Novel Statistical Designs for Phase I/II and Phase II Clinical Trials With Dose-Finding Objectives

Major Considerations and Objectives of Seamless Phase I/II Trials

Throughout, we assume that we are given a set D of K doses (ie, D = {d ₁, … ,d _K} ) of the selected agent to investigate in the study. We shall first consider bivariate binary efficacy and toxicity outcomes. Let X denote the dose and (Y_T , Y_E ) denote binary indicators of toxicity (Y_T ) and efficacy (Y_E ). Let p(d) = Pr(Y_T = 1|X = d) and q d = Pr Y E = 1 | X = d denote, respectively, the marginal probabilities of toxicity and efficacy for a patient treated at dose d ∈ D . Let 0 < p ˉ T < 1 and 0 < q _ E < 1 be some prespecified threshold values for the maximum acceptable toxicity rate ( p ˉ T ) and the minimum acceptable efficacy rate ( q ˉ E ). The MTD is defined as the highest dose from D with p d ≤ p ˉ T , and the minimum effective dose (MED) is defined as the lowest dose from D with q d ≥ q ˉ E . We shall use d_L and d_U to denote the MED and the MTD, respectively. In most practical situations, d L < d U . The interval [d_L , d_U ] is called a therapeutic window (TW). Doses within TW should be suitable for further investigation. Of particular interest is the dose that maximizes the probability of “success” (ie, probability of efficacy without toxicity). Let q ˜ d = P r ( Y E = 1 | Y T = 0 , X = d ) denote the conditional probability of efficacy given no toxicity at dose d. Then the probability of success at dose d is

In many dose-response settings, the function α(d) in equation (1) is unimodal. The most successful dose is the dose that maximizes equation (1). The conditions for existence of MSD are given in Durham et al. ²⁸ Note that the MSD may fall outside TW (eg, the risk of toxicity at MSD may be higher than maximally acceptable). This leads to a definition of the optimal safe dose (OSD), ²⁹ the dose that maximizes equation (1) under an additional restriction on the marginal probability of toxicity:

Similar concepts can be developed from a Bayesian perspective. ³⁰ Let θ denote a parameter vector that characterizes dose-toxicity and dose-efficacy probability relationships, and let π_T(d, θ) and π_E(d, θ) denote, respectively, prior distributions for the marginal probability of toxicity and efficacy at dose d. Based on data from the trial, the corresponding posterior distributions are derived. A dose d is said to be acceptable if

for some prespecified probability cutoffs c_T and c_E . Provided that the set of acceptable doses is nonempty, one can define the most desirable dose as one that satisfies conditions (3) and for which the posterior probability of success (efficacy without toxicity) is maximum.

Overall, the objectives of a phase I/II trial may include the following:

Various adaptive designs, including algorithm- and model-based approaches, have been proposed to achieve these objectives in practice. Given small sample sizes in phase I/II trials, simulations under various experimental scenarios are necessary to assess design performance. Some measures of goodness of the design include percentage of correct dose identification, number of subjects treated at doses within a TW, and estimation efficiency of the target dose and other parameters of interest.

Phase I/II Up-and-Down Designs

Up-and-down designs are nonparametric design procedures that have been extensively studied in the context of phase I oncology trials with toxicity outcomes. ³¹ They can be also applied in phase I/II trials with efficacy and toxicity outcomes. In essence, an up-and-down design generates dose assignments sequentially, taking into account the most recent outcome at a current dose level (Markovian design). The next patient’s dose assignment can be one dose higher, one dose lower, or the same dose.

