Outcome-Adaptive Allocation using Auxiliary and Primary Outcomes

Abstract

Studies with delayed outcomes generally receive little benefit from adaptive allocation procedures. In this manuscript we present an optimal design for outcome-adaptive allocation by combining information from delayed primary outcomes and more quickly observed auxiliary outcomes. Bayesian methods are used to construct the joint distribution of these outcomes, which is used to estimate the components of the optimal allocation ratio. Simulation studies show this approach to be effective at achieving adaption even before the delayed outcome is observed.

Keywords: Randomization; Adaptive clinical trials; Study design; Bayesian methods

Introduction

Optimal response-adaptive allocation designs are intended to minimize the overall number of treatment failures observed in a trial. In cases with sufficient evidence of some treatment outperforming another, the allocation algorithm will increase the probability that subjects are allocated to the superior treatment. These designs thus can exhibit fewer treatment failures, then balanced designs [1].

In practice, some primary outcomes – such as survival or relapse – require months or years before they are observed. With these outcomes, there can be a delay in updating the allocation rate for the next patient or group of patients. However, the efficiency of the response adaptive design highly depends on the immediacy of observed data: if few primary end points are observed in early stages of the trial, adaptation will not occur. Bai et al. [2] have shown that moderately delayed responses will not affect asymptotic properties of the adaptive procedure under certain delay mechanisms, though there could be a higher risk of assigning more patients to some inferior treatment. If the rate at which outcomes are observed is too slow relative to the rate of patient accrual, then the benefits of adaptive allocation may not be realized.

In this paper, we introduce an adaptive allocation design that incorporates an auxiliary outcome that is positively correlated with the primary outcome yet is more quickly observed. Rather than use a second outcome as a surrogate or replacement of the primary outcome in the allocation algorithm, our procedure aggregates information from both the auxiliary and primary outcomes, based on the classical response adaptive design framework for binary data. The goals of this paper are to: 1) introduce a response adaptive design framework that simultaneously uses both primary and auxiliary outcomes, and 2) incorporate a bivariate beta distribution [3] as the prior distribution of correlated binomial data to account for dependence between the two outcomes. Relevant background is provided in the next Section, after which the methodological set-up and allocation algorithm are introduced. A simulation study comparing the joint approach with both balanced and optimal allocation is then presented, and the manuscript concludes with a brief discussion.

Materials and Methods

Background

The goal of classical response-adaptive procedures is to minimize the loss function given that the information level at each stage is constant [4]. This loss function contains the difference, in treatment success rates (θ=P_A-P_B, where P_A and P_B are the success rates for treatment A and B) and sample size (n_i=n_A,i+n_B,_A):

L(θ)=u(θ)n_A,i+υ(θ)n_B,i (1)

where n_A,i and n_B,i are the cumulative number of patients assigned to groups A and B at the i^th stage of the study, υ(θ) is the loss for a patient allocated to treatment A, and υ(θ) is the loss for a patient allocated to treatment B. We also assume${\sigma }_A^2{n}_{(A,i)}+{\sigma }_B^2{n}_{(B,i)}=K$ where ${\sigma }_A^2$ and ${\sigma }_B^2$ are the outcome variance in groups A and B, respectively, and K is some constant.

Patients are generally exposed to two risks in randomized trials: treatment failure and assignment to an inferior treatment. Let θ<0 indicate treatment A is inferior (p_A<pB

) and θ>0 indicate treatment B is inferior theta;<0 indicate treatment A is inferior (). The treatment failure risks are described by υ(θ) and υ(θ). The function υ(θ) increases as θ decreases and υ(θ) increases as θ increases. The allocation ratio (n_A,i/n_B,i) determines the probability of assigning patients to the inferior treatment. The loss function, then integrates these two risks, and our goal is to minimize this loss function subject to the constant variability at each stage of the trial. Minimization of the equation (1) can be solved for the allocation ratio using the delta method (Appendix A.1), and the minimized allocation ratio is:

