Special Article - Biostatistics Theory and Methods

Austin Biom and Biostat. 2015;2(3): 1023.

# Outcome-Adaptive Allocation using Auxiliary and Primary Outcomes

Sinks S¹, Sabo RT²* and Mukhopadhyay N²

¹Office of Biostatistics, Center for Drug Evaluation and Research, Food and Drug Agency, USA

²Department of Biostatistics, Virginia Commonwealth University, USA

***Corresponding author: ** Sabo RT, Department of
Biostatistics, Virginia Commonwealth University, 830
East Main Street, Richmond, VA 23298-0032, USA

**Received: **June 01, 2015; **Accepted: **June 11, 2015; **Published: **July 02, 2015

## 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*and

_{A}*P*are the success rates for treatment

_{B}*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}<p*B*

*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*) 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,i}/n_{B,i}*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, $\text{R}={\text{\sigma}}_{\text{A}}/{\text{\sigma}}_{\text{B}}=\sqrt{{\text{p}}_{\text{A}}{\text{q}}_{\text{A}}/{\text{p}}_{\text{B}}{\text{q}}_{\text{B}}}$
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*is an auxiliary outcome for treatment

_{j}*j*and

*Y*is a primary outcome, where

*j**X*and

_{j}*Y*both are binary variables. According to the observed outcome sequence, we denote P

*j*_{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˜

*B*IN(nX,j, P

_{X,j}) and Yj|P

_{y,j}˜

*B*IN(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

_{j}and YJ is explained through the association between P

_{X,j}and P

_{X,j}Thus, the posterior distribution of P

_{X}and

*P*(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,

_{Y}*and*

*u(θ)**are positive functions that measure the risk of assigning patients to treatment*

*υ(θ)**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

*and*

*u(θ)**over all possible sets of*

*υ(θ)**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 n

*A*,I and n

*B*,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*) is the product of the conditional distributions of X

_{j},Y,_{j},*P*_{X,j},P_{y,j}_{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)*

$\left(\begin{array}{c}{n}_{x}\\ x\end{array}\right){P}_{X}^{x}{(1-{P}_{X})}^{{n}_{x}-x}*\left(\begin{array}{c}{n}_{y}\\ y\end{array}\right){P}_{Y}^{y}{(1-{P}_{Y})}^{{n}_{y}-y}$

$\frac{\Gamma ({\alpha}_{1}+{\alpha}_{2}+\text{\beta}){P}_{X}^{{\alpha}_{1}-1}{(1-{P}_{X})}^{{\alpha}_{2}+\beta -1}{P}_{Y}^{{\alpha}_{2}-1}{(1-{P}_{Y})}^{{\alpha}_{2}+\beta -1}}{\text{\Gamma}({\alpha}_{1})\text{\Gamma}({\alpha}_{2})\text{\Gamma}(\beta ){(1-{P}_{X}{P}_{Y})}^{{\alpha}_{1}+{\alpha}_{2}+\beta}}$

Integrating with respect to * P_{Y}*, the joint distribution of (X,Y,

*P*) is:

_{X}$f(X,Y,{P}_{X})=\left(\begin{array}{c}{n}_{x}\\ x\end{array}\right)\left(\begin{array}{c}{n}_{y}\\ y\end{array}\right)\frac{\text{\Gamma}({\alpha}_{1}+{\alpha}_{2}+\text{\beta})}{\text{\Gamma}({\alpha}_{1})\text{\Gamma}({\alpha}_{2})\text{\Gamma}(\beta )}{P}_{x}^{x+{\alpha}_{1}-1}{(1-{P}_{X})}^{{\alpha}_{2}+\beta -1+{n}_{x}-x}\text{(6)}$

$\frac{\Gamma (\text{y}+{\alpha}_{2})\text{\Gamma}({\alpha}_{1}+\text{\beta}+{n}_{y}-\text{y})}{\text{\Gamma}({\alpha}_{1}+{\alpha}_{2}+\beta +{n}_{y})}$

**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*and the data (X,Y) is obtained through division:

_{X}*f(P_{Y}| X,Y,P_{X})=f(X,Y,P_{X},P_{Y})/f(X,Y,P_{X}) (7)*

$\frac{\Gamma ({\alpha}_{1}+{\alpha}_{2}+\beta +{n}_{y}){P}_{y}^{y+{\alpha}_{2}-1}{(1-{P}_{Y})}^{{\alpha}_{1}+\beta -1+{n}_{y}-y}}{\text{\Gamma}(y+{\alpha}_{2})\text{\Gamma}({\alpha}_{1}+\beta +{n}_{y}-y)}$

$\frac{{(1-{P}_{X}{P}_{Y})}^{{\alpha}_{1}+{\alpha}_{2}+\beta}}{{}_{2}{}^{}F{}_{1}({\alpha}_{1}+{\alpha}_{2}+\beta ;y+{\alpha}_{2};{\alpha}_{1}+{\alpha}_{2}+\beta +{n}_{y};{P}_{X})}.$

As presented in the defined loss function, *u(θ)* and

*υ(θ)*are functions of

*,*

*P*_{Y}*A*and

*,B. Also, we know that treatment A is independent from treatment B, which indicates the distributions for treatment*

*P*_{Y}*A*

*(f(X,Y,*and for treatment

*P*,_{X}*)**P*_{Y}*A*)*B*

*(f(X,Y,*are independent. As long as we know the conditional distribution (f

*P*,_{X}*)**P*_{Y}*A*)*(*) and (

*|X,Y,**P*_{Y}*P*)A_{X}*f(*for treatment

*|X,Y,**P*_{Y}*P*)_{X}*B*)*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

*=1-*

*υ(θ)**,*

*P*_{Y}*B*, recalling that

*,j is the primary efficiency rate in the jth treatment. Then the loss function (4) is reduced to*

*P*_{Y}*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 (

*α*,

_{1}*α*,β) (Appendix

_{2}*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[

*|X,Y,*

*P*_{Y}*P*] is guaranteed to reside within the range (0,1). The correlation of X and Y is then proportional to the correlation of

_{X}*P*and

_{X}*and takes the following form (Appendix*

*P*_{Y}*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 )» (*α _{1}*,

*α*,β ) . Therefore, we intend to have a less informative prior by choosing

_{2}*r*no greater than 15 when

*α*,

_{1}*α*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

_{2}*α*,

_{1}*α*and are large.

_{2}## 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 (*α _{1}*,

*α*,β) reveal the knowledge about the correlation between the auxiliary and primary outcomes (X and Y) and efficacies of the outcomes (

_{2}*P*and

_{X}*) for a specific treatment. Based on clinician experience or pilot studies, we can determine an appropriate combination of (*

*P*_{Y}*α*,

_{1}*α*,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

_{2}*F(○)*be the history of the first

_{i-1}=*F((x*_{1},_{y1},Iy_{1},T_{1})…(x_{i}-1,_{yi}-1,Iyi-_{1},T_{i-1}))*i – 1*patients. Based on

*F(○)*

*i-1*, then we have the results listed in Table 1.

**Table 1:**Data calculations after

*i*patient are accrued.

^{th}Table 1:Data calculations afteripatient are accrued.^{th}

## Allocation algorithm

The first two steps in the algorithm for conducting the proposed adaptive design are (1) to set the initial allocation rate to 0.5 for the
first patient, and (2) to update the auxiliary efficacy for treatment
*A* and *B* with posterior means , ${\stackrel{\u0303}{p}}_{x,A}=({\alpha}_{1,A}+{x}_{A,i-1})/({\alpha}_{1,A}+{\beta}_{A}+{n}_{A,i-1})$
and
${\stackrel{\u0303}{p}}_{x,B}=({\alpha}_{1,B}+{x}_{B,i-1})/({\alpha}_{1,B}+{\beta}_{B}+{n}_{B,i-1})$
for the ith stage. These posterior means
are weighted averages of the sample proportion and prior mean, and
approximate the sample proportion as the sample size increases.
Notice that the posterior distribution of the auxiliary outcome is
given by the beta-binomial distribution.