For bivariate binary toxicity and efficacy outcomes, Kpamegan and Flournoy ^32,33 proposed an “optimizing” up-and-down design to target MSD under the assumption that α(d) in equation (1) is unimodal. The design induces an irreducible Markov chain on a lattice of doses and converges to a stationary distribution around the MSD. The empirical treatment mode converges to the mode of the stationary distribution. Ivanova ²⁹ proposed another up-and-down design to target OSD. Ivanova’s design was shown via simulation to yield a high percentage of correct dose selection ^29,34 ; however, it was also shown to have lower estimation efficiency than model-based penalized optimal adaptive designs. ³⁵ Hardwick et al ³⁶ proposed “directed walk” designs to target MSD. At each step, an estimate of the MSD is obtained based on data accrued to date (non-Markovian design), and the dose is modified (increased or decreased by one level or is kept unchanged) in the direction of the updated optimum. At the end of the trial, the optimum dose is estimated according to the curve-estimation scheme employed in the design. Seven parametric and nonparametric methods of estimation of the efficacy-toxicity relationship were explored. The design was evaluated through simulation in terms of probability of correct selection, decision efficiency, sampling efficiency, and expected successes lost. The major conclusion is that “the smoothed shape-constrained methods performed roughly as well as the parametric techniques for each model, while requiring fewer assumptions”.³⁶ Durham et al ²⁸ discussed various sequential designs to target MSD, including parametric designs, up-and-down designs, the Kiefer-Wolfowitz procedure, ³⁷ and randomized Pólya urn designs. For the latter approach, dose assignments converge almost surely to a global maximum, which is attractive in cases when α(d) is not unimodal. An application of the generalized Pólya urn for dose-finding efficacy-toxicity studies is also discussed by Rosenberger. ³⁸

Model-Based and Bayesian Adaptive Phase I/II Designs

The development of model-based adaptive designs starts with the formulation of a statistical model for the joint probabilities:

where d ∈ { d 1 , … , d K } , θ is a vector of unknown parameters and ψ y z = ψ y z d , θ are some nonlinear functions ( 0 < ψ y z < 1 and ∑ y = 0 1 ∑ z = 0 1 ψ y z = 1 ). The likelihood for a single observation at dose d is

and the full likelihood based on independent individual observations from m patients is L m θ = ∏ l = 1 m L ( θ , y l , z l , x l ) . Assuming θ has a prior density g(θ), the posterior density of θ is g θ | D a t a ∝ L m θ g ( θ ) , and the posterior joint probabilities can be updated accordingly. Given these posteriors, a Bayesian adaptive dose-finding algorithm determines a set of acceptable doses satisfying conditions (3) and then selects the “most desirable” dose among the acceptable ones for the next patient or cohort of patients. Thall and Russell ³⁹ proposed a Bayesian dose-finding design based on a 3-parameter proportional odds model for trinary outcomes (no effect; success; toxicity). Thall and Cook ⁴⁰ considered a 4-parameter continuation-ratio model for trinary outcomes and a 6-parameter Gumbell model for a situation when all 4 efficacy-toxicity outcome combinations are observable. An application of their method is presented in Thall et al. ⁴¹ Thall et al ⁴² extended the method ⁴⁰ to account for patient important prognostic covariates.

