)(cid:117)(cid:116) k (cid:88) i = n i (cid:88) j = (cid:110) ( y i j − ¯ y i ) + ( Y i j − ¯ Y i ) (cid:111) (cid:44) ν = U ∗ m (cid:114) V ∗ ν + q ∗ , m = , . . . , M , where U ∗ m = k (cid:88) i = c mi (¯ y i + ¯ Y i ) σ (cid:118)(cid:116) k (cid:88) i = c mi n i , V ∗ = k (cid:88) i = n i (cid:88) j = ( Y i j − ¯ Y i ) (cid:46) σ , and the constant q ∗ = k (cid:88) i = n i (cid:88) j = ( y i j − ¯ y i ) (cid:46) ( ν σ ) . Under H , the joint distribution of ( U ∗ , . . . , U ∗ M ) (cid:48) is multivariate normal with mean vector( b ∗ , . . . , b ∗ M ) (cid:48) and variance-covariance matrix RRR ∗ , where b ∗ m = k (cid:88) i = c mi ¯ y i (cid:44) σ (cid:118)(cid:116) k (cid:88) i = c mi n i , m = , . . . , M . Since V ∗ ∼ χ ν and is independent of ( U ∗ , . . . , U ∗ M ) (cid:48) , the joint density function of ( U ∗ , . . . , U ∗ M , V ∗ ) (cid:48) f ( U ∗ ,..., U ∗ M , V ∗ ) ( u ∗ , . . . , u ∗ M , v ∗ ) = π ) M / | RRR ∗ | / Γ ( ν / ν / × ( v ∗ ) ν / − e − v ∗ / exp (cid:40) −
12 ( u ∗ − b ∗ , . . . , u ∗ M − b ∗ M )( RRR ∗ ) − ( u ∗ − b ∗ , . . . , u ∗ M − b ∗ M ) (cid:48) (cid:41) , where Γ ( · ) is the Gamma function. Now make the transformation T ∗ m | YYY = U ∗ m (cid:114) V ∗ ν + q ∗ , m = , . . . , M , and W ∗ = V ∗ with Jacobian ( W ∗ /ν + q ∗ ) M / . The joint density function of TTT ∗ | YYY is f TTT ∗ | YYY (cid:0) ( t ∗ , . . . , t ∗ M ) | yyy (cid:1) = π ) M / | RRR ∗ | / Γ ( ν / ν / (cid:90) + ∞ (cid:32) w ∗ ν + q ∗ (cid:33) M / ( w ∗ ) ν / − e − w ∗ / × exp (cid:34) − t ∗ (cid:32) w ∗ ν + q ∗ (cid:33) / − b ∗ , . . . , t ∗ M (cid:32) w ∗ ν + q ∗ (cid:33) / − b ∗ M ( RRR ∗ ) − t ∗ (cid:32) w ∗ ν + q ∗ (cid:33) / − b ∗ , . . . , t ∗ M (cid:32) w ∗ ν + q ∗ (cid:33) / − b ∗ M (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) yyy (cid:35) dw ∗ . We then obtain the conditional Type I error probability A = − P H (cid:0) TTT ∗ ≤ ( c ∗ − α , . . . , c ∗ − α ) (cid:48) | YYY (cid:1) = − (cid:90) · · · (cid:90) ( t ∗ ,..., t ∗ M ) ≤ ( c ∗ − α ,..., c ∗ − α ) f TTT ∗ | YYY (cid:0) ( t ∗ , . . . , t ∗ M ) | yyy (cid:1) dt ∗ · · · dt ∗ M . After making the adaptations at the interim analysis, from the conditional distribution of
TTT = ( T , . . . , T M ) (cid:48) given YYY , the adaptive critical value ˜ c − α can be determined as a solution tothe following equation:˜ A = − P H (cid:0) TTT ≤ (˜ c − α , . . . , ˜ c − α ) (cid:48) | YYY (cid:1) = − (cid:90) · · · (cid:90) ( t ,..., t M ) ≤ (˜ c − α ,..., ˜ c − α ) f TTT | YYY (( t , . . . , t M ) | yyy ) dt · · · dt M = A , f TTT | YYY (( t , . . . , t M ) | yyy ) = π ) M / | ˜ R ˜ R ˜ R | / Γ ( ν / ν / (cid:90) + ∞ (cid:18) w ν + q (cid:19) M / w ν / − e − w / × exp (cid:34) − (cid:40) t (cid:18) w ν + q (cid:19) / − b , . . . , t M (cid:18) w ν + q (cid:19) / − b M (cid:41) ˜ R ˜ R ˜ R − (cid:40) t (cid:18) w ν + q (cid:19) / − b , . . . , t M (cid:18) w ν + q (cid:19) / − b M (cid:41) (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) yyy (cid:35) dw ,ν = k (cid:88) i = n i − k , ν = ν + ν , q = k (cid:88) i = n i (cid:88) j = ( y i j − ¯ y i ) (cid:46) ( νσ ) , and b m = k (cid:88) i = c mi ¯ y i (cid:44) σ (cid:118)(cid:116) k (cid:88) i = c mi / n i + k (cid:88) i = c mi n i , m = , . . . , M . H is rejected if T max = max { TTT } ≥ ˜ c − α . Use of the critical value ˜ c − α provides control of theType I error probability at level α according to the CRP principle (M¨uller and Sch¨afer, 2001,2004). To illustrate the adaptive generalized multiple contrast tests (AGMCTs), we generated a nu-merical example. Suppose that there are k = ddd Stage1 = (0 , . , . , . , . (cid:48) . The total sample sizes in two stages are the same ( N = N = n = · · · = n = N / = M = θθθ are shown in Table 1.We assume that the true dose-response model is the Exponential 1 model: f Exp1 ( d , θθθ ) = E + E exp( d /δ ) = . + .
017 exp( d / . . We generate the Stage 1 data from a multivariate normal distribution with mean f Exp1 ( ddd Stage1 , θθθ ) = (0 . , . , . , . , . (cid:48) and covariance matrix σ III = . III . The sample mean and vari-ance estimates from the Stage 1 data are ¯ yyy = (0 . , . , . , . , . (cid:48) and s = . ,respectively. 13he optimal contrast vectors in Stage 1 based on the M = ccc = ( − . , − . , . , . , . (cid:48) ccc = ( − . , − . , − . , . , . (cid:48) ccc = ( − . , − . , − . , . , . (cid:48) ccc = ( − . , − . , . , . , . (cid:48) ccc = ( − . , − . , − . , . , . (cid:48) . After conducting three di ff erent GMCTs using Tippett’s, Fisher’s, and inverse normal combi-nation statistics, we obtain the following Stage 1 p-values: p T = . p F = . p N = . θ ˆ θ ˆ θ are shown in Table 2.Since we only adapt the candidate dose-response models and do not adapt the dosagegroups, the latter remain the same as in Stage 1: ddd Stage2 = (0 , . , . , . , . (cid:48) and n = · · · = n = N / =
24. The Stage 2 data are then generated from a multivariate normaldistribution with the same mean and covariance matrix as in Stage 1. The sample mean andvariance estimates from the Stage 2 data under adaptation are ¯ yyy = (0 . , . , . , . , . (cid:48) and s = . , respectively.The optimal contrast vectors in Stage 2 based on the M = ccc = ( − . , − . , . , . , . (cid:48) ccc = ( − . , − . , − . , . , . (cid:48) ccc = ( − . , − . , − . , . , . (cid:48) ccc = ( − . , − . , . , . , − . (cid:48) ccc = ( − . , − . , . , . , − . (cid:48) . After conducting three di ff erent GMCTs using Tippett’s, Fisher’s, and inverse normal combi-nation statistics, we obtain the following Stage 2 p-values: p T = . p F = . p N = . We use the same simulated data as in Section 4.1 to illustrate the adaptive multiple contrast test(AMCT) for the known variance case (for purposes of this illustration, we use σ = . ). Wefirst obtain the non-adaptive critical value u ∗ − α . The joint null distribution of ZZZ ∗ = ( Z ∗ , . . . , Z ∗ ) (cid:48) is multivariate normal with mean 000 and covariance matrix RRR ∗ , where RRR ∗ = .
977 0 .
912 0 .
842 0 . .
977 1 0 .
977 0 .
750 0 . .
912 0 .
977 1 0 .