The next step is (3) to calculate R* using, $E({P}_{Y,A}|{\stackrel{\u0303}{p}}_{x,A},F{(\circ )}_{i-1}),E({P}_{Y,B}|{\stackrel{\u0303}{p}}_{x,B},F{(\circ )}_{i-1}),{\stackrel{\u0303}{\sigma}}_{Y,A}^{2},{\stackrel{\u0303}{\sigma}}_{Y,B}^{2}$ with respect to obtaining estimates of ${\sigma}_{Y,A}^{2}$
and ${\sigma}_{Y,B}^{2}$
both posterior
means and sample estimates have disadvantages. Posterior variance
estimates are functions of the posterior means and are computationally
intensive, while sample variances might not be estimable in small
sample sizes due to response delays, no events being observed, or
patients clustering in one sample. As a compromise, the posterior
conditional variance $V\left[{P}_{Y,j}\text{|}{\stackrel{\u0303}{p}}_{x,j},F{(\circ )}_{i-1}\right]$
is used for *i* = *k,* and the
empirical sample variance $({y}_{j,i-1}({n}_{j,i-1}-{y}_{j,i-1}))/{n}_{j,i-1}^{2}$
is used for i > *k*,
where *k* is the minimum number of accrued patients after which the
sample variance is available for both treatment groups. Steps (2) and
(3) are then repeated, and randomization is terminated depending on
some specified stopping criterion (final sample size achieved, early
termination threshold exceeded, etc.).

## Results and Discussion

## Simulation study sampling methodology

In this simulation study, we are interested in modeling different
clinical scenarios to see the performance of our bivariate allocation
method compared with both optimal and balanced allocation.
Specifically, we examine: 1) how different primary outcomes between
treatment *A* and B affect simulation results, in terms of allocation
proportions, the number of patients assigned to each treatment, error
rate, number of treatment failures, and 2) how different auxiliary
efficacies affect the simulation results.

In many situations, simulations are conducted separately for each
method with a large number of repetitions, where each simulated
data set represents a single trial. However, this approach could lead to
scenarios where one or more of the approaches are exposed to more
instances of rarely occurring samples than are other approaches.
Thus, it may be more realistic to generate **N _{A}** random observations from the treatment

*A*population and

*N*random observations from treatment B population for each trial, where each method would then sample from the same pool of subjects. Suppose N is the total sample size of the clinical trial, then

_{B}*N*and

_{A}*N*should both be greater than N. Within a trial, three allocation methods will actually share the same sample pool to simulate from populations of treatment

_{B}*A*and

*B,*and the sample pool is regenerated after each trial. In this manner, we are able to reduce variation between the samples used by each method.

The sample size for balanced allocation is fixed in advance, while the sample size of the adaptive methods is allowed to adjust during the trial. The total sample size for each combination of parameters (discussed below) is selected to yield 90% power for a two-sided Z-test assuming balanced allocation; in cases with no true difference, sample sizes of 100 subjects per group were created. Correlated binomial responses are, sampled from a multinomial distribution given both auxiliary and primary efficacies with a specified correlation [7]; note that this correlation does not vary freely in the range (-1,1) due to restrictions of the joint probability distribution of auxiliary and primary outcomes. We assume the correlation between auxiliary and primary outcomes is fixed regardless of treatment effect.

## Simulation settings

Optimal allocation utilizes the primary outcome to update the
allocation ratio, which is calculated based on the sample proportions.
As mentioned earlier, these sample estimates may not be estimable
in early stages of the trial, when no variability exists in treatment
responses or no primary responses are available. *A* lead-in is
introduced to the simulation process for optimal allocation during
which patients are assigned to treatments with equal probability.