A number adaptive designs for phase I/II trials were proposed on the basis of extensions of the celebrated continual reassessment method of O’Quigley et al. ⁴³ O’Quigley et al ²⁰ developed a phase I/II dose-finding design in the context of a pediatric HIV trial where efficacy is quantified by viral load reduction and toxicity is defined as some drug-related side effect. Initially, a CRM design is used to target a dose with an acceptable toxicity rate. The probability of success is estimated as well, and if it is lower than the minimum desired amount, the target toxicity rate is increased. This is continued until a dose with an acceptable probability of success is found or until all doses have been exhausted. As more patients are treated in the trial, a sequential probability ratio test is applied to determine whether the success rate at a particular dose is sufficiently high. Indeterminate results lead to further experimentation; otherwise, the trial may stop early for efficacy or futility. Zohar and O’Quigley ⁴⁴ proposed a similar likelihood-based CRM design but with stricter monitoring rules for toxicity. Their design was tested via simulations under 6 efficacy-toxicity scenarios, along with the procedure of O’Quigley et al. ²⁰ When appropriate design calibrations are applied, the 2 methods result in high percentages of correct dose selection and lower than the maximum allowed sample sizes at stopping. Braun ⁴⁵ proposed “bivariate CRM” for phase I/II trials using 1-parameter working models for marginal probabilities of efficacy and toxicity and a joint model that includes within-subject association between efficacy and toxicity. The bivariate CRM is likely to identify the correct dose when the dose-response curve is steep around the true target and the starting dose is close to this target; however, it may be inferior to some nonparametric designs (eg, the design of Gooley et al ²⁷ ) in terms of proportion of correct dose selection. Zhang et al ⁴⁶ developed a CRM-type design under an assumption of a continuation ratio model. Seegers et al ⁴⁷ proposed an approach based on maximizing the marginal probability of efficacy subject to constraint on the marginal probability of toxicity. Zhong et al ⁴⁸ extended the CRM to 3 outcomes: toxicity, efficacy, and surrogate efficacy. In their dose-finding algorithm, after posterior distributions are obtained, one first checks the criteria for early termination (for excess toxicity or futility). If the stopping criteria are not met, a dose estimated to be the most likely closest to the target probabilities of efficacy and toxicity is selected for the next cohort of patients. No dosage shift by more than one level is permitted. The trivariate CRM exhibits better performance than the corresponding bivariate CRM based on surrogate efficacy data, both in terms of proportion of correct dose identification and in terms of proportions of patients treated at the correct dose.

Whitehead et al ^49,50 proposed Bayesian decision-theoretic designs for phase I/II trials. Two-parameter logistic regression models are used for the marginal dose-toxicity probability and the conditional probability of efficacy given no toxicity. The goal is to estimate TW. Optimal dose escalation is performed according to the variance gain based on a weighted combination of the posterior variances of MED and MTD. Several other Bayesian phase I/II adaptive designs maximizing various utility functions related to the target probability of efficacy without toxicity were proposed by Loke et al, ⁵¹ Wang and Day, ⁵² and Houede et al. ⁵³

Phase I/II Optimal Adaptive Designs

Dragalin and Fedorov ³⁵ used convex optimal design theory to develop efficient and ethical phase I/II trial designs. They assumed Gumbel bivariate logistic and Cox bivariate binary models for the joint probabilities of efficacy-toxicity. Let ξ = { ( d k , ρ k ) , k = 1 , … , K } denote the design measure (d_k are distinct dose levels, 0 ≤ ρ k ≤ 1 is the probability mass at dose d_k , and ∑ k = 1 K ρ k = 1 ). Assuming that we have independent observations, the Fisher information matrix for design ξ is

where μ d k , θ = − E ∂ 2 ∂ θ ∂ θ T log L ( θ , y , z , d k ) is the information of a single observation at dose d_k . Importantly, M = M ξ , θ is inversely proportional to the asymptotic variance-covariance matrix of an efficient estimator of θ; therefore, maximizing M, in some sense, by choice of ξ, an experimenter can achieve accurate estimation of various parameters of interest. Let Ψ ( M ) be some convex criterion that reflects the study objective; let ϕ d , θ > 0 be some convex penalty function that reflects ethical and/or economical loss associated with dose d (eg, ϕ d , θ = ψ 10 d , θ − 1 1 − ψ 01 d , θ − ψ 11 d , θ − 1 to encourage efficacious doses and penalize toxic ones); and let Φ ξ , θ = ∑ k = 1 K ρ k ϕ ( d k , θ ) be the expected penalty for design ξ. Dragalin and Fedorov ³⁵ obtained solutions to several local optimal design problems, including D-optimal design, which minimizes the criterion Ψ M = − l o g | M | (maximum information per unit), and a penalized D-optimal design, which minimizes − ( log M − log Φ ( ξ , θ ) ) (maximum information per cost unit). The local optimal designs depend on the true value of θ, which is unknown in practice. However, one can estimate θ sequentially using emerging data from the trial and construct optimal adaptive designs. Dragalin and Fedorov ³⁵ studied adaptive D-optimal design and adaptive penalized D-optimal design via simulation and showed that the latter design is advantageous over Ivanova’s up-and-down design ²⁹ both in terms of accuracy of estimation and percentage of correct selection of the optimal dose. Dragalin et al. ⁵⁴ gave an application of an adaptive penalized D-optimal design in the context of dose-finding study of an anticoagulant compound for prevention of venous thromboembolism.