$R=\frac{n_{A,i}}{n_{B,i}}=\frac{{\sigma }_A}{{\sigma }_B}\sqrt{\frac{v(\theta )}{u(\theta )}.}\text{ (2)}$

Consequently, we need only model υ(θ)) and υ(θ) to realize some specific objective. For binary response trials, if υ(θ))=υ(θ)=1, the allocation ratio, $\text{R}={\text{σ}}_{\text{A}}/{\text{σ}}_{\text{B}}=\sqrt{{\text{p}}_{\text{A}}{\text{q}}_{\text{A}}/{\text{p}}_{\text{B}}{\text{q}}_{\text{B}}}$ which is the so-called Neyman allocation rule [5], which minimizes estimator variance. If υ(θ)=1-p_A and υ(θ))=1-p_B, the allocation ratio A B R = p p turns out to be the socalled optimal allocation ratio, which minimizes the expected number of treatment failures [1]. Loss functions υ(θ and υ(θ can be treated as functions of unknown parameter p_A and p_B, which can be estimated based on patient responses using a sequential estimation method. If our primary response is delayed, we may not have information to estimate υ(θ), υ(θ) and R appropriately.

Allocation ratio derivation with two outcomes

For treatments j=A or B, suppose X_j is an auxiliary outcome for treatment j and Yj is a primary outcome, where X_j and Yj both are binary variables. According to the observed outcome sequence, we denote P_X,j as the “success” rate for the auxiliary outcome in treatment j, and P_X,j as the success rate for the primary outcome. We assume that 1) P_X,j and P_X,j are random variables with some joint distribution, 2) the conditional random variables X_j|P_X,j˜BIN(nX,j, P_X,j) and Yj|P_y,j˜BIN(nY,j,P_y,j) are independent, where nX,i and nY,i are the number of observed auxiliary and primary outcomes, and 3) the association between X_jand YJ is explained through the association between P_X,j and P_X,j Thus, the posterior distribution of P_X and P_Y (we remove the subscripts for simplicity) can be expressed as:$f(P_X,P_Y|X,Y)\propto f(X|P_X)f(Y|P_Y)f\left(P_X,P_Y\right) \text{ 13}$As mentioned earlier, u(θ) and υ(θ) are positive functions that measure the risk of assigning patients to treatment A and B given primary efficacies (P_Y,A,P_Y,B). In addition, we also have auxiliary efficacies P_X,A and P_X,B, which offer some information about P_Y,A and P_Y,B, respectively, since they are associated. Therefore, it is reasonable to average u(θ) and υ(θ) over all possible sets of P_Y,A and P_Y,B given (P_X,X,Y)A and (P_X,X,Y)B. Based on the loss function (1) of the classical adaptive design framework, the loss function of the procedure using auxiliary and primary outcomes takes the following form:

L(θ)=E[u(θ)|(P_X,X,Y)A, (P_X,X,Y)B]nA,i +E[υ(θ)|(P_X,X,Y)A, (P_X,X,Y)B]nB,I (4) where nA,I and nB,i are the number of patients in treatment A and B at i^th stage of the trial. The two conditional expectations in (4) can be calculated through the conditional posterior distribution from (3). The minimization of the function (4) is the same as that of the loss function (1) in classical response-adaptive design framework, since the conditional expectations are assumed to be known. Therefore, the allocation ratio is

$R^{*}={\sigma }_{Y,A}/{\sigma }_{Y,B}\sqrt{E\left[v\left(\theta \right)\text{|}{(P_X,X,Y)}_A,{(P_X,X,Y)}_B\right]/E[u(\theta )|{(P_X,X,Y)}_A,{(P_X,X,Y)}_B}$

Two-dimensional beta-binomial model

Martin and Vaeth [6] proposed a two-dimensional beta binomial distribution that can model the association between two count variables. We use a similar approach to model the association between the auxiliary and primary outcomes, which is done through modeling the dependence between their respective success rates. Olkin and Liu [3] derived a bivariate beta distribution from three marginal gamma distributions. We use this distribution as a prior for (P_X,j,P_y,j). Given the assumptions about the design, the joint distribution of (X_j,Y,_j,P_X,j,P_y,j) is the product of the conditional distributions of X_j|P_X,j and Y,j|P_y,j and prior distribution of (P_X,j,P_y,j).

To simplify our notation, the following distributions are generalized to any (X,Y,P_X,P_Y) given a specific treatment.

f(X,Y,P_X,P_Y)=f(X,Y|P_X,P_Y)*f(P_X,P_Y|(α1,α2,β) (5)

$\left(\begin{array}{c}{n}_x\\ x\end{array}\right)P_X^x{(1-P_X)}^{n_x-x}*\left(\begin{array}{c}{n}_y\\ y\end{array}\right)P_Y^y{(1-P_Y)}^{n_y-y}$

$\frac{\Gamma ({\alpha }_1+{\alpha }_2+\text{β})P_X^{{\alpha }_1-1}{(1-P_X)}^{{\alpha }_2+\beta -1}{P}_Y^{{\alpha }_2-1}{(1-P_Y)}^{{\alpha }_2+\beta -1}}{\text{Γ}({\alpha }_1)\text{Γ}({\alpha }_2)\text{Γ}(\beta ){(1-P_X{P}_Y)}^{{\alpha }_1+{\alpha }_2+\beta }}$

Integrating with respect to P_Y, the joint distribution of (X,Y,P_X) is:

$f(X,Y,P_X)=\left(\begin{array}{c}{n}_x\\ x\end{array}\right)\left(\begin{array}{c}{n}_y\\ y\end{array}\right)\frac{\text{Γ}({\alpha }_1+{\alpha }_2+\text{β})}{\text{Γ}({\alpha }_1)\text{Γ}({\alpha }_2)\text{Γ}(\beta )}{P}_x^{x+{\alpha }_1-1}{(1-P_X)}^{{\alpha }_2+\beta -1+n_x-x}\text{ (6)}$

$\frac{\Gamma (\text{y}+{\alpha }_2)\text{Γ}({\alpha }_1+\text{β}+n_y-\text{y})}{\text{Γ}({\alpha }_1+{\alpha }_2+\beta +n_y)}$

*2F1(α1+α2+β;y+α2; α1+α2+β+n2;P_X)

Where 2F1 is the Gaussian hyper geometric function. Therefore, the conditional distribution of P_Y given P_X and the data (X,Y) is obtained through division:

f(P_Y| X,Y,P_X)=f(X,Y,P_X,P_Y)/f(X,Y,P_X) (7)

$\frac{\Gamma ({\alpha }_1+{\alpha }_2+\beta +n_y)P_y^{y+{\alpha }_2-1}{(1-P_Y)}^{{\alpha }_1+\beta -1+n_y-y}}{\text{Γ}(y+{\alpha }_2)\text{Γ}({\alpha }_1+\beta +n_y-y)}$

$\frac{{(1-P_X{P}_Y)}^{{\alpha }_1+{\alpha }_2+\beta }}{{}_2{}^{}F{}_1({\alpha }_1+{\alpha }_2+\beta ;y+{\alpha }_2;{\alpha }_1+{\alpha }_2+\beta +n_y;P_X)}.$

As presented in the defined loss function, u(θ) and υ(θ) are functions of P_Y,A and P_Y,B. Also, we know that treatment A is independent from treatment B, which indicates the distributions for treatment A (f(X,Y,P_X,P_Y)A) and for treatment B (f(X,Y,P_X,P_Y)A) are independent. As long as we know the conditional distribution (f(P_Y|X,Y,P_X)A) and (f(P_Y|X,Y,P_X)B) for treatment A and B, we are able to calculate the conditional expectation from the loss function (4).

As we are interested only in optimal allocation, we focus solely on the case when u(θ)=1-P_Y,A and υ(θ) =1-P_Y,B, recalling that P_Y,j is the primary efficiency rate in the jth treatment. Then the loss function (4) is reduced to

L(θ)=(1-E[P_Y,A|( P_X,X,Y)A])nA,i+(1-E[P_Y,B|( P_X,X,Y)B])nB,i* (8)

The optimal allocation ratio can then be rewritten as

$R^{*}=\frac{{\sigma }_A^2}{{\sigma }_B^2}\sqrt{\frac{E[1-P_{Y,B}|{(X,Y,P_X)}_B]}{E[1-P_{Y,A}|{(X,Y,P_X)}_A]}}$

For a given treatment, the conditional expectation is a function of X,Y,P_X with prior parameters (α₁,α₂,β) (Appendix A.2).

$E\left[P_Y\text{|}X,Y,P_X\right]=\frac{y+{\alpha }_2}{{\alpha }_1+{\alpha }_2+\beta +n_y}*\frac{{}_2{}^{}F{}_1({\alpha }_1+{\alpha }_2+\beta ;y+{\alpha }_2+1;{\alpha }_1+{\alpha }_2+\beta +n_y;P_X)}{{}_2{}^{}F{}_1({\alpha }_1+{\alpha }_2+\beta ;y+{\alpha }_2;{\alpha }_1+{\alpha }_2+\beta +n_y;P_X)}\text{ (9)}$

The expression on the right side of equation (9) is the Gauss continued function. The continued function of the Gauss hyper geometric function converges uniformly for 0<P_X<1. Therefore, E[P_Y|X,Y,P_X] is guaranteed to reside within the range (0,1). The correlation of X and Y is then proportional to the correlation of P_X and P_Y and takes the following form (Appendix A.3):

$Corr\left(X,Y\right)=Corr\left(P_X,P_Y\right)\sqrt{\frac{n_x{n}_y}{\left(n_x-{\alpha }_1-\beta \right)\left(n_y-{\alpha }_2-\beta \right)}}.\text{ (10)}$

Prior density selection

In the beta-binomial model, subject matter expertise can be used to provide some information to assess the probability of having a successful outcome, which then determines the mean or mode of the beta distribution. The sum (r) of a and β determines the variance of the beta distribution given some desired marginal mean. As r increases, the more compact and informative will be the prior distribution. The sum r indicates how confident we are on the expert advice or literature information, and r-2 is known as the effective sample size. If we lack confidence in the prior belief of success probability, we can weigh the data more by selecting a wide unimodal beta density function (i.e. by selecting low r).

For the bivariate beta distribution, we adopt the same logic in selecting the marginal densities, which follow beta distributions. It can be shown that the prior correlation of the Olkin and Liu [3] distribution is narrowly bounded when the marginal means are given, which may diminish the ability of the bivariate prior distribution to adequately model the association between success rates. According to Equation (10), the correlations of auxiliary and primary outcomes is approximately equal to the correlation of auxiliary and primary efficacy as (nx ,ny )» (α₁,α₂ ,β ) . Therefore, we intend to have a less informative prior by choosing r no greater than 15 when α₁, α₂ and β are greater than 1. As studied in Olkin and Liu [3], the bivariate beta distribution tends to have a bivariate normal density when α₁,α₂ and are large.

Estimation rule for allocation rate

Although the allocation rate depends on unknown parameters, we will apply the sequential sampling rule following the trend of optimal adaptive design to update the allocation rate. The prior parameters (α₁,α₂,β) reveal the knowledge about the correlation between the auxiliary and primary outcomes (X and Y) and efficacies of the outcomes (P_X and P_Y) for a specific treatment. Based on clinician experience or pilot studies, we can determine an appropriate combination of (α₁,α₂,β) that satisfies ${\alpha }_1/({\alpha }_1+\beta )\approx E(P_X)$ and ${\alpha }_2/({\alpha }_2+\beta )\approx E(P_Y)$ . Let (xk,yk) be the paired auxiliary and primary binary outcomes for the kth subject, and let Tk be that subject’s treatment indicator. Let Iyk indicate whether the primary response for the kth patient has become accessible when a new patient is enrolled in the study. Let F(○)_i-1=F((x₁,_y1,Iy₁,T₁)…(x_i-1,_yi-1,Iyi-₁,T_i-1)) be the history of the first i – 1 patients. Based on F(○)i-1, then we have the results listed in Table 1.