Prior distributions take into account the uncertainty of *P _{X}* and

*before observed data is considered. Recalling that auxiliary and primary efficacies (*

*P*_{Y}*P*and

_{X}*) follow beta distributions,*

*P*_{Y}*α*and β are the shape and scale parameters for the auxiliary efficacy (

_{1}*P*), and

_{X}*α*and β are the shape parameters for the primary efficacy (

_{2}*). We assume that the mean of each prior distribution is equal to some value (pX, pY), which gives us ${\alpha}_{1}/({\alpha}_{1}+\beta )\approx E({p}_{x})$ and $\frac{{\alpha}_{2}}{{\alpha}_{2}+\beta}\approx E({p}_{y})$ . Given these two equations, the relationship among (α*

*P*_{Y}_{1}, α

_{2}, β) can be determined, and with a known correlation between the

*P*and

_{X}*, the exact combination of α*

*P*_{Y}_{1}, α

_{2}, β) can be found.

Our goal is to model scenarios where the primary outcome has a
rare event rate and the auxiliary outcome has a moderate event rate.
We thus select primary efficacies between * P_{Y}∈[0.1,0.3]*, and auxiliary
efficacies between

*P*. Due to restrictions of the correlation in the bivariate beta distribution (see Discussion), we assume the correlation between auxiliary and primary outcomes is 0.5 in all cases. In order to incorporate delayed observations of the primary outcome, we assume the primary outcome for each subject is not observed until 30 additional subjects have accrued into the trial. Alternatively, we assume that the auxiliary outcome is immediately observed.

_{X}∈[0.4,0.7]## Simulation results

In what follows, we refer to our proposed method as the bivariate
approach and the traditional optimal allocation method as the
univariate approach. Table 2 presents the number of patients assigned
to treatment B (the more effective treatment) for a given sample size.
We first note that in cases of differences in treatment success rates,
the bivariate approach accounting for auxiliary information assigns
more subjects to the more effective treatment than does the univariate
approach. With an effect size of 0.1 between the primary success rates
(n = 526), the bivariate optimal method assigned approximately 50
more patients to treatment B than does balanced allocation, while the
univariate optimal approach assigned approximately 42 more, which
are 19% and 16% improvement over balanced allocation, respectively.
When the effect size increases to 0.2 (n = 162), the bivariate approach,
allocated on average 23 more patients to treatment *B* than balanced
allocation (a 28% increase), while the univariate approach assigned
only 14 more (a 17% increase). We also see that the two adaptive
approaches perform similarly to balanced allocation when there is
no difference in primary success rates, with the bivariate optimal
approach performing similarly even when the auxiliary outcomes
have different success rates between treatments.

**Table 2:**Summary Number of patients in group B (receiving more effective treatment).

Sample

Size

Primary

Auxiliary

MethodTRT A

TRT B

TRT A

TRT B

Bivariate

Univariate

Balance

5260.1

0.2

0.4

0.7

313.4

304.8

263.5

0.4

0.6

312.2

304.4

262.8

0.5

0.7

312.1

305.0

263.0

0.5

0.6

313.6

305.4

263.2

0.6

0.6

313.0

305.0

263.1

0.4

0.7

104.0

95.4

80.9

0.4

0.6

104.7

96.0

81.1

1620.1

0.3

0.5

0.7

104.0

95.5

81.1

0.5

0.6

104.8

95.2

80.8

0.6

0.6

104.9

95.4

81.6

0.4

0.6

99.7

99.9

100.3

2000.3

0.3

0.6

0.5

100.8

100.1

99.8

0.5

0.5

100.0

100.3

99.8

0.7

0.5

97.1

99.9

100.0

Table 2:Summary Number of patients in group B (receiving more effective treatment).

In Table 3 we see the average number of treatment failures for each of the three allocation strategies. Both optimal allocation methods produced slightly fewer failures than balanced allocation, though the improvements were small due to the low success rates for the primary outcomes. In addition, the bivariate approach averaged nearly 1 fewer failure than the univariate approach when the effect size was 0.1 and nearly 2 fewer failures when the effect size was 0.2. While modest (especially compared to the reported standard deviations), this improvement shows that incorporating the more quickly realized auxiliary information in the manner described for the bivariate method can lead to real gains compared to the standard univariate approach. Table 4 shows the estimated empirical power and type-one error rates for each approach. Though the two optimal allocation approaches lead to imbalanced treatment groups, such imbalances did not affect either power or the level of the resulting hypothesis tests.