Other Phase I/II Adaptive Designs

So far, we have discussed designs for which both efficacy and toxicity outcomes are binary. However, other types of outcomes (eg, categorical, continuous, time to event) are frequently encountered in practice.

Several papers ^{55 –57} proposed Bayesian adaptive designs based on joint modeling of binary toxicity and continuous efficacy outcome. Specifically, Bekele and Shen ⁵⁵ used latent modeling to account for the correlation between binary toxicity and continuous efficacy (biomarker expression) and proposed a Bayesian design for identifying a dose with minimal toxicity and sufficiently high level of biomarker expression. The authors proposed randomizing patients adaptively among acceptable doses such that doses with higher “preference” scores are assigned more frequently. Zhou et al ⁵⁶ assumed a 2-parameter logistic model for a binary adverse event and a linear mixed effects model (conditional on the absence of adverse event) for a continuous efficacy outcome. Prior and posterior mode estimates are used to facilitate adaptive dose-escalation, and a safety constraint is imposed to avoid assignments to unsafe doses. As an illustrative example, the authors used a clinical trial of an anti-factor Xa compound. Hirakawa ⁵⁷ considered a 4-parameter logistic model with multiplicative heteroscedastic variance for continuous efficacy (conditional on binary toxicity) and a 2-parameter logistic model for the marginal probability of toxicity. A sequential dose allocation procedure with appropriate stopping rules is based on minimizing the weighted Mahalanobis distance from the posterior mean of the outcomes to the “most desirable point.” Hirakawa’s procedure ⁵⁷ was compared with the method of Bekele and Shen ⁵⁵ via simulations and was found to be advantageous over the latter—similar or higher percentage selection of correct dose and generally lower average number of patients assigned to unacceptably toxic or futile doses.

Yuan and Yin ⁵⁸ proposed Bayesian adaptive designs based on joint time-to-event modeling of efficacy and toxicity. They used marginal Cox proportional hazards model for time-to-toxicity and the cure rate model for time-to-efficacy, with Clayton copula to account for the correlation. The objective is to identify the treatment arm that meets the marginal safety and efficacy requirements and maximizes the desirability of the dose expressed as the ratio of areas under survival curves of the times to toxicity and efficacy. Lei et al ⁵⁹ considered a setting with time-to-event efficacy and binary toxicity, with Cox proportional hazards model as the marginal model for efficacy and a random-effect probit model for toxicity. The marginal models share common random effects to introduce correlation between efficacy and toxicity. Bayesian adaptive randomization based on an efficacy-toxicity tradeoff index is applied to allocate patients among the doses, with additional stopping rules that allow early termination of unsafe or inefficacious dose levels. Operating characteristics of the proposed design were assessed by simulation under several experimental scenarios, including misspecification of the model or the prior distribution. The design was found to assign more patients to dose levels with most appropriate efficacy-toxicity profiles and yield higher correct selection probabilities compared to the adaptive randomization design based solely on efficacy outcomes. Yin et al ¹⁹ proposed a 2-stage design for a phase I trial of cytostatic agent. At the first stage, dose escalation with a traditional 3+3 design is applied to determine the upper dose-searching bound. At the second stage, the dose search continues on the basis of efficacy data while monitoring for toxicity as well. Efficacy is modeled as a time-to-event endpoint that may be censored by the decision-making time to avoid enrollment suspensions. A dose with the highest estimated efficacy rate is selected for the next cohort of patients. The proposed design was compared with a 2-stage design that requires the use of complete (uncensored) efficacy data. The 2 designs have a similar percentage of correct dose selection, while the former results in substantial reduction of the trial duration.

