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Abstract

In the past decade, the pharmaceutical industry experienced a paradigm shift from classical to adaptive clinical trial design. The high NDA failure rate and the increasing cost in pharmaceutical R & D is the motivation behind the innovation. Biostatisticians in collaboration with physicians and other major stockholders in pharmaceutical R & D are the driving force in this revolution. In this review paper, we provide an overview of adaptive trial design, covering majority of types of adaptive designs, and the opportunities, challenges, and controversies surrounding adaptive trials. We cover the topic broadly as there have been explosions of research papers that consider adaptive design over the past decade. Adaptive designs have become very popular, making it impossible to cover them all in a single overview paper.

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Keywords

Adaptive clinical trial; Adaptive design; Winner design add-arm design; Adaptive randomization; CRM; Sample-size reestimation; Group sequential design; Error-spending

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Adaptive Design Methods in Clinical Trials

Investment in pharmaceutical research and development (R & D) has more than doubled in the past decade. The increase in spending for biomedical research does not reflect an increased success rate of pharmaceutical development. There are several critical areas for improvement in drug development. One of the obvious areas for improvement is the design, conduct, and analysis of clinical trials. Improvement of the clinical trials process includes [1] the development and utilization of biomarkers or genomic markers [2], the establishment of quantitative disease models, and [3] the use of more informative designs such as adaptive and/or Bayesian designs. We should not use the evaluation tools and infrastructure of the last century to develop this century’s advances. Instead, an innovative approach using adaptive design methods for clinical development must be implemented.

In general, an adaptive design consists of multiple stages. At each stage data analyses are conducted and adaptations take place based on pdated information to maximize the probability of success of a trial. An adaptive design is a clinical trial design that allows adaptations or modifications to aspects of the trial after its initiation without undermining the validity and integrity of the trial [1-3].

The adaptations may include, but are not limited to, (1) group sequential designs, (2) sample-size adjustable designs, (3) pickthe- winner and add-arm designs, (4) adaptive treatment allocation designs, (5) adaptive dose-escalation designs, (6) biomarker-adaptive designs, (7) adaptive treatment-switching designs, (8) adaptive dose-finding designs, and (9) adaptive error-spending designs. An adaptive design usually consists of multiple stages. At each stage, data analyses are conducted, and adaptations are taken based on updated information to maximize the probability of success.

An adaptive design must preserve the validity and integrity of the trial. The validity can be classified as internal and external. Internal validity is the degree to which we are successful in eliminating confounding variables and establishing a cause-effect relationship (treatment effect) within the study itself. External validity is the degree to which findings can generalize to the population at large. Integrity involves minimizing operational bias; creating a scientifically sound rotocol design; adhering firmly to the study protocol and standard perating procedures (SOPs); executing the trial consistently over time and across sites or country; providing comprehensive analyses of trial data and unbiased interpretations of the results; and maintaining the confidentiality of the data.

The term “adaptive design” in clinical trial may have been first introduced by Bauer as early as 1989 [4-8]. Adaptive Randomization designs were early studied by Hoel, Simon and Weiss, Simon, Wei, Wei & Durham, Hardwick and Stout, Rosenberger & Lachin, and more recently by Rosenberger & Lachin, Rosenberger and Seshaiyer, Hu, and Ivonova under different names [9-20]. The recent developments in adaptive randomization were covered in the book edited by Sverdlov [21]. Bayesian adaptive dose-escalation design was first studied by O'Quigley, Pepe and Fisher [22]. Chang and Chowstudied a hybrid approach with adaptive randomization [1]. Adaptive dose-escalation and dose-finding design were further studied by vanova & Kim, Yin & Yuan, and Thall [23-25]. Winner design with three groups [26] and drop-loser and add-arm designs [27] were also developed for later stage trial designs.

A randomized concentration-controlled trial (RCCT) is one in which subjects are randomly assigned to predetermined levels of average plasma drug concentration. The dose adaptation takes place based upon observed concentrations that are used in each patient to modify the starting dose to achieve the pre-specified randomized concentration. The RCCT is designed to minimize the inter individual pharmacokinetic (PK) variability within comparison groups and consequently decrease the variability in clinical response within these groups [28,29].

Adaptive methods were further developed to allow sample size reestimation [30-32], which has caused controversies surrounding the unequal weights for patients from different stages and efficiency of the design compared to GSD [33].Biomarker utilization in clinical trial was studied by Simon & Maittournam, Mandrekar & Sargent, Weir & Walley, Simon, and Baker et al. [34-38]. Population Enrichment design and adaptive designs using biomarker or genomic markers are adaptive designs that allow us to select target population based on interim data [39]. Simon & Wang and Freidlin, Jiang & Simon studied a genomic signature design, Jiang, Freidlin & Simon proposed Biomarkeradaptive threshold design, Chang and Wang, Hung & O'Neill studies population enrichment design using biomarker, which allow interim decision on the target population based on power or utility [40-45]. Song & Pepe studied markers for selecting a patient's treatment [46]. Further studies on biomarker-adaptive design were done by Beckman, Clark, and Chen for oncology trials, and more recently by Wang, Wang, Chang, and Menon using a two-level relationship between biomarker-primary endpoint [47-50].

Recently, adaptive design has been developed into multiregional clinical trials [51], in which sample size can be redistributed among different regions depending on the interim data in different regions to maximize the probability of success.

For any particular compound, there are many adaptive designs can be used. The optimal design fits particular needs should be based on so-called evaluation matrix. Menon and Chang studied the optimization of the adaptive design was using simulations [52]. Chang provides a practical approach [53]. For adaptive trial interim data monitoring and adaptation-making, readers can refer to books Proschan, Lan, and Wittes, DeMets, Furberg, and Friedman, and Chang [54-56].

The analyses of data from adaptive trials are complicated with many unsolved controversies, provoking discussions of fundamental statistical principles or even scientific principles in general [57,58]. Different proposed methods have been studied by Chang, Emerson, Liu and Hall, Lui and Xiong, Chang, Chang, Wieand, and Chang, Chang, Gould, and Snapinn, and Pickard and Chang [59-71].

In a recent investigation on the use of adaptive trial designs [72], it is reported that among all adaptive design they investigated, 29% is GSD, 16% SSR, 21% Phase-I/II or Phase-II/III seamless designs, and 41% dose-escalation, dose-selection and others.

The adaptive design has gained popularity since 2005 when a group of people from the pharmaceutical industry, academia, and government started a series of conferences and workshops on adaptive clinical trials in the United States and Internationally. There are two working groups who have made significant contributions to the popularity of adaptive clinical trial designs: PhRMA Adaptive Design Working Group [2] and Biotechnology Industry Organization (BIO) Adaptive Design Work Group [73]. The popular book by Jennison & Turnbull has greatly popularized group sequential design [74,75]. The first book that covers a broad of adaptive design methods by Chow and Chang has helped the community to understand what adaptive designs really mean [76]. In the following year Chang [44] published Adaptive Design Theory and Implementation using SAS and R, which enable statisticians to get hands on experiences on adaptive designs and simulations, and help them practically implement different adaptive trials. The Bayesian Adaptive Method in Clinical Trials by Berry et al. greatly popularized the Bayesian adaptive design in oncology trials [76].

Chow and Chang give a review of adaptive designs [77]. Bauer et al. also give a recent review on adaptive design [8]. Our purposes here are different: we give an overview of adaptive design and clarify common confusions to statisticians who have limited exposure to adaptive trials. For this reason, we cannot avoid some statistical formulations to clarify the important concepts.

In particular, we aim to answer the following common sources of confusion in adaptive trial design: (1) Adaptive design can be a frequentist approach or Bayesian approach, which are based on very different philosophies or two different statistical paradigms, (2) adaptive design is a relatively new methodology with vast amount publications in a short period of time with inconsistent terminologies and (3) different test statistics, different error-spending parametric function, different stopping boundaries, different adaptations can all lead different methods.

In this paper we will describe trial adaptations including: (1) group sequential designs, (2) error-spending designs (3) sample size re-estimation designs, (4) pick-the-winner and add-arm designs, (5) daptive randomization designs, (6) adaptive dose-escalation designs, and (7) biomarker-adaptive designs. This list is not exhaustive but will provide a foundation for readers for further exploration into adaptive design literature.

A group sequential design (GSD) is an adaptive design that allows for premature termination of a trial due to efficacy or futility based on the results of interim analyses. The simplest adaptive trial design is the group sequential design (GSD). The idea is no difference from the full sequential analysis, first developed by Abraham Wald as a tool for more efficient industrial quality control during World War I [78]. A similar approach was independently developed at the same time by Alan Turin, for messages decoding [79,80]. GSD for clinical trials was early suggested by several researchers, including Elfring & Schultz, McPherson, but more impactful papers are papers of Pocock, O'Brien & Fleming, Lan & DeMets, Wang & Tsiatis, Whitehead & Stratton, and Lan & Demets [81-88]. For the practical reasons, in group sequential tests the accumulating data are analyzed at intervals rather than after every new observation [74].

GSD was originally developed to ensure clinical trial efficiency under economic constraints. For a trial with a positive result, early stopping ensures that a new drug product can be exploited sooner. If a negative result is indicated, early stopping avoids wasting resources and exposing patients to an ineffective therapy. Sequential methods typically lead to savings in sample-size, time, and cost when compared with the classical design with a fixed sample-size. Interim analyses also enable management to make appropriate decisions regarding the allocation of limited resources for continued development of a promising treatment. GSD is probably one of the most commonly used adaptive designs in clinical trials.

There are three different types of GSDs: early efficacy stopping design, early futility stopping design, and early efficacy/futility stopping design. If we believe (based on prior knowledge) that the test treatment is very promising, then an early efficacy stopping design may be used. If we are concerned that the test treatment may not work, an early futility stopping design may be employed. If we are not certain about the magnitude of the effect size, a GSD permitting both early stopping for efficacy and futility should be considered. In practice, if we have reasonable knowledge regarding the effect size, then a classical design with a fixed sample-size would be more efficient.

The statistical determination of drug efficacy in a Phase-III trial is typically through hypothesis test:

where z₂ and z₂ are the common z-statistics formulated based on data collected from each stage (not cumulative data), and I₁, I₂ are the information times ( bounded by 0 and 1) or sample size fractions at stage one and two, respectively.

Since I₁ and I₂ are predetermined, √I₁z₁ and √I₁z₂ in the test statistic are independent. As a result, the probability distribution of T₁ = z₁ and T₂ = √I₁z₁ + √I₂z₂ are standard multivariate normal under null hypothesis H₀.

To control Type-I error, stopping rules of a GSD can be specified as

Reject H₀ (stop for efficacy) if T_k ≥ α_k

Accept H₀ (stop for futility) if T_k < β_k,

Continue trial to the next stage if β_k ≤ T_k < α_k

where the stopping boundary β_k < (k=1,...,K-1) and β_k=α_k. For convenience, are called the efficacy and futility boundaries, respectively.

To reach the k^th stage, a trial has to pass the 1^st to (k-1)^th stages. Therefore, the probability of rejecting the null hypothesis H₀ at the k^th stage is given by Ψ_k(α_k), where probability function

ψ_k(t) = Prβ₁ < T₁ < α₁,...,α_k-1 < T_k-1 < α_k-1,T_k ≥ t)

The acceptance probability at the kth stage is given by Φ_k(β_k), whereψ_k(t) = Prβ₁ < T₁ < α₁,...,α_k-1 < T_k-1 < α_k-1,T_k < t)

At the final stage K, the stopping boundary α_k=β_k.

For continuous, binary, and survival endpoints, the test statistics on z-scale at different stages follow a multivariate normal distribution under the large sample size assumption. This multivariate normal distribution becomes standard one under the null hypothesis, H₀. Stopping boundaries can be determined by: (1) determine how much α (Type-I error) we are willing to spend at each stage, denoted by π_k, π₁+ π₂+... π_k=α (the overall significance level for the test), (2) select a futility stopping boundaries β₁,β₂,..., β_k-1, and a small α₁=π₁<α (the overall significance level of the test), (3) try different t, through multiple integration until ψ_k(t)= π_k, k=2,3,.... The α_k values are determined progressively from α₂ to α_k.

If β_k=0, the stopping rule in (3) implies if the observed treatment effect is zero, the trial will be stopped, if β_k= -∞, it implies the trial will never stop for futility; if β_k =1, it implies when the (unadjusted) p-value p >1-Φ(1) = 0.1587 , the trial will stop for futility.

When the futility boundary, β_k (>-∞), is used, the efficacy boundary, α_k, is smaller making it easier to reject H₀ in comparison to the situation without a futility boundary. This is because with β_k, some of the Type-I errors when T_k>β_k are eliminated. In practice, however, the FDA so far has taken a conservative approach, making it more difficult to claim efficacy (i.e., α_k reduced). The FDA has requested that α_k be calculated based on the scenario that there is no any futility boundary and the assumption that the trial will never stop for futility at an interim analysis. The basis for the recommendation of this approach is that a company often does not stop for futility as specified in the protocol, i.e., the futility boundaries β_k are not considered formally binding.

GSD is the simplest and well accepted approach for adaptive design [89], but it is major limitation is that the timing (information time) of the interim analyses is fixed, which is inconvenient in practice since the scheduled time of interim analysis (IA) often needs to change due to the availability of data monitor committee (DMC) members. If the information time changes for the IA, then the stopping boundaries are not valid anymore. Because of this practical issue, Lan and DeMets developed so-called error-spending approach that allows to change the timing and the number of analyses in a GSD [85]. For GSD, how to incorporate safety information in the design is an area that needs further study.

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Error-Spending Approach

The Error-Spending Approach to adaptive trials uses a prespecified error-spending function to recalculate the stopping boundaries based on the actual timing of the IA. As we discussed in Section 2, to determine the efficacy stopping boundaries α_k, we need to select an amount of the total α that will be spent at each stage: π₁,π₂,...,π_k. If we want the error-spending to follow certain patterns, e.g., monotonically increasing, constant, or monotonically decreasing, we can specify a function for the π to follow. This function can be based on information time I (sample size fraction) at the interim analyses, or simply as a function of stage sequence number.

The main advantage of using an error-spending function of information time π(I) is that it allows us to change the timings and the number of analyses without α-inflation, as long as such a change is not based on the observed treatment difference and an errorspending procedure is followed. For example, if an interim analysis is conducted at information time I=0.3, we will spend π(0.3) at the interim analysis regardless of the scheduled time. The error-spending approach is an adaptive design method that can be useful when DMC members cannot make the schedule time to review the interim results. The method that prespecified a error-spending function and recalculate the stopping boundary based on the actual interim analysis time is called error-spending approach [44,56,85,88,90].

Commonly used classes of error-spending functions are O’Brien-Fleming-like spending function, the Pocock-like function, and the power-function family. The power-function family, π(I) =αt^b is valuable as it, can approximate the others error-spending functions well (Figure 1).

Error-spending approach solved the practical problems with GSD about IA schedule, but the maximum sample size in a GSD may not be larger enough when the treatment effect is over-estimated. If we use a larger maximum sample size in GSD, it may not be costeffective. The sample size re-estimation design is developed to handle the situation when there is a great uncertainty about treatment effect, but we don’t want to make a large, unnecessary commitment before seeing the interim data.

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Sample-Size Re-estimation Design

A sample-size reestimation (SSR) design refers to an adaptive design that allows for sample-size adjustment or re-estimation based on the review of interim analysis results. The sample-size requirement for a trial is sensitive to the treatment effect and its variability. An inaccurate estimation of the effect size or its variability could lead to an underpowered or overpowered design, neither of which is desirable. If a trial is underpowered, it will not be able to detect a clinically meaningful difference, and consequently could prevent a potentially effective drug from being delivered to patients. On the other hand, if a trial is overpowered, it could lead to unnecessary exposure of many patients to a potentially harmful compound when the drug, in fact, is not effective. In practice, it is often difficult to estimate the effect size and variability because of many uncertainties during protocol development. Thus, it is desirable to have the flexibility to re-estimate the sample-size in the middle of the trial.

There are two types of sample-size reestimation procedures, namely, sample-size reestimation based on blinded data and samplesize reestimation based on unblinded data. In the first scenario, the sample adjustment is based on the (observed) pooled variance at the interim analysis to recalculate the required sample-size, which does not require unblinding the data. In this scenario, the Type-I error adjustment is practically negligible; in fact, FDA and other regulatory agencies typically regard this type of sample size adjustment to be unbiased and without any statistical penalty. In the second scenario, both the effect size and its variability are re-assessed, and sample-size is adjusted based on the updated information. The statistical method for adjustment could be based on the observed effect size or the calculated conditional power.

The main statistical challenge of SSR compared to GSD is that unlike in GSD, in which the √I₁z₁ and √I₁z₂ in the test statistic T2 are independent, in SSR, because √I₁z₂ is a function of the second stage sample size n₂ that depends on the observed treatment difference or z₁ from the first stage data. The joint distribution of T₁ and T₂ is much more complicated than in GSD. Several solutions have been roposed, including the fixed weight method [30], the method of adjusting stopping boundary through simulations [44,56], and promising-zone methods [31,32].

The fixed weight method for combining test statistics usually takes the square-root of the information time at the interim analysis with the original final sample size N₀ as the denominator, i.e.,

T₂=√I₀z₁+√1-I₀z₂

The weighted Z-statistic approach (7) is flexible in that the decision to increase the sample size and the magnitude of sample size increment are not mandated by pre-specified rules. Chen, DeMets, and Lan proposed the Promising-Zone method, where the unblinded interim result is considered &#8220promising” and eligible for SSR if the conditional power is greater than 50 percent, equivalently, the sample size increment needed to achieve a desired power does not exceed a prespecified upper bound [31]. The Promising-Zone method may be justifiable to effectively put resources on drug candidates that are likely effective or that might fail only marginally at the final analysis with the original sample size, based on the information seen at the interim analysis.

Some researchers prefer using an equal weight for each patient, i.e., “one patient, one vote”. The “one patient, one vote” policy for the test statistic at the second stage does not follow the above formulation because the patients from the first stage (T₁) have potentially two opportunities to vote in the decision-making (rejecting or not rejecting the H₀), one at the interim analysis and the other at the final analysis, while patients from the second stage have at most one time to vote when the trial continues to the second stage.

Chen, DeMets, and Lan proposed a variation of the Promising-Zone method, using a conservative test statistic for the second stage [31]:

where z₁ and z₂ are the usual Z-statistic based on the subsamples from stages 1 and 2, respectively. n₁ is the sample size at the interim analysis, N₀ is the original final sample size and N is the new final sample size after adjustment based on the interim analysis.

Using this modified test statistic, the authors proved that when the conditional probability based original sample size is larger than 50%, then adjusted sample size, i.e., increasing the sample size when the unblinded interim result is promising will not inflate the type I error rate and therefore no statistical adjustment is necessary.

Mehta and Pocock proposed another Promising-Zone method, in which they define the promising zone as the interim p-value p₁ between 0.1587 (z₁=1.206) and 0.0213 (z₁=2.027) [32]. They chose these values because when one-sided p-value p₁ is larger but close to 0.1 at the interim analysis with information time Τ=0.5, the trial is likely to marginally fail to reject H₀ at the final analysis with p-value around 0.06 to 0.15. Those drug candidates are likely clinically effective, thus we want to “save” those trials and allow them a better chance to show statistical significance by increasing the sample size. If the p₁ is, say, larger than 0.2 (0.4 for two-sided p-value), we will stop the trial earlier at the first stage. For practical purposes, we recommend the following approach: combine the futility boundary with the upper bound of the promising zone and recommend using β₁ =0.2 for the one-sided p-value at the interim analysis with information time 0.5.

Current SSR methods [30], down weighting patients from the second stage, are not very efficient. Tsiatis and Mehta argue that GSD is more efficient than SSR design [33]. On the other hand, GSD is just a special case of SSR design and “the part” cannot be larger than “the whole”. More efficient SSR designs are desirable and need more research.

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Pick-Winner Design

The multiple-arm dose response study has been studied since the early 1990s. Under the assumption of monotonicity in dose response, Williams proposed a test to determine the lowest dose level at which there is evidence for a difference from the control [91]. Cochran, Amitage, and Nam developed the Cochran-Amitage test for monotonic trend. For the strong familywise Type-I error control, Dunnett’s test and Dunnett & Tamhane based on the multivariate normal distribution is most often used and Rom, Costello, Connell proposed a closed set test procedure [92-98]. A recent addition to the adaptive design arsenal is multiple-arm adaptive designs. A typical multiple-arm confirmatory adaptive design (variously called dropthe-loser, drop-arm, pick-the-winner, adaptive dose-finding design, or Phase II/III seamless design) consists of two stages: a selection stage and a confirmation stage. For the selection stage, a randomized parallel design with several doses and a placebo group is employed. After the best dose (the winner) is chosen based on numerically “better” observed responses, the patients of the selected dose group and placebo group continue to enter the confirmation stage. New atients are recruited and randomized to receive the selected dose or placebo. The final analysis is performed with the cumulative data of atients from both stages [44,99].

Recent additions to the literature include Bretz et al. who studied confirmatory seamless Phase II/III clinical trials with hypothesis selection at interim. Huang, Liu, and Hsiao proposed a seamless design to allow pre-specifying probabilities of rejecting the drug at each stage to improve the efficiency of the trial [100,101]. Posch, Maurer, and Bretz described two approaches to control the Type I error rate in adaptive designs with sample size reassessment and/or treatment selection [102]. The first method adjusts the critical value using a simulation-based approach that incorporates the number of patients at an interim analysis, the true response rates, the treatment selection rule, etc. The second method is an adaptive Bonferroni-Holm test procedure based on conditional error rates of the individual treatment-control comparisons. They showed that this procedure controls the Type I error rate, even if a deviation from a pre-planned adaptation rule or the time point of such a decision is necessary. Shun, Lan and Soo considered a study starting with two treatment groups and a control group with a planned interim analysis [26]. The inferior treatment group will be dropped after the interim analysis. Such an interim analysis can be based on the clinical endpoint or a biomarker. The unconditional distribution of the final test statistic from the ‘winner’ treatment is skewed and requires numerical integration or simulations for the calculation. To avoid complex computations, they proposed a normal approximation approach to calculate the Type-I error, the power, the point estimate, and the confidence intervals. Heritier, Lo and Morgan studied the Type-I error control of seamless unbalanced designs, issues of noninferiority comparison, multiplicity of secondary endpoints, and covariance adjusted analyses [103]. Further extensions of seamless designs that allow adaptive designs to continue seamlessly either in a subpopulation of patients or in the whole population on the basis of data obtained from the first stage f a Phase II/III design have also been developed. Jenkins, Stone, and Jennison proposed design adds extra flexibility by also allowing the trial to continue in all patients but with both the subgroup and the full population as coprimary populations when the Phase II and III endpoints are different but correlated time-to-event endpoints [104].

Shun, Lan and Soo found that under the global null hypothesis, common z statistic Z_w from the winner group is approximately normal distributed with mean [26]. Therefore, they proposed a the test statistic

which has approximately the standard normal distribution.

The approximate p-value can be easily obtained: p_A=1-Φ (Z^*).

The exact p-value, based on the exact distribution of Z_w, is given byp =p_A+ 0.0003130571(4.444461^T) - 0.00033.

For a general K-group trial, we define the global null hypothesis as H_G: μ₀ = μ₁ = μ₂...μ_k and the hypothesis tesm (winner) and the control as

H₀: μ₀ = μ_w, w = winner (selected arm)

Under larger sample size assumptions, given the observed valuesx_i and standard deviation in group i (i=0 and w), placebo and w for the winner) has a standard normal distribution. The test statistic for the comparison between the placebo group and the winner group can be defined as

Chang and Chang and Wang prove that the distribution of T under HG is given by

where the standard normal distribution pdf and cdf, annotated by φ and Φ, respectively [27,56].

If the purpose of the interim analysis is only for picking the winner, the stopping boundary c_α can be determined using F_T (cα ) =1-α for a one-sided significance level α. Numerical integration or simulation can be used to determine the stopping boundary and power. For Τ=0.5, the numerical integrations give cα=2.352, 2.408, 2.451, and 2.487 for K=4, 5, 6, and 7, respectively. The power of winner designs can be easily obtained from simulations.

Pick-winner design can improve the trial efficiency, but how to set the selection criteria for winner to optimize the adaptive trial is still unclear and requires further investigation.

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Add-Arm Design

In a winner design, patients are randomized into to all arms (doses) and at the interim analysis, inferior arms will be dropped. Compared to traditional dose-finding design, this adaptive design can reduce the sample size by not carrying over all doses to the end of the trial or dropping the losers earlier. However, it does require all the doses have to be explored. For unimodal response curves, Chang, Chang and Wang propose an effective adaptive dose-finding design that allows adding arms at interim analysis [27,56]. The trial design starts with two arms and depending on the response of the two arms and the unimodality assumption, we can decide which new arms to be added. This design does not require exploration of all arms (doses) to find the best responsive dose, therefore may further reduce the sample size from the dropping-loser design by as much as 20%.

Under the H_G, the cdf of the test statistic T, formulated from the placebo and the finally selected arm is given by

where cdf F_T* is given by

If we examine the design that only allows us to reject H_G at the final analysis for the moment: if the T≥c_α, test we would reject H₀. In such a case, the stopping boundary can be determined using T_F (c_α) for an one-sided significance level α. Numerical integration or simulation can be used to determine the stopping boundary and power. For example, the simulation shows rejection boundary to be c_α=2.267 for a one-sided α=0.025.

Under unimodal response, Chang-Wang’s three-stage add-arm design is usually more powerful that the 2-stage drop-arm design primarily because the former takes the advantage of the knowledge of unimodility of response. If the response is not unimodal, we can use that prior knowledge to rearrange the dose sequence so that it becomes unimodal response.

In an add-arm design, all arms are usually selected with equal chance when all arms have the same expected response, but the probability of rejection is different even when all arms are equally effective [27]. This feature allows us to effectively use the prior information to place more effective treatments in the middle but at the same time the two arms in the first stage have a large enough separation in terms of response to increase the power.

In an add-arm design, dose levels don’t have to be equally placed or based on the dose amount. The arms can be virtual dose levels or combinations of different drugs. Furthermore, the number of arms and the actual dose levels do not have to be prespecified, it can be decided after the interim analyses.

An add-arm design can further reduce the sample size from a pick-winner design, but it adds a complexity to the trial: there are three stages in an add-arm design, but two stages in a pick-winner design. On one hand, reducing sample size will reduce the time for the trial. On the other hand, a staggered patient enrollment in three stages will take longer than to enroll patients than with two stages.

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Adaptive Randomization Design

Response-adaptive randomization or allocation is a randomization technique in which the allocation of patients to treatment groups is based on the responses (outcomes) of the previous patients. The main purpose is to provide a better chance of randomizing the patients to a superior treatment group based on the knowledge about the treatment effect at the time of randomization. As a result, responseadaptive randomization takes ethical concerns into consideration.

In the response-adaptive randomization, the response does not have to be on the primary endpoint of the trial; instead, the randomization can be based the response on a biomarker. Scholars including Zelen, Wei and Durham, Wei, et al., Stallard and Rosenberger, Hu and Rosenberger, and many others have contributed in this area [19,105-108].

Fiore et al. use a response-adaptive randomization based on Bayesian posterior distribution of treatment effect given the observed response data and discuss the application is a point-of-care clinical trial with insulin administration [109]. Fava et al., Walsh et al. and Doros, et al. studied two-stage re-randomization adaptive designs and analyses in trials with high placebo effect [110-112].

The well-known response-adaptive models include the playthe-winner (PW) model and the randomized play-the-winner (RPW) model. A RPW model is a simple probabilistic model used to sequentially randomize subjects in a clinical trial [106,113]. The RPW model is useful for clinical trials comparing two treatments with binary outcomes. In the RPW model, it is assumed that the previous subject’s outcome will be available before the next patient is randomized. At the start of the clinical trial, an urn contains a₀ balls representing treatment A and b₀ balls representing treatment B, where a₀ and b₀ are positive integers. We denote these balls as either type A or type B balls. When a subject is recruited, a ball is drawn and replaced. If it is a type A ball, the subject receives treatment A; if it is a type B ball, the subject receives treatment B. When a subject’s outcome is available, the urn is updated as follows: A success on treatment A (B) or a failure on treatment B (A) will generate an additional a₁ (b₁) type-B balls in the urn. In this way, the urn builds up more balls representing the more successful treatment.

RPW can reduce the number of subjects assigned to the inferior arm, however, for Phase three trials, imbalanced randomization can reduce the power [44,114]. In small studies, response-adaptive randomization can be more problematic in causing potential confounders imbalance than a classical design. Ning and Huang proposed response-adaptive randomization with adjustment for covariate imbalance [115]. There are also several optimal RPW designs have been proposed based on proportion difference, odd ratio, and relative risk [116]. However, optimal RPW designs in terms of power require further development.

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Adaptive Dose-Escalation Design

Adaptive Dose-escalation design is a type of adaptive designs that is not based on Type-I error control and it is often used for earlyphase studies. Dose-escalation is often considered in early phases of clinical development for identifying maximum tolerated dose (MTD), which is often considered the optimal dose for later phases of clinical development. An adaptive dose-finding (or dose-escalation) design is a design at which the dose level used to treat the next-entered patient is dependent on the toxicity of the previous patients, based on some traditional escalation rules. Many early dose-escalation rules are adaptive, but the adaptation algorithm is somewhat ad hoc. Recently more advanced dose-escalation rules have been developed using modeling approaches (frequentist or Bayesian framework) such as the continual reassessment method (CRM) [1,22] and other accelerated escalation algorithms. These algorithms can reduce the sample-size and overall toxicity in a trial and improve the accuracy and precision of the estimation of the MTD.

The CRM is often presented in an alternative forms [24]. In practice, we are usually prespecify the doses of interest, instead of any dose. Let (d₁,...,d_k) be a set of dose and (p₁,...,p_k) be the corresponding prespecified probability, called the “skeleton”, satisfying p₁<p₂,...<p_k. The dose-toxicity model of the CRM is assumed to be

Pr (toxicity at d_i)=π_i(α)=P_i^exp(α),i=1,2,...,K,

where α is an unknown parameter. Parabolic tangent or logistic structures can also be used to model the dose-toxicity curve.

Let D be the observed data: y_i out of n_i patients treated at dose level i have experienced the dose-limiting toxicity (DLT). Based on the binomial distribution, the likelihood function is

Using Bayes’ theorem, the posterior means of the toxicity probabilities at dose j can be computed by

where g₀ (α) is a prior distribution for α, for example, α∼N(0,σ₂).

Ivanova and Kim study dose-finding for binary ordinal and continuous outcomes with monotone objective function or utility function [23]. Ivanova, Flournoy, Chung proposed a method for cumulative cohort design for dose-finding. Lee and Cheung proposed a model calibration in the continual reassessment method [117,118]. CRM designs with efficacy-safety trade-offs are studied by Thall and Cook, Yin, Li, and Ji, and Zhang, Sargent, and Mandrekar [119-122]. Bayesian adaptive designs with two agents in Phase-I oncology trial are studied by Thall, et al., Yin and Yuan [123-125]. Bayesian adaptive designs for different phases of clinical trials are covered in the book by Berry, Carlin, Lee, Muller [76].

CRM is relatively monitor-intensive adaptive design since it is requires calculation to determine the next patients assignment based on the real-time data. When there is a delayed response, the efficiency of the design will be largely lost. There also rules to prevent too fast dose-jump to protect patient’s safety in the trial, which also further reduce the efficiency CRM designs.

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Randomized Concentration-Controlled Trial

Sanathanan and Peck published a simulation study where they investigated the improvement in sample size and efficiency that can be gained from the randomized concentration-controlled trial (RCCT) design [28]. An RCCT can be considered “individualized treatment” or “personalized medicine” because, for each patient, dosage is adjusted according to the targeted concentration. Such dose modification or adaptation is based on the observed plasma concentration of the previous doses. The idea is that since the randomization point is moved closer to the clinical endpoint in the chain of causal events, randomizing patients to the defined target concentrations, as opposed to doses, makes it possible to reduce the within-group variability in the response variable in relation to the randomization variable [126]. The statistical methods for the dosage prediction and adjustments (so-called adaptive feedback strategy) can be based on the simple dose-proportionality assumption (AUC is proportional to the dose) or to the maximization of the Bayesian posterior probability of achieving the target concentration. Research showed that RCCT has the following advantages: (1) power will increase or sample size will reduce, (2) study power will be less sensitive to high variability in PK, (3) less confounded estimation of exposure-response, and (4) increased safety with narrow therapeutic window drugs [28,29,127]. The idea of RCCT can also be used for adjusting doses for each patient based on the target biomarker or PD response. Such a trial is called randomized biomarker-controlled trial (RBCT).

Since pharmacological effects are driven by the concentration at the site of action, the systemic drug concentrations are clearly in the causal pathway of drug action. Biomarkers and PD responses, on the other hand, may or may not be in the causal pathway, and thus can be very misleading markers when they are not (e.g. when the biomarker reflects an off-target effect). Moreover, it is well known that many biomarkers and PD responses are inherently more variable than drug concentration measurements. Thus, there is no “conceptual” advantage of the RBCT over the RCCT --- blind employment of the RBCT can even lead to greater inefficiency than that of the RCT design. Note that both RCCT and RBCT trial outcomes present statistical inferential challenges when the frequentist approach is employed to assess dose-response, since dose assignment is not randomized in either design.

Some common algorithms for adjusting dosage are:

(1) Adjustment based dose proportionality (Cella, Danhof, andPasqua, 2012):

(2) Adjustment based on linearity on log-scale (Power model):

LogC = β₀+β₁log d+ ε,

where d is dose and ε is the within subject variability.

The parameters β₀ and β₁ can be frequentist or Bayesian posterior estimates. After the expected (average) value of dose d to achieve the target concentration C can be determined.

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Biomarker-Adaptive Design

Biomarker-adaptive design (BAD) refers to a design that allows for adaptations using information obtained from biomarkers. A biomarker is a characteristic that is objectively measured and evaluated as an indicator of normal biologic or pathogenic processes or pharmacologic response to a therapeutic intervention [128]. A biomarker can be a classifier, prognostic, or predictive marker.

A classifier biomarker is a marker that usually does not change over the course of the study, like DNA markers. Classifier biomarkers can be used to select the most appropriate target population, or even for personalized treatment. Classifier markers can also be used in other situations. For example, it is often the case that a pharmaceutical company has to make a decision whether to target a very selective population for whom the test drug likely works well or to target a broader population for whom the test drug is less likely to work well. However, the size of the selective population may be too small to justify the overall benefit to the patient population. In this case, a BAD may be used, where the biomarker response at interim analysis can be used to determine which target populations should be focused on.

A prognostic biomarker informs the clinical outcomes, independent of treatment. They provide information about the natural course of the disease in individuals who have or have not received the treatment under study. Prognostic markers can be used to separate good- and poor-prognosis patients at the time of diagnosis. If expression of the marker clearly separates patients with an excellent prognosis from those with a poor prognosis, then the marker can be used to aid the decision about how aggressive the therapy needs to be.

A predictive biomarker informs the treatment effect on the clinical endpoint. Compared to a gold-standard endpoint, such as survival, a biomarker can often be measured earlier, easier, and more frequently. A biomarker is less subject to competing risks and less affected by other treatment modalities, which may reduce samplesize. A biomarker could lead to faster decision-making. In a BAD, “softly” validated biomarkers are used at the interim analysis to assist in decision-making, while the final decision can still be based on a gold-standard endpoint, such as survival, to preserve the Type-I error [56].

We briefly discuss the population enrichment design. Denote the treatment difference between the test and control groups by δ₊, δ_-, and δ, for biomarker-positive, biomarker-negative, and overall patient populations, respectively. The test statistics for the treatment effects in biomarker-positive, biomarker-negative, and the overall patient populations are denoted by Z₊, Z_-, and Z, respectively.

The test statistic for overall population is

where n and N are the sample size for the biomarker-positive patients and overall patients in the trial, the correlation coefficient between Z and Z₊ is

The rejection boundary and power calculation can be found in the text by Chang [44,56].

For adaptive designs, the information of Z-, Z+, and Z can be used to determine/change the randomization - biomarker enrichment design [37]. Wang studied the utility of predictive biomarker in adaptive design and discovered that the correlation coefficient ρ between the biomarker and primary endpoint in a clinical trial is not necessarily the main predicting factor [48]. For this reason, Wang and Chang developed a two-level hierarchical model, called biomarker informed adaptive design [50]. Wang, Chang, and Sandeep developed a biomarker-informed add-arm design for unimodal response [49].

The main challenges in utilization of biomarkers in clinical trials with classical or adaptive design are (1) usually a small set of data available with great uncertainties, and (2) there are often set of markers, not single marker, that affect the outcomes of the primary endpoint, which will make the model more complicated and require more data points or subjects to produce a valid model.

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Clinical Trial Simulation

Due to the complexity of adaptive trials, extensive simulations are often required during the design and conduct of clinical trials, and even in the analysis of trial data, simulation can be used to evaluate the robustness of the results.

Clinical trial simulation (CTS), also called Monte Carlo Similation, is a process that mimics clinical trials using computer programs. CTS is particularly important in adaptive designs for several reasons: (1) the statistical theory of adaptive design is complicated with limited analytical solutions available under certain assumptions; (2) the concept of CTS is very intuitive and easy to implement; (3) CTS can be used to model very complicated situations with minimum assumptions, and Type-I error can be strongly controlled; (4) using CTS, we can not only calculate the power of an adaptive design, but we can also generate many other important operating characteristics such as expected sample-size, conditional power, and repeated confidence interval - ultimately this leads to the selection of an optimal trial design or clinical development plan; (5) CTS can be used to study the validity and robustness of an adaptive design in different hypothetical clinical settings, or with protocol deviations; (6) CTS can be used to monitor trials, project outcomes, anticipate problems, and suggest remedies before it is too late to make modifications; (7) CTS can be used to visualize the dynamic trial process from patient recruitment, drug distribution, treatment administration, and pharmacokinetic processes to biomarkers and clinical responses; and finally, (8) CTS has minimal cost associated with it and can be done in a short time.

In protocol design, simulations are used to select the most appropriate trial design with all necessary scenarios considered. In trial monitoring, simulations are often used to guide the a necessary adaptations, such as stop or continue the trial, sample-size adjustment, evaluation of conditional power or probability of success given the interim observed data, shift of patient randomization probability, change of target patient population, etc. In the analysis of trial data, CTS can be used for evaluation of missing data impact and effect of other factors to evaluate the internal validity and external validity of the conclusions.

Chang provide comprehensive CTS techniques in classical and adaptive designs using SAS and he also provide introductory to adaptive design using R [53,56]. For broad coverage of Monte Carlo simulations in the pharmaceutical industry, we recommend the text “Monte Carlo Simulations in the Pharmaceutical Industry” by Chang [129]. The scope covered in the 13 chapters includes: (1) Meta-simulation in pharmaceutical industry game, (2) macrosimulation in pharmaceutical R & D, (3) Clinical Trial Simulation, (4) simulation in clinical trial management and execution, (5) simulation for prescription drug commercialization, Molecular design and simulation, (6) Disease modeling and biological pathway simulation, (7) Pharmacokinetic simulation, (8) Pharmacodynamic simulation, (9) Monte Carlo for inference and beyond, and others.

A recent book edited by Menon and Zink covers a range ofadaptive design simulation using SAS. Many adaptive design practicians have contributed to the books. It is a good reference in adaptive trial designs [130].

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Deciding Which Adaptive Design to Use

When we design our first adaptive trial, we will need to know how to start and we may even wonder if the adaptive design would really be better than a classic design. What if we miss something or something goes wrong?

The first step is to choose an appropriate type of adaptive design based on the trial objective(s). If it is a Phase III confirmatory trial, we may consider group sequential design or SSR design. If the timing of the analyses or the number of analyses are expected to change for practical reasons (e.g., the availability of DMC committee members or safety concern requires more interim analyses), we should choose GSD with error-spending approach. If the estimation of effect size is very unreliable, we should use SSR. If the number of doses (arms) to be considered is more than two (include the control arm) and, for example, we don’t know exactly which dose (treatment regimens or drug combinations of drugs) is the best (or good enough), we should use a winner or add-arm design. Note that an add-arm design is usually statistically more efficient than a drop-arm design, but also adds a little more to the complexity of the trial design and conduct. If a screening test of a stable biomarker is available (and practical) and we expect the test drug may have different efficacy and/or safety profiles for patients with and without the biomarker, we can use a biomarker enrichment design, in which the interim analysis will be used for deciding which population should be the target. If the biomarker is expected to respond to the drug and it can be measured earlier at the interim analysis than the primary endpoint of the trial, then we can use a biomarker-informed adaptive design. The response-adaptive design can be based on the primary endpoint or the biomarker response if the former is not available at the time of randomization.

In any setting, we should perform the clinical trial simulations to further evaluate and compare the operating characteristics of different designs or designs with different parameters. If the trial is an early stage design for a progressive disease such as cancer, we can use dose-escalation design in which the dose gradually increases to protect the patients’ safety. The next step is to determine whether a superiority, non-inferiority, or equivalence trial is needed based on the primary trial objective and the regulatory requirement and to determine the number of analyses points. The timing and number of interim analysis will be dependent on the safety requirement (some need more frequent safety monitoring), statistical efficiency (more interim analyses might reduce the sample size, need CTS to check), and practicality (complexity of the trial conduct and associated time and cost). We may need to consider some special issues, such as paired data or missing data, or other parameters. We would again conduct broad simulations with various design features/parameters and choose the most appropriate one based on proposed evaluation matrix. Finally, we need to consider the practical issues: How long will the trial take? Should we halt the trial when we performed the interim analysis? How fast can we do the interim analysis including the data cleaning. Will the recruitment speed and delayed response jeopardize the adaptive design? Who will perform the interim analysis and write the interim monitoring plan or DMC Charter? Dose the regulatory agency agree on the non-inferiority margin if a noninferiority adaptive trial? How the randomization be done? Can IVRS (interactive voice response system) support the adaptive design? How will the drug be distributed to the clinic sites? How will the primary analysis for the adaptive trial data be done?

The trial objectives are generally determined by the study team at a pharmaceutical company. The prioritization of the objectives is usually based on the clinical and commercial objectives and the trial design is optimized to increase the likelihood of success.After the objectives are determined, we choose the primary endpoint (a normal, binary, survival endpoint, etc.) to best measure the objectives and the timing of when the endpoints should be measured, and decide whether it is based on the change from baseline or the raw measure at a post-baseline time point. Next, we estimate the expected value for the endpoint in each treatment group based on prior information, which might be an early clinical trial, a preclinical study, and/or a published clinical trial results by other companies for the same class drug. The reliability of the historical data will affect our choice of the adaptive design.For Phase III trials, the Type-I error rate must typically be controlled at one-sided significance level α=0.025 for most of drugs. For early phase trials, α can be more flexible. A smaller α will reduce the false positive drug candidate entering late phase trials, but the same time increase the chance of eliminating the effective drug candidates from further studies, unless we increase the sample size to keep the Type-II error rate β unchanged.The most important factors (operation characteristics) for selecting a design with interim analyses are the expected sample size and maximum sample size for a fixed power [56]. If we wish to reduce the expected cost, we might want to choose a design with minimum expected sample size; if we wish to reduce the maximum possible cost, we might want to consider a design with minimum total sample size. In any case, we should carefully compare all the stopping probabilities between different designs before determining an appropriate design. O‘Brien-Fleming’s boundary is very conservative in early rejecting the null hypothesis. Pocock boundary applies a constant boundary on p-scale across different stages [131]. To increase the probability of accepting the null hypothesis at earlier stages, a futility boundary can be used. In general, if we expect the effect size of the treatment is less than what we have assumed and the cost is your main concern, we should use an aggressive futility boundary to give a larger stopping probability at the interim in case the drug is ineffective or less effective. On the other hand if we worry about the effect size might be under-estimated, we should design trial with aggressive efficacy stopping boundary (Pocock-like rather than O‘Brian-Fleming-like) to boost the probability early efficacy stopping in case the drug is very effective.

To choose an adaptive design among several options, we have to consider the trial objective, the cost, the duration of the study, the recruitment challenges, execution challenges, how to get the team and the regulatory authority buy-in. We can consider the impact of failure and success of the trial using a utility function [56]:U = ∫ R(θ)w(R)f(θ)dθwhere θ is the design input parameter vector, R is outcome or result vector component or operating characteristics, w is the corresponding weight function w(R), measuring the relative importance of R among all possible Rs, and f(θ) is the prior distribution of θ.Practically, we may select three or more typical scenarios for the main evaluation: H₀, H_s, and H_a. Their weighted average N, the maximum sample size N_max, number of analyses, power, utility, safety management, complexity of the trial design and conduct can compose the evaluation matrix and it is convenient to summarized in a Table like (Table 1).