602 0 . .
842 0 .
750 0 .
602 1 0 . .
896 0 .
956 0 .
957 0 .
715 1 . The value of u ∗ − α is obtained using the qmvnorm function in the R -package mvtnorm , re-15ulting in u ∗ − α = . ZZZ ∗ given YYY , k (cid:88) i = c i ¯ y i σ (cid:118)(cid:116) k (cid:88) i = c i n i , . . . , k (cid:88) i = c Mi ¯ y i σ (cid:118)(cid:116) k (cid:88) i = c Mi n i (cid:48) = (0 . , . , . , . , . (cid:48) , and the conditional covariance matrix R R R ∗ = RRR ∗ /
2. The conditional error is obtained using the pmvnorm function in the R -package mvtnorm as A = − P H (cid:0) ZZZ ∗ ≤ ( u ∗ − α , . . . , u ∗ − α ) (cid:48) | YYY (cid:1) = . . After adapting the dose-response models as in Section 4.1 above, we obtain the conditionaldistribution of
ZZZ | YYY , which is multivariate normal with mean k (cid:88) i = c i ¯ y i σ (cid:118)(cid:116) k (cid:88) i = c i n i + k (cid:88) i = c i n i , . . . , k (cid:88) i = c Mi ¯ y i σ (cid:118)(cid:116) k (cid:88) i = c Mi n i + k (cid:88) i = c Mi n i , (cid:48) = (0 . , . , . , . , . (cid:48) and covariance matrix ˜ R ˜ R ˜ R = .
500 0 .
487 0 .
454 0 .
339 0 . .
487 0 .
500 0 .
489 0 .
272 0 . .
454 0 .
489 0 .
500 0 .
182 0 . .
339 0 .
272 0 .
182 0 .
500 0 . .
266 0 .
233 0 .
164 0 .
450 0 . . Finally, we obtain the adaptive critical value ˜ u − α = .
978 and the combined test statistics
ZZZ = ( Z , . . . , Z M ) (cid:48) = (2 . , . , . , . , . (cid:48) . We reject H since Z max = . ≥ ˜ u − α . To illustrate the AMCT in the unknown variance case (Section 3.2), we use the same exampledata as in Section 4.1 for M = max and Linear-log candidate dose-response models in Table 1. Other settings are the same asin Section 4.1, including the optimal contrasts for both Stage 1 and Stage 2, and the updateddose-response models for Stage 2.We first obtain the non-adaptive critical value c ∗ − α . The joint null distribution of TTT ∗ = ( T ∗ , T ∗ ) (cid:48) is bivariate t with degrees of freedom 2 ν and correlation matrix RRR ∗ , where ν = N − =
115 and
RRR ∗ = . .
977 1 . The value of c ∗ − α is obtained using the qmvt function in the R -package mvtnorm , resulting in c ∗ − α = . adaptIntegrate function in the R -package cubature . A = − π ) M / | RRR ∗ | / Γ ( ν / ν / (cid:90) + ∞ (cid:90) c ∗ − α −∞ (cid:90) c ∗ − α −∞ (cid:32) w ∗ ν + q ∗ (cid:33) M / ( w ∗ ) ν / − e − w ∗ / × exp (cid:34) − t ∗ (cid:32) w ∗ ν + q ∗ (cid:33) / − b ∗ , t ∗ (cid:32) w ∗ ν + q ∗ (cid:33) / − b ∗ ( RRR ∗ ) − t ∗ (cid:32) w ∗ ν + q ∗ (cid:33) / − b ∗ , t ∗ (cid:32) w ∗ ν + q ∗ (cid:33) / − b ∗ (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) yyy (cid:35) dw ∗ dt ∗ dt ∗ = . . After adapting the dose-response models at the end of Stage 1, we consider the conditionaldistribution of
TTT | YYY . The adaptive critical value ˜ c − α can be obtained by solving the followingequation using a binary search algorithm:˜ A = π ) M / | ˜ R ˜ R ˜ R | / Γ ( ν / ν / (cid:90) + ∞ (cid:90) ˜ c − α −∞ (cid:90) ˜ c − α −∞ (cid:18) w ν + q (cid:19) M / w ν / − e − w / × exp (cid:34) − (cid:40) t (cid:18) w ν + q (cid:19) / − b , t (cid:18) w ν + q (cid:19) / − b (cid:41) ˜ R ˜ R ˜ R − (cid:40) t (cid:18) w ν + q (cid:19) / − b , t (cid:18) w ν + q (cid:19) / − b (cid:41) (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) yyy (cid:35) dw dt dt = A , R ˜ R ˜ R is ˜ R ˜ R ˜ R = .