Several papers provide adaptive designs for phase I/II drug combination trials. ^{60 –62} Bayesian adaptive phase I/II dose-schedule finding designs were proposed in Li et al ⁶³ and Zhang and Braun. ⁶⁴

In summary, adaptive phase I/II designs hold much potential and promise to improve dose finding in early development studies. The list of the references given here is by no means exhaustive. More novel methods are being developed, and more trials with these designs are anticipated to be conducted in the future. A successful implementation of such designs relies on validated statistical software. The MD Anderson center website (https://biostatistics.mdanderson.org/SoftwareDownload/) contains many software packages implementing adaptive phase I/II designs, and they are freely downloadable.

Trend Tests and Multiple Comparison Procedures

Consider a parallel group design to compare the effects of increasing dose levels of a drug (treatments 1 , … , K ) versus placebo (treatment 0). Let Y_jk denote the response of the j th patient in the k th group, where j = 1 , … , n k , k = 0 , 1 , … , K , and ∑ k = 0 K n k = n . Assume Y_jk ’s are independent random variables with E ( Y j k ) = μ k and v a r Y j k = σ k 2 . Therefore, we have a saturated model with no assumption on the functional form of the dose-response curve. As the first step for comparison of treatment effects, one may want to test the homogeneity hypothesis,

versus the alternative,

Assuming normal responses with homogeneous variance ( σ 2 ), an analysis of variance F test can be used for this purpose. ⁶⁷ More specific single degree of freedom tests can be constructed with linear contrasts. ⁶⁶ For binary outcomes, one can use the Wald chi-square test with K degrees of freedom or a likelihood ratio test. Often times, it is feasible to assume that a dose-response curve is monotone increasing, in which case one may want to test the hypothesis in equation (6) versus an ordered alternative:

Statistical methodologies for testing H ₀ versus H ₂ for continuous responses include both nonparametric ⁶⁹ and parametric tests. ^70,71 For binary responses, the popular trend tests include the Cochran-Armitage test and multiple contrast tests ⁷² to name a few. The power of a trend test depends on the functional from of an underlying dose-response relationship that is unknown at the outset. A formal approach to incorporate model uncertainty in the selection of a trend test is given in Bretz et al. ²⁴

Once an overall evidence of the drug effect has been established, the next step is to identify doses that are different from placebo. One important goal is to identify the MED, “the smallest dose producing a clinically important response that can be declared statistically, significantly different from the placebo response.”⁶⁵ Assuming an analysis of variance model Y j k = μ k + ϵ j k with ϵ j k ~ N ( 0 , σ 2 ) where larger response implies better treatment efficacy, the MED can be quantified as

for some prespecified clinically relevant treatment difference Δ > 0 . Dunnett’s method ⁷³ can be used to obtain 100 1 − α % simultaneous lower confidence bounds on μ k − μ 0 for k = 1 , … , K , and the lowest dose with the confidence bound greater than Δ can be chosen as a conservative estimate of the MED. ²¹ One can also apply stepwise procedures by testing either a predefined sequence of hypotheses (“closed testing” approach) or the sequence of hypotheses defined by the order of responses to identify doses that are significantly different from the placebo while maintaining the family-wise error rate. ⁶⁶ See Tamhane and Logan ⁷⁴ for a thorough overview of modern multiple comparison procedures in dose-response studies.