**Table 3:**Summary of Expected Number of Patient Failures (Standard Deviation).

Sample

Size

Primary

Auxiliary

MethodTRT A

TRT B

TRT A

TRT B

Bivariate

Univariate

Balance

5260.1

0.2

0.4

0.7

442.4 (8.6)

443.2 (8.5)

447.3 (7.9)

0.4

0.6

442.1 (8.6)

443.0 (8.5)

447.1 (8.2)

0.5

0.7

441.8 (8.5)

442.5 (8.4)

447.5 (8.1)

0.5

0.6

441.8 (8.9)

442.5 (8.8)

447.3 (8.1)

0.6

0.6

441.2 (8.6)

443.1 (8.7)

447.4 (8.3)

0.4

0.7

125.0 (5.6)

126.7 (5.6)

129.9 (5.0)

0.4

0.6

124.6 (5.5)

126.4 (5.6)

129.5 (5.1)

1620.1

0.3

0.5

0.7

125.1 (5.4)

126.8 (5.4)

129.5 (5.0)

0.5

0.6

124.8 (5.4)

126.7 (5.3)

129.5 (5.2)

0.6

0.6

124.8 (5.5)

126.6 (5.4)

129.7 (5.2)

0.4

0.6

140.0 (6.8)

140.1 (6.8)

140.2 (6.8)

2000.3

0.3

0.6

0.5

140.0 (6.4)

140.0 (6.4)

140.0 (6.4)

0.5

0.5

139.8 (6.4)

139.9 (6.3)

139.9 (6.3)

0.7

0.5

139.9 (6.5)

139.9 (6.5)

139.9 (6.4)

Table 3:Summary of Expected Number of Patient Failures (Standard Deviation).

**Table 4:**Summary of power/error rate.

Sample

Size

Primary

Auxiliary

MethodTRT A

TRT B

TRT A

TRT B

Bivariate

Univariate

Balance

5260.1

0.2

0.4

0.7

0.92

0.91

0.91

0.4

0.6

0.90

0.92

0.89

0.5

0.7

0.90

0.89

0.92

0.5

0.6

0.91

0.91

0.90

0.6

0.6

0.90

0.90

0.88

0.4

0.7

0.91

0.91

0.91

0.4

0.6

0.91

0.92

0.91

1620.1

0.3

0.5

0.7

0.90

0.90

0.91

0.5

0.6

0.92

0.92

0.91

0.6

0.6

0.91

0.91

0.91

0.4

0.6

0.06

0.06

0.06

2000.3

0.3

0.6

0.5

0.05

0.05

0.05

0.5

0.5

0.05

0.05

0.04

0.7

0.5

0.04

0.04

0.05

Table 4:Summary of power/error rate.

Tables 5, 6 and 7 provide estimates of the allocation rates as well as a measure of their variability (Inter quartile Range, IQR) after 25%, 50% and 75% of patients have been accrued. In the low effect size (n = 526) and no difference (n = 200) cases after 25% of the trial has been concluded Table 4, we see that the bivariate and univariate methods have similar average allocation ratios and IQRs. Interestingly, in the large effect size case with a smaller total sample size (n = 162), we see that the univariate approach has not yet begun adapting, since few primary outcomes have been observed at this point. These results show that while the bivariate and univariate allocation approaches behave similarly after enough observations are in hand, the ability of the bivariate approach to begin the adaptation sooner is the determining factor in any gains it exhibits over the univariate approach. We see that the allocation ratios and their variability’s did not change much after 50% (Table 6) and 75% (Table 7) of the patients are accrued. At this point both the bivariate and univariate procedures allocate patients between treatments in almost identical manners, regardless of effect size.

**Table 5:**Summary of Allocation Rate (IQR) at the 25

^{th}percentile visit for treatment A.

Sample

Size

Primary

Auxiliary

MethodTRT A

TRT B

TRT A

TRT B

Bivariate

Univariate

5260.1

0.2

0.4

0.7

0.41 (0.36, 0.45)

0.41 (0.37, 0.45)

0.4

0.6

0.41 (0.36, 0.46)

0.41 (0.36, 0.46)

0.5

0.7

0.41 (0.36, 0.46)

0.41 (0.37, 0.46)

0.5

0.6

0.41 (0.37, 0.45)

0.41 (0.37, 0.45)

0.6

0.6

0.41 (0.36, 0.45)

0.41 (0.37, 0.45)

0.4

0.7

0.34 (0.32, 0.37)

0.50 (0.50, 0.50)

0.4

0.6

0.33 (0.32, 0.36)

0.50 (0.50, 0.50)

1620.1

0.3

0.5

0.7

0.33 (0.32, 0.37)

0.50 (0.50, 0.50)

0.5

0.6

0.33 (0.31, 0.36)

0.50 (0.50, 0.50)

0.6

0.6

0.33 (0.31, 0.35)

0.50 (0.50, 0.50)

0.4

0.6

0.50 (0.46, 0.54)

0.50 (0.45, 0.55)

2000.3

0.3

0.6

0.5

0.50 (0.46, 0.54)

0.50 (0.45, 0.55)

0.5

0.5

0.50 (0.47, 0.53)

0.50 (0.45, 0.55)

0.7

0.5

0.51 (0.46, 0.55)

0.50 (0.45, 0.55)

Table 5:Summary of Allocation Rate (IQR) at the 25^{th}percentile visit for treatment A.

**Table 6:**Summary of Allocation Rate (IQR) at the 50

^{th}percentile visit for treatment A.

Sample

Size

Primary

Auxiliary

MethodTRT A

TRT B

TRT A

TRT B

Bivariate

Univariate

5260.1

0.2

0.4

0.7

0.41 (0.38, 0.44)

0.41 (0.38, 0.44)

0.4

0.6

0.41 (0.38, 0.44)

0.41 (0.38, 0.44)

0.5

0.7

0.41 (0.36, 0.44)

0.41 (0.38, 0.44)

0.5

0.6

0.41 (0.36, 0.44)

0.41 (0.38, 0.44)

0.6

0.6

0.41 (0.36, 0.44)

0.41 (0.38, 0.44)

0.4

0.7

0.35 (0.32, 0.41)

0.38 (0.32, 0.44)

0.4

0.6

0.35 (0.32, 0.41)

0.37 (0.32, 0.43)

1620.1

0.3

0.5

0.7

0.36 (0.32, 0.41)

0.38 (0.32, 0.43)

0.5

0.6

0.36 (0.32, 0.41)

0.38 (0.32, 0.44)

0.6

0.6

0.36 (0.32, 0.41)

0.38 (0.32, 0.43)

0.4

0.6

0.50 (0.47, 0.53)

0.50 (0.47, 0.53)

2000.3

0.3

0.6

0.5

0.50 (0.47, 0.53)

0.50 (0.47, 0.53)

0.5

0.5

0.50 (0.47, 0.53)

0.50 (0.47, 0.53)

0.7

0.5

0.51 (0.47, 0.53)

0.50 (0.47, 0.53)

Table 6:Summary of Allocation Rate (IQR) at the 50^{th}percentile visit for treatment A.

**Table 7:**Summary of Allocation Rate (IQR) at the 75

^{th}percentile visit for treatment A.