500 0 . .
487 0 . . Finally, we obtain the adaptive critical value ˜ c − α = .
734 with tolerance 10 − . The com-bined test statistics are TTT = ( T , T ) (cid:48) = (2 . , . (cid:48) and we reject H since T max = . ≥ ˜ c − α . In this section, we conduct simulation studies to compare the operating characteristics of theAGMCTs with those of the AMCT in the setting of a design that adapts the candidate dose-response models based on data from Stage 1. We also compare these with the operating char-acteristics of the corresponding tests in a non-adaptive design.Assume k = ddd Stage1 = (0 , . , . , . , . (cid:48) . The total sample size is the samefor each of the two stages ( N = N ) and the group sample sizes within each stage are equal,with N = N =
60, 120, 180, and 240. The M = θθθ are shown in Table 1. The outcome for each patient is distributedas N ( µ ( d ) , σ ), where the true mean configuration µ ( d ) follows one of the eight di ff erent dose-response models in Table 4, and σ = . E max E max
3, Exponential 1, Exponen-tial 2, Quadratic 2, Step and Truncated-logistic models, the optimal contrasts are not highlycorrelated with those of the candidate models (Figure 2).For the AGMCTs, we use three GMCTs to combine the M = p -values within each stage: Tippett’s ( T ), Fisher’s ( F ) and inverse normal ( N ) combination methods (Ma andMcDermott, 2020). The same GMCT is used in both Stage 1 and Stage 2. To perform theoverall test, only the inverse normal ( Ψ N ) combination statistic is used to combine p and p across stages since our preliminary simulation studies showed that, in general, using Ψ N to18ombine p and p yielded greater power than using Ψ F . The reason for this is that under thealternative hypothesis, p and p both tend to be small and the rejection region of Ψ N is largerthan that of Ψ F when p and p are both small (Wassmer and Brannath 2016, Section 6.2).For the AGMCTs, we report the results of the operating characteristics for both the knownand unknown variance cases. The results for the corresponding GMCTs in a non-adaptivedesign are also reported. For the AMCT, the simulation studies of the operating characteristicsare presented only for the known variance case. The corresponding test in a non-adaptivedesign is just the MCP-Mod procedure, which is equivalent to the GMCT based on Tippett’scombination method in a non-adaptive design.All estimated values of Type I error probability and power are based on 10,000 replicationsof the simulations. The Type I error probabilities for the AGMCTs and the AMCT (Tables A1and A2 in the Appendix) agree with theory that the tests being considered all exhibit control ofthe Type I error probability at α = . E max E max
3, Exponential 1, Exponential 2,Quadratic 2, Step and Truncated-logistic models), however, the AGMCTs and AMCT are morepowerful than the corresponding tests in a non-adaptive design. Another observation is that theoverall performance of the AMCT is similar to those of the AGMCTs.For the unknown variance case, the power curves of the competing tests are shown in Figure4. The results for these comparisons are very similar to those for the known variance case.