Modeling Techniques

Let d 1 , … , d K denote the dose levels of a drug measured on a continuous scale, and let d ₀ = 0 denote the placebo. For a continuous response variable Y, assume a statistical model

where θ is a vector of model parameters, f d k , θ is the mean response at dose d_k ( k = 0 , 1 , … , K ), j indexes the patient within dose d_k , and ϵ j k are independent random errors, distributed as N ( 0 , σ 2 ) . Various linear and nonlinear functions for the mean response have been proposed in the literature. For instance, a 4-parameter sigmoid E_max model,

is increasingly popular. ^75,76 In equation (10), θ = ( E 0 , E m a x , E D 50 , r ) , where E ₀ is the effect at the zero dose, E_max is the maximum change in effect attributable to the drug (at an infinite dose), ED ₅₀ is the dose that produces 50% of E_max , and r is a parameter that determines the shape of the dose-response curve (the Hill parameter). A nonlinear least squares estimator of θ can be obtained by minimizing the deviance function ∑ k = 0 K ∑ j = 1 n k ( Y j k − f d k , θ ) 2 ; numerical algorithms for this purpose have been implemented in SAS and R. In addition to the vector θ itself, one may be interested in estimating functions of θ. For instance, the MED for model (10) can be defined as

One can estimate MED in equation (11) by replacing θ with its efficient estimator and use bootstrap methods to obtain the corresponding confidence intervals.

The main advantage of using the modeling approach in dose-response studies is borrowing strength across dose levels and extrapolation beyond the doses actually studied. The target dose that corresponds to a certain degree of the drug effect can be formally quantified and estimated on the basis of data from the trial. A disadvantage is susceptibility of the methods to modeling bias. Since at the beginning of the trial there is little information about the underlying dose-response relationship, the postulated model is just a best guess. In the regulatory environment, it is important to prespecify the analysis strategy in advance. One possibility is to specify several candidate models and utilize one that provides best fit given observed data. ⁷⁷ Another approach is a combination of multiple comparisons with modeling techniques. ²⁴

A Combination of Multiple Comparisons With Modeling Techniques (MCPMod)

Bretz et al ²⁴ proposed a novel approach to utilize various candidate models at the trial design stage and select the most appropriate ones for subsequent studies while controlling the familywise error rate. At the beginning of the trial, a set of candidate models is posited. When experimental data become available, each model is fitted and based on the obtained estimates the significance of the dose-response relationship for the given model is tested. The contrast coefficients for each test are chosen in an “optimal” way to account for the assumed dose-response relationship. The appropriate statistical adjustments are made for multiple testing. If the null hypothesis of a flat dose-response cannot be rejected, the procedure stops; otherwise, the significant models are considered for subsequent studies. Based on the chosen models, one can perform various statistical inference procedures. The MCPMod methodology can be thought of as an adaptive analysis strategy. An R package for MCPMod implementation is available. ⁷⁸

Optimal Designs for Phase II Dose-Finding Studies

So far, we have discussed statistical methodologies for addressing specific research questions in dose-finding studies. Another important aspect is optimization of the trial design. One can distinguish optimal allocation designs for studies with multiple comparisons and optimal dose-finding designs for studies with modeling techniques.

The setup for a study with multiple comparisons includes K + 1 treatment groups (treatment 0 is placebo and treatments 1 , … , K are increasing dose levels of the drug). Define a vector of treatment allocation proportions ρ = ( ρ 0 , ρ 1 , … , ρ K ) , where 0 ≤ ρ k ≤ 1 and ∑ k = 0 K ρ k = 1 . In practice, for a study of size n, a rounding procedure is applied to have n k = n ρ k subjects assigned to the k th group such that ∑ k = 0 K n k = n . An optimal design problem is to find an optimal allocation vector ρ ∗ as a solution to some formal optimization problem. For instance, in the binary response case, the goal may be to minimize the expected number of treatment failures in the study subject to a lower bound on the power of the homogeneity test. ⁷⁹ An overview of optimal allocation designs in clinical trials is given in Sverdlov and Rosenberger. ⁸⁰ Depending on the optimization problem, optimal allocation proportions may be unbalanced across the treatment arms. At least 3 factors may contribute to an unbalanced allocation: heterogeneity in the response variance, unequal interest in specific treatment comparisons, and ethical constraints. Frequently, optimal allocation designs depend on unknown model parameters and cannot be implemented directly. This motivates the development of optimal response-adaptive randomization. ⁸¹ Two types of response-adaptive randomization procedures with established statistical properties are the doubly adaptive biased coin design ⁸² and optimal randomized urn models. ⁸³

Optimal designs for model-based dose-finding problems involve searches over a continuum of doses. Given a dose-response model parameterized by vector θ, the design is characterized by a probability measure ξ = { x k , ρ k , < , where x_k ’s are distinct points in the dose space (eg, in an interval [d ₀, d_K ). An optimum design problem is to determine the probability measure ξ = { x k , ρ k , k = 1 , … , K ; ∑ k = 1 K ρ k = 1 } to optimize some convex optimality criterion. Several recent papers provided optimal designs for different models with various dose-finding objectives. ^{84 –90} While many of these optimal designs are based on nonlinear models and depend on unknown model parameters, they do provide insight for practical use. One can also construct sequential or multistage adaptive designs utilizing likelihood-based or Bayesian methods for parameter estimation to generate dose assignments near the “true” optimal design.

Adaptive Dose-Ranging Studies

Adaptive dose-ranging (ADR) studies are multistage adaptive designs that are aimed at achieving selected dose-finding objectives. The key objectives—including testing the overall dose-response, identification of the MED, and estimation of the dose-response profile—have been discussed in the previous sections. At the conclusion of phase II, some key decisions are to be made: whether to proceed to phase III, conduct an additional study, or terminate the development of the compound. Recently, the PhRMA Working Group published 2 papers that compare various ADR designs with traditional analysis of variance–type designs through simulation. ^91,92 The major finding from these papers is that ADR designs can generally improve the trial efficiency compared to conventional single-stage equal allocation designs. However, no design is uniformly better than others when assessed in terms of multiple efficiency criteria across different experimental setups. This reinforces the importance of the clear definition of study objectives and the assessment of design characteristics via simulation under a variety of standard to worst-case scenarios. Some important considerations for selecting a phase II dose-ranging study design are as follows:

What is the primary objective of the study? If the goal is to select a dose (eg, MED) for subsequent testing, then ADR studies that optimize estimation efficiency of the target dose, such as an adaptive MCPMod ^24,25,93 or a general adaptive dose allocation, ⁹⁴ may be considered. For a trial with dose-response estimation objective, the methods based on D-optimality ⁷⁵ may be useful. For a trial with multiple objectives, some useful ADR designs can be found in Miller et al ⁷⁶ and Padmanabhan and Dragalin. ⁹⁵

What is the design working model? If one is willing to assume a monotone dose-response relationship, one can consider adaptive designs based on sigmoidal E_max model. ^75,76 A popular model that does not require the monotonicity assumption is the normal dynamic linear model. Adaptive designs based on normal dynamic linear model can be found in Smith et al, ⁹⁶ Weir et al, ⁹⁷ and Berry et al. ⁹⁸ For a completely nonparametric approach, one can consider an adaptive t-statistic design. ²⁶

How many stages are there in the design? An adaptive design can be fully sequential or have 2 or more stages. In regard to the ADR designs studied by PhRMA Working Group, a note was made in Pinheiro et al: “there was little benefit in more than two IA (Interim Analyses) for most, but not all, of the methods.” ^99(p440)

Additional design considerations include the choice of statistical philosophy to perform design adaptations (Bayesian or likelihood based), the choice of stopping criteria (eg, for futility, safety, or efficacy), and the choice of the maximum sample size. Also, regarding experimental parameters and design operating characteristics for simulation studies, careful selection is essential.

In summary, a well-designed and well-conducted ADR study can help investigators reach correct decisions and ensure that the “right” (safe and efficacious) doses of an investigational drug are carried forward for testing in pivotal phase III studies.