Covariate-Adjusted Response–Adaptive (CARA) randomization is applica- ble in clinical trials with heteroscedastic and nonlinear models with possi- bly treatment–covariate interactions where balanced designs may be subopti- mal. As noted by Sverdlov, Rosenberger and Ryeznik [146], there are at least three important reasons why CARA randomization merits consideration in clinical trial practice. These include: 1) ethical considerations; 2) nonlinearity and heteroscedasticity of statistical models; and 3) presence of treatment- by-covariate interactions (when the magnitude and direction of the treatment effect may differ for patient subgroups within a treatment). CARA randomiza- tion designs can be viewed as an important step toward personalized medicine [85]. Recently, theoretical properties of a very general class of CARA proce- dures were established by Zhang et al. [184] and Zhang and Hu [182]. Rosen- berger and Sverdlov [129] discuss the appropriateness of CARA randomization in clinical trials. In§1.5.1–§1.5.4 we describe major types of CARA random- ization designs.
1.5.1 Treatment Effect Mapping and Urn-Based CARA Randomization Designs
The idea of treatment effect mapping was introduced by Rosenberger [124].
This heuristic approach has an intuitive appeal of skewing randomization probability toward an empirically better treatment according to the current estimate of the treatment difference (using covariate adjustments as appropri- ate). Rosenberger, Vidyashankar and Agarwal [131] proposed a CARA proce- dure based on a logistic regression model for which treatment randomization probabilities are set proportional to the estimated treatment odds. Bandy- opadhyay and Biswas [24] proposed a CARA procedure for a linear model using probit mapping of a covariate-adjusted estimate of the mean treatment difference. Unfortunately, both procedures [24, 131] are not optimal in any sense and can result in highly unbalanced treatment groups and loss in power of statistical tests.
Another class of heuristic CARA procedures is based on urn models.
Covariate-adjusted extensions of the randomized play-the-winner rule of Wei and Durham [166] were proposed in the papers [22, 23, 109]. Covariate- adjusted extensions of Ivanova’s [95] drop-the-loser rule were proposed in the papers [26, 40]. All these designs can assign more patients to the better treat- ment within covariate subgroups. Some of these designs are more variable than others and more research is needed to make a definitive recommendation for practice.
1.5.2 Target-Based CARA Randomization Designs
Zhang et al. [184] and Zhang and Hu [182] developed a framework for CARA randomization designs that can target a covariate-adjusted version of an opti- mal allocation derived under a framework without covariates. This approach ensures asymptotically the desired allocation proportions for different treat- ment groups and different covariate values. Under widely satisfied condi- tions, target-based CARA designs maintain strong consistency and asymp- totic normality of both parameter estimators and treatment allocation pro- portions [184], and the designs have similar power and estimation efficiency to balanced randomization designs [129, 146]. Zhang and Hu [182] and Che- ung et al. [56] showed that this general methodology is applicable for responses from a generalized linear model. However, the major question is what alloca- tion to target given that the individual patient covariate values are unknown at the trial outset. Target-based CARA designs with various allocation targets were developed for linear models [18, 20, 190], logistic models [56, 129], expo- nential survival models [38, 146] and longitudinal models [43, 94]. Chambaz and van der Laan [54] proposed group sequential CARA designs analyzed via targeted maximum likelihood estimation methodology (cf. Chapter 16).
1.5.3 Utility-Based CARA Randomization Designs
Atkinson and Biswas [7, 8] proposed a class of CARA designs for which ran- domization probabilities are determined sequentially by maximizing a utility function that combines inferential and ethical criteria. Let φk denote some measure of information from applying treatmentTk to a new eligible patient with covariate vectorz, and letπk=πk(θ,z) denote the allocation proportion for treatmentTk for a given value ofz (0< πk <1 and PK
k=1πk = 1) such thatπk’s are skewed in favor of superior treatment arms. Then the treatment allocation probabilitiesP1, . . . , Pk (PK
k=1Pk= 1) are obtained by maximizing the utility function
U =
K
X
k=1
Pkφk−γ
K
X
k=1
PklogPk
πk
,
whereγ ≥0 is the tradeoff parameter (γ = 0 is “most efficient” andγ→ ∞ is “most ethical” design). The optimal randomization probabilities are
Pk= πkexp(φk/γ) PK
j=1πjexp(φj/γ), k= 1, . . . , K.
Atkinson and Biswas [7, 8] studied this class of allocation designs in the con- text of a linear model withDA-optimality as an inference criterion and probit mapping of a covariate-adjusted treatment difference as an ethical criterion.
Further extensions were developed for logistic regression models [25, 119, 129], exponential survival models [27, 146], and longitudinal models [43]. Most re- cently, Hu, Zhu and Hu [92] proposed a very broad class of CARA designs based on efficiency and ethics (CARAEE) which extend Atkinson and Biswas’s [7, 8] methodology and unify several important designs in the literature. See Chapter 14 for examples of CARAEE designs for a logistic regression model.
1.5.4 Bayesian CARA Randomization
The idea of Bayesian CARA randomization is to skew randomization proba- bility in favor of superior treatments while adjusting for patient heterogeneity according to some Bayesian criterion (e.g., the posterior probability that a given treatment is most successful for a patient’s covariate profile). Unlike CARA designs discussed thus far, Bayesian CARA randomization procedures are selection designs—at the end of the trial the treatment with highest pos- terior probability of the criterion is selected. Some important examples of Bayesian CARA designs can be found in Thall and Wathen [154] and Che- ung et al. [57]. In these papers, the authors showed via extensive simulations that their designs allocate on average substantially more patients to superior treatments (within patient subgroups when there are treatment–covariate in- teractions) and are similar to non-adaptive balanced randomization designs
in terms of correct selection probability. Another interesting idea iscovariate- balanced response–adaptiveBayesian randomization designs proposed by Ning and Huang [117] and Yuan, Huang and Liu [175]. These designs share the advantages of Bayesian adaptive randomization and covariate–adaptive ran- domization (cf. Chapter 17).