In this article, we extend the MCP-Mod procedure with GMCTs (Bretz et al ., 2005; Ma andMcDermott, 2020) to two-stage adaptive designs. We perform a GMCT within each stage and19ombine the stage-wise p -values using a specified combination method to test the overall nullhypothesis of no dose-response relationship. We also consider and extend an alternative AMCTapproach proposed by Miller (2010), which uses the maximum standardized stratified contrastacross Stage 1 and Stage 2 as the test statistic. One issue that deserves further exploration ishow to best determine the “base test” for the AMCT. Our development in Sections 3.1 and3.2 is based on pre-specification of the contrasts, number of candidate dose-response models,and group sample sizes to be the same in Stage 2 as they were in Stage 1. While this isnot necessarily the best choice, in the absence of the ability to enumerate all possible two-stage designs being considered, it might be quite reasonable in practice. An issue that remainsunresolved is that of accurately computing the conditional error and adaptive critical value forthe AMCT when the variance is unknown since these involve multidimensional integrals.Simulation studies demonstrate that the AGMCTs and AMCT are generally more powerfulfor PoC testing than the corresponding tests in a non-adaptive design if the true dose-responsemodel is, in a sense, not “close” to the models included in the initial candidate set. Thismight occur, for example, if the selection of the candidate set of dose-response models is notwell informed by evidence from preclinical and early-phase studies. This is consistent withintuition: if the dose-response models are badly misspecified at the design stage, using datafrom Stage 1 to get a better sense of the true dose-response model and using data from bothStage 1 and Stage 2 to perform an overall test for H should result in increased power. Onthe other hand, if the true dose-response model is “close” to the models specified in the initialcandidate set, the non-adaptive design is su ffi cient to detect the PoC signal. In this case, theadaptive design does not provide any benefit and results in a small loss of e ffi ciency.Comparisons among the di ff erent AGMCTs and the AMCT did not reveal major di ff er-ences in their operating characteristics in general. Di ff erences among the AGMCTs tended tobe larger in the setting of a non-adaptive design (Ma and McDermott, 2020). In principle, theAGMCTs proposed here for two-stage adaptive designs could be extended to multiple stages,although the circumstances under which that would be beneficial are not clear.20 eferences
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Conflict of Interest
The authors have declared no conflict of interest. M = E max f ( d , θθθ ) = E + E max d / ( ED + d ) = . + . d / (0 . + d )Linear-log f ( d , θθθ ) = θ + θ log(5 d + = . + { . / log(6) } log(5 d + f ( d , θθθ ) = θ + θ d = . + . d Quadratic f ( d , θθθ ) = θ + θ d + θ d = . + . d − . d Logistic f ( d , θθθ ) = E + E max / [1 + exp { ( ED − d ) /δ } ] = . + . / [1 + exp { (0 . − d ) / . } ] Table 2: M = E max f ( d , θθθ ) = . + . d / (0 . + d )Linear-log f ( d , ˆ θ ˆ θ ˆ θ ) = . + .
24 log(5 d + f ( d , ˆ θ ˆ θ ˆ θ ) = . + . d Quadratic f ( d , ˆ θ ˆ θ ˆ θ ) = . + . d − . d Logistic (polynomial approximation) f poly5 ( d , ˆ θ ˆ θ ˆ θ ) = . − . d + . d − . d + . d Table 3: Combining p and p across stages using Fisher’s and inverse normal combinationmethods. Fisher Inverse NormalWithin-stagecombinationstatistic Acrossstages Ψ F Overallp-value Reject H Within-stagecombinationstatistic Acrossstages Ψ N Overallp-value Reject H Ψ T Ψ T Ψ F Ψ F Ψ N Ψ N Table 4: Eight di ff erent true dose-response models considered in the simulation studies. E max . + . d / (0 . + d ) E max . + . d / (0 . + d )Exponential 1 0 . + .
017 exp { d log(6) } Exponential 2 0 . + . d / . . + . d − . d Double-logistic (cid:34) . + . + exp { . − d ) } (cid:41) I ( d ≤ . + (cid:40) . + . + exp { d − . } (cid:35) I ( d > . . + . I ( d ≥ . . + . / (cid:2) + exp { . − d ) } (cid:3) ppendix In this section, we display the Type I error probabilities of the AGMCTs and the AMCT for theknown and unknown variance cases in Tables A1 and A2, respectively.Table A1: Type I error probabilities of the AGMCTs and the AMCT in the known variancecase for designs that adapt the candidate dose-response models, as well as the correspondingtests in a non-adaptive design. N = N = N = N = N = N = N = N = Table A2: Type I error probabilities of the AGMCTs in the unknown variance case for designsthat adapt the candidate dose-response models, as well as the corresponding tests in a non-adaptive design. N = N =
60T F NAdaptive 0.0491 0.0509 0.0519Non-adaptive 0.0484 0.0512 0.0512 N = N = N = N = N = N =240T F NAdaptive 0.0487 0.0519 0.0527Non-adaptive 0.0507 0.0510 0.0515