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Research article

An Alternative Perspective on Consensus Priors with Applications to Phase I Clinical Trials

Steven B. Kim1*, Daniel L. Gillen2

1Department of Mathematics and Statistics, California State University, Monterey Bay
2Department of Statistics, University of California, Irvine

*Corresponding author: Dr. Steven B. Kim, California State University, Monterey Bay, Tel: 831-582-3954,

Email: stkim@csumb.edu

Submitted: 08-22-2016 Accepted: 10-25-2016 Published: 11-03-2016

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Abstract

We occasionally need to make a decision or a series of decisions based on a small sample. In some cases, an investigator is knowledgeable about a parameter of interest in some degrees or is accessible to various sources of prior information. Yet, two or more experts cannot have an identical prior distribution for the parameter. In this manuscript, we discuss the use of a consensus prior and compare two classes of Bayes estimators. In the first class of Bayes estimators, the contribution of each prior opinion is determined by observing data. In the second class, the contribution of each prior opinion is determined after observing data. Bayesian designs for Phase I clinical trials allocate trial participants at new experimental doses based on accumulated information, while the typical sample sizes are fairly small. Using simulations, we illustrate the usefulness of a combined estimate in the early phase clinical trials.

Keywords: Consensus prior; Weighted Bayes estimator; Small-sample studies; Phase I clinical trials

Introduction

In small-sample studies, statisticians often rely on parametric assumptions to gain efficiency for estimation and testing. In a Bayesian framework, prior assumptions are also incorporated into the model. Statistical inference regarding parameters of scientific interest is based on the resulting posterior distribution which is obtained by updating the prior assumption with observed data. While large sample sizes are preferred for statistical precision, there are many situations when we need to make a decision or a series of decisions based on a small amount of empirical evidence together with a prior. In this case, an investigator’s prior opinion can be highly influential, yet it is still essential.

When eliciting prior assumptions, opinions from multiple subject area experts are generally more informative than the opinion of a single expert. On the other hand, when multiple experts have divergent prior opinions, we face the dilemma of how each prior opinion should be incorporated into inferential procedures. This decision is important particularly in sparse data settings where inference may change substantially depending upon the way each prior is incorporated into posterior inference. A combination of multiple sources of information has been practiced, and it has been known that simple combination methods perform reasonably well relative to more complicated approaches [1]. One simple method to combine multiple prior opinions is a consensus prior, as discussed by Samaniego among others [2]. Samaniego focused on a class of estimators which combine individual posterior estimation. When each individual has a unique posterior estimate originated from a unique prior distribution, the class of estimators average the multiple posterior estimates based on the credibility of each prior elicitation determined before observing data. This approach seems more robust than relying on a single posterior estimate.

When we have multiple candidate models, accounting for model uncertainty has been widely discussed in the literature, and the idea of model averaging has been popular in dose-response modeling in small sample studies [3-7]. It is a robust approach which allows multiple models to have a simultaneous contribution to inference, where the weight of contribution is determined by observed data. In addition to model uncertainty, we sometimes face divergent prior beliefs among experiments on the same parameter of interest [8]. When it is fairly evident that one expert’s prior guess is far more plausible relative to the other prior opinions that were obtained, it is tempting to determine the credibility of each prior based on observed data. In this manuscript, we provide an alternative perspective of a consensus prior and the resulting inference obtained from the use of a consensus prior. Specifically, we show that a natural posterior weighting scheme with weights based upon observed data arising naturally in the case of a consensus prior.

This manuscript is organized as follow. In Section 2, we briefly review a consensus prior and the notion of self-consistency in the context of Bayesian estimation [2]. We then provide an alternative perspective of a consensus prior with a Bayes estimator in which the contribution of each investigator is determined after observing data. We also comment on Occam’s window criterion in the context of a consensus prior. In Section 3, we provide two examples. The first example is a normal model to enhance the understanding of the alternative perspective of a consensus prior. The second example is a logistic model applied to early phase clinical trials. In this type of application, experimentalists are generally required to make a series of decisions for trial patients based on sparse data together with all available prior information. In Section 4, we present the simulation results, and we conclude with a discussion on the utility of consensus priors in scientific investigations in Section 5.

Bayesian Decisions Based on a Consensus Prior

A Consensus Prior

In small-sample studies, a single prior opinion that is contrary to observed data may substantially influence posterior inference. To guard against this undesirable situation, we may rely on multiple opinions. In this setting, it is necessary to incorporate prior disagreements in the resulting inferential procedure. We initially focus on a case when all investigators specify conjugate priors through the same model (i.e. the same likelihood function). Let Q={Q1,…,QK} denote a set of priors and ɡk (θ | Qk ) denote the prior density function of θ from Qk (i.e. prior specification by the kth investigator). As discussed by Samaniego, one possible approach for a compromise is to gather the multiple investigators and formulate a single prior through conversations. This process may not be systematic and is logistically infeasible in many practical settings. One simple alternative and systematic approach to account for multiple priors is to specify a K-fold mixture

biostat eqstn 6.1


which is referred to as a consensus prior. The assigned prior probabilities P(Qk ) for Qk∈Q reflect the plausibility of each investigator’s prior opinion, perhaps depending on various  degrees of knowledge and experience. Multiple advantages stemming from the use of a consensus prior based on individual conjugate priors are listed by Samaniego [2]. In this manuscript, one particular advantage of interest is that the posterior distribution is also a K-fold mixture of gk (θ |biostat line 6.1 ,Qk ) by conjugacy, where denotes observed data. Its mathematical property will be further discussed in a later section (see Section 2.3).

Self-Consistency

We let denote a fixed sufficient and unbiased estimator of a scalar parameter θ. The Bayes estimator of θ which minimizes square error loss is said to be self-consistent if E(θ| = θ*)=θ*, where θ*=E(θ) is the prior expectation [2]. If such an estimator exists, the property of self-consistency is appealing in the sense that the posterior estimate is θ* when both a prior estimate and an unbiased estimator are θ*. Under an exponential family, we may find a self-consistent estimator of the form (1-η) θ*+ηθ̂. However, self-consistency is not guaranteed in many cases, particularly when the estimand of interest is a nonlinear function of regression parameters.

Using the form of a consensus prior in Equation (2.1), Samaniego noted that self-consistency is not preserved in general. He instead discussed the class of estimators of the form

biostat eqstn 6.2

where an individual posterior estimator θ̂k is of the form (1-ηkk*kθ̂. Here θ k*=E(θ|Qk) is an individual prior mean and ak does not depend on data. This class of estimators is a natural compromise among disagreeing posterior estimates, where the contribution of each investigator is determined by the prior plausibility, say ak=P(Qk ).

If ηk is restricted to be constant (i.e. ηk=η) and θ̂=θ*= the property of self-consistency can be preserved because

biostat eqstn 6.3

If ηk depends on k, self-consistency is not generally guaranteed.

An Alternative Perspective for a Consensus Prior

While the notion of self-consistency is mathematically appealing, there are cases when we wish to use empirical evidence to determine each individual’s contribution to the posterior estimation. For instance, when θ̂1=-1 and θ̂2=1 after observing θ̂=2, it may be a natural reaction to desire a2>a1. Any data-dependent weighting scheme initially appears to be foundational unsound from a Bayesian perspective since one would seemingly be required to change their prior assumptions following the observation of empirical results. While briefly noted by Samaniego [2] we show in this section that at least one data- dependent weighting scheme can be formally justified by showing its equivalence to the use of a consensus prior that is specified prior to the observation of any experimental data. The result not only provides formal justification for the use of adaptively weighted priors in sequential experimentation and also yields further intuitive appeal for the use of consensus priors in Bayesian inference.

Let f( biostat line 6.1| θ) be the likelihood function which is common for all Qk∈Q. The individual marginal likelihood functionbiostat line 6.1

biostat eqstn 6.4


Therefore, the posterior inference initialized by a consensus prior and the posterior inference based on the weighted posterior density function are equivalent. In addition, the posterior contribution of the kth investigator is exactly equal to P(Qk | biostat line 6.1) in Equation (2.5). This procedure is very similar to the procedure of Bayesian model averaging which allows multiple likelihoods to account for model uncertainty [3].

Relative to the consensus posterior g2 (θ |biostat line 6.1 ), the Bayes estimator of θ which minimizes square error loss is

biostat eqstn 6.5

where ak*=P(Qk|biostat line 6.1 ) is the posterior weight given in Equation (2.5) and θ̂k=E(θ | biostat line 6.1, Qk) is the individual Bayes estimator which minimizes square error loss with respect to the individual posterior. At first glance, the data-dependent weighting scheme in this estimator may appear incoherent from a Bayesian perspective because each posterior weight ak* depends on the same data biostat line 6.1 . However, the proposition supports that it is a natural reaction after observing biostat line 6.1 in a Bayesian framework. We also note that self-consistency of θ̂** is not simple to be achieved even if θ̂k is of the form (1-η) θk*+ηθ̂. In the following section, we compare the characteristics of θ̂* and θ̂**.

Examples and Simulation Studies

A Normal Model

We consