1.6.1 Randomized Phase I Trial Designs
Adaptive randomization can be also useful in phase I trials where the pri- mary objective is to determine the maximum tolerated dose. The maximum tolerated dose is formally defined as the (100×Γ)th quantile of an unknown monotone dose–toxicity probability curve (Γ is typically set between 0.1 and 0.35 in phase I oncology studies). One famous design that can target any pre- specified toxicity level Γ∈(0,1) is therandom walk ruledeveloped by Durham and Flournoy [62]. For the random walk rule targeting Γ∈(0,0.5], the dose and the binary toxicity outcome of a current patient is used to determine the dose assignment for a new patient (hence it is a Markovian procedure). If a toxicity outcome is observed for the current patient, the dose for the new pa- tient is decreased by one level; otherwise the dose assignment is determined at random: the next highest dose is assigned with probability b= Γ/(1−Γ) or the current dose is assigned with probability 1−b, with appropriate ad- justments at the lowest and highest dose levels. The random walk rule has well-established exact and asymptotic properties [63]. One important exten- sion of the random walk rule is thegroup up-and-down design proposed by Gezmu and Flournoy [77], a non-randomized procedure which makes dose as- signments to cohorts of patients such that patients in the same cohort receive the same dose, but the doses may differ across the cohorts. For a cohort of size k≥1, the group up-and-down design can target toxicity probabilities of the form Γ = 1−(0.5)1/k. By adding a randomization component to the group up-and-down design, one can construct more flexible designs that can target any Γ∈(0,1) [14].
Finally, adaptive randomization can be a useful tool to reduce “stickiness”
in sequential dose-finding algorithms that are based on maximization of some utility functions (cf.§18.5 of the current volume).
1.6.2 Adaptive Optimal Dose-Finding Designs
A typical phase II dose–response study is a randomized placebo- and/or active- controlled parallel group design with several doses of an investigational drug.
The primary objectives include an assessment of the dose–response relation- ship with respect to some efficacy outcome and identification of dose(s) with desirable benefit–risk ratio for subsequent testing in phase III trials. A single- stage balanced randomization design would randomize study patients equally amongK doses and the data would be analyzed only once the primary out- comes have been observed from all patients in the study. While this approach is scientifically sound, it lacks flexibility and it may not be formally optimal for nonlinear dose–response models.
An adaptive optimal design with one or more interim analyses can be a more efficient option for a phase II dose-finding trial. Suppose the pri- mary outcome follows a regression model E(Y) = f(d,θ), where f is some nonlinear function, d is a dose measured on a continuous scale, and θ is a vector of parameters of interest. Let M(ξ,θ) denote the Fisher informa- tion for θ using design ξ. The D-optimal design minimizes the volume of the confidence ellipsoid forθ. Mathematically, the problem is to find the de- signξ∗ ={(d∗i, ρ∗i(θ)), i= 1, . . . , m;Pm
i=1ρ∗i(θ) = 1} (a set of optimal doses and the probability mass at these doses) such that ξ∗ minimizes the crite- rion−log|M(ξ,θ)|. Ifθ were known, theD-optimal designξ∗ could be easily computed. Since in practiceθ is unknown, one can construct a two-stageD- optimal adaptive design as follows. At the first stage, a pilot sample ofN(1) patients are randomized using equal allocationρ(1)= (1/K, . . . ,1/K)0 among K pre-determined doses that span the design space Ω = [0, dmax], where 0 anddmax correspond to the placebo and the maximum dose, respectively. At interim, one fits the model to obtainbθ, an estimate of θ based on the data fromN(1)patients. This estimate is used to approximate the true unknownξ∗, and the treatment allocation proportions for the second stage are set adap- tively to ρ(2) = (ρ1(bθ), . . . , ρm(bθ))0, where ρi(bθ) is the probability mass at the estimated optimum dose d∗i, i = 1, . . . , m (here we use m instead of K to emphasize that the optimum doses for the second stage may be different from the doses in the first stage). The estimated target allocation ρ(2) can be implemented by means of brick tunnel randomization [106]. Chapter 20 of this volume gives a nice example of a two-stage adaptiveD-optimal design for a 3-parameter Emax model.
1.6.3 Randomized Designs with Treatment Selection
An important design adaptation rule in multi-arm randomized controlled tri- als is treatment selection—based on observed data at interim, only a sub- set of “most promising” experimental treatments along with the control are carried forward to the next stage. This allows eliminating inefficient arms early thereby potentially reducing the size of the experiment. Treatment se- lection can be viewed as a subclass as adaptive randomization. Consider, for example, the following two-stage treatment selection design. At the first stage,N(1) patients are equally randomized amongK−1 experimental arms (T2, . . . , TK) and placebo (T1); therefore the target allocation for the 1st
stage is ρ(1) = (1/K, . . . ,1/K)0. Based on treatment assignments and out- comes from the N(1) patients, two “best” experimental treatments are se- lected according to some pre-specified quantitative criterion (e.g., the esti- mated treatment–placebo difference is greater than a threshold) for further definitive comparison with the placebo. At the second stage, additionalN(2) patients are equally randomized among the placebo and the selected treat- ments; therefore the target allocation for the second stage is set adaptively to
ρ(2) = 1
3,0, . . . ,0,1
3,0, . . . ,0,1 3
0 ,
where the probability mass 1/3 is for the placebo and the two selected exper- imental treatment arms. The goal of the second stage is to formally test each of the selected experimental treatments versus the placebo. Proper statistical adjustments in the final analysis are required to ensure strong control of the type I error rate. The adaptive randomization part of treatment selection de- signs involves the reassessment of the target allocation for the second stage given data from the first stage (allocation proportions to some treatment arms may be set to 0). Some notable examples of treatment selection designs are the seamless phase II/III designs [45, 113], multi-arm multi-stage (MAMS) designs (cf. Chapter 18), and sequential elimination designs (cf. Chapter 19), among others. A recent successful application of a seamless phase II/III trial is discussed in Chapter 21.
1.6.4 Group Sequential Adaptive Randomization
Many contemporary clinical trials include sequential or group sequential (GS) monitoring of trial data with the goal to potentially stop the trial early for futility (if there is lack of any treatment effect), efficacy (if treatment effect is pronounced), or safety reasons. Such designs are attractive from ethical, administrative, and economic perspectives. The GS design methodology en- sures that usual frequentist properties of statistical tests (e.g., strong control of the type I error rate) are maintained. The GS designs are viewed as “well- understood” statistical designs and their use is encouraged by the FDA [74].
An excellent treatise on statistical methods of GS designs can be found in the book by Jennison and Turnbull [97]. The general setup of a GS de- sign includes specification of the number of treatment arms, the type of early stopping, the number of interim analyses, the choice of stopping boundaries, and the maximum sample size in the study. At each analysis, the standard- ized test statistic is computed and compared with the pre-specified critical values, and a decision is made whether to stop or continue the trial. On ter- mination, a properly performed statistical analysis ensures valid conclusions on the treatment effect.
Most GS designs use fixed randomization for treatment assignments, and most adaptive randomization designs use a fixed sample size. Combining GS monitoring with adaptive randomization creates a class of more flexible de- signs which utilize the advantages of two types of adaptation. Such GS adap-
tive randomization designs are technically more complex and their theoretical properties may be elusive.
Zhang and Rosenberger [186] and Plamadeala and Rosenberger [120] es- tablished statistical properties of sequentially monitored conditional random- ization tests following Smith’s [141] and Efron’s [64] biased coin designs, re- spectively. Zhu and Hu [188, 189] considered sequential monitoring of RAR clinical trials with the DBCD [90] and with RAR urn models. They showed that under widely satisfied conditions, the sequential test statistics asymptot- ically satisfy the canonical joint distribution defined in Jennison and Turnbull [97]. This ensures control of type I error rate and other important asymptotic properties of GS RAR designs. Chambaz and van der Laan [52, 53] studied GS RAR designs for two-arm binary response trials and also demonstrated advan- tages of such combination designs. Chambaz and van der Laan [54] developed GS CARA designs which use targeted maximum likelihood methodology for data analysis. Lai and Liao [107] proposed another interesting GS RAR design with certain optimal properties.
1.6.5 Complex Adaptive Design Strategies
So far we have discussed adaptive randomization designs applied on the trial level. Another possibility is to consider more complex designs where adapta- tions are applied at the program level of a compound or even at the portfolio level of several compounds with the goal to optimize the overall drug devel- opment process.
Krams and Dragalin [104] distinguish four types of complex adaptive de- sign strategies that are becoming increasingly popular in modern drug devel- opment. For each type, the general principle is to keep one or more aspects of the study fixed and let some other aspects be modified adaptively accord- ing to some pre-determined criteria. The four types of complex adaptive de- sign strategies are [104, p. 71]: 1) adaptive “population” finder; 2) adaptive
“compound” finder; 3) adaptive “indication” finder; and 4) adaptive “com- pound/population” finder.
As an example, let us consider the first type, the adaptive “population”
finder. The fixed aspects of the trial design are the indication (e.g., breast cancer) and the treatment (e.g., epidermal growth factor receptor inhibitor), and the objective is to establish which subset of the population benefits most.
One example in this category is the design proposed by Follman [71] which adaptively changes the subgroup proportions in the trial to increase the repre- sentation of the more responsive subset of patients in the study and decrease the representation of patients for whom the drug is not efficacious or harm- ful. Technically, the adaptive “population” finder can be viewed as a CARA randomization design (cf.§1.5).
As another example, let us consider the fourth, most complex type, the adaptive “compound/population” finder. The fixed aspect of the design is the patient population (e.g., women with moderate to high-risk primary breast
cancer), but this population is known to be heterogeneous at the outset (e.g., there are some predictive biomarkers such as the human epidermal growth factor receptor 2 (HER2) status (+/–) which are known to affect the patient’s response to a given treatment). There are multiple experimental compounds which are to be tested in parallel in this patient population, with the goal to identify the compound(s) with the most desirable benefit–risk ratio in differ- ent subpopulations. Two examples of adaptive “compound/population” finder trials are the I-SPY 2 trial [30] and the BATTLE trial in lung cancer [102].
These trials included multiple adaptive elements—the key ones were the rules for Bayesian adaptive randomization of new patients with a given biomarker signature to different compounds (Bayesian CARA randomization), the in- terim treatment selection rules based on Bayesian criteria, and the stopping rules for futility or efficacy.
Overall, there is an important link between adaptive randomization (in the broad sense) and complex adaptive design strategies described by Krams and Dragalin [104]. One can expect to see a broader application of these ideas in clinical trials for diseases with high unmet medical need.