Sample

Size

Primary

Auxiliary

MethodTRT A

TRT B

TRT A

TRT B

Bivariate

Univariate

5260.1

0.2

0.4

0.7

0.41 (0.39, 0.43)

0.41 (0.39, 0.43)

0.4

0.6

0.41 (0.39, 0.43)

0.41 (0.39, 0.43)

0.5

0.7

0.41 (0.39, 0.43)

0.41 (0.39, 0.43)

0.5

0.6

0.42 (0.39, 0.44)

0.42 (0.39, 0.44)

0.6

0.6

0.41 (0.39, 0.44)

0.41 (0.39, 0.44)

0.4

0.7

0.36 (0.32, 0.40)

0.36 (0.31, 0.40)

0.4

0.6

0.36 (0.32, 0.40)

0.36 (0.31, 0.40)

1620.1

0.3

0.5

0.7

0.36 (0.32, 0.40)

0.36 (0.32, 0.40)

0.5

0.6

0.36 (0.31, 0.40)

0.36 (0.32, 0.40)

0.6

0.6

0.36 (0.31, 0.40)

0.36 (0.32, 0.41)

0.4

0.6

0.50 (0.48, 0.52)

0.50 (0.48, 0.53)

2000.3

0.3

0.6

0.5

0.50 (0.48, 0.52)

0.50 (0.47, 0.52)

0.5

0.5

0.50 (0.48, 0.52)

0.50 (0.48, 0.52)

0.7

0.5

0.50 (0.48, 0.52)

0.50 (0.48, 0.53)

Table 7:Summary of Allocation Rate (IQR) at the 75^{th}percentile visit for treatment A.

## Conclusion

In this manuscript we introduced an optimal allocation strategy for reducing binary treatment failures when the primary outcome is delayed in measurement. Our approach compliments the lagged primary outcome with a quickly observed auxiliary binary outcome, assuming that both outcomes are positively correlated. The information from both outcomes is combined using a Bayesian approach, where we use a bivariate beta prior to model the dependence between the success rates of the primary and auxiliary outcomes. We have shown the dependence between the rates to be proportional to the dependence between primary and auxiliary observations. This Bayesian approach also allows researchers to incorporate information from other sources, including results from previous studies, pilot data, or even hypotheses based on clinical expertise.

One limitation of the presented work is that we only considered
trials with two groups. While optimal allocation designs exist for
three-group trials [8], the allocation rations are more complex, and
have not been expressed in closed form for any case greater than three
groups [8,9]. Our focus was also solely upon binary outcomes. We
have assumed that the auxiliary and primary outcomes are *known* to
be positively correlated *a priori*. Hence we did not study scenarios
where the correlation between the two outcomes was positive,
but weak (we assumed a correlation of 0.5), where there was no
correlation, or where the correlation was negative.

Natural extensions of this approach would be to account for more
than two groups, and also to account for continuous outcomes, or
even cases where primary and auxiliary outcomes are of different
modeling types (e.g., continuous and discrete). The loss-function
approach used in Sections 2 and 3 could be used again to find the
bivariate optimal allocation ratio for cases of three or more treatment
arms, as Jeon and Hu [8] derived the univariate optimal allocation
ratio for minimizing treatment failures in trials with three arms and
binary outcomes. *A* similar approach could be used to jointly model
continuous primary and auxiliary outcomes, though incorporating
the association between such outcomes with a bivariate normal
distribution will be more straightforward than the current approach. Jointly modeling heterogeneous modeling types can follow the general
outline provided here, though selecting a bivariate prior distribution
may become more challenging.

The choice of modeling the association between primary and auxiliary outcomes through modeling the association between primary and auxiliary success rates with prior information was done for simplicity. We selected the bivariate beta distribution for this prior [3], though other distributions can certainly be used. An alternative approach would be to jointly model the outcomes in a way that directly incorporates their dependence, which would then allow separate prior elicitation for the marginal parameters of each outcome. In cases where we are interested in either different prior specification or in jointly modeling the outcomes, copula functions offer flexible alternatives.

## Acknowledgement

We are grateful for the support and encouragement provided by Dr. Karl Peace. Services in support of the research project were generated by the VCU Massey Cancer Center Biostatistics Shared Resource, supported, in part, with funding from NIH-NCI Cancer Center Support Grant P30 CA016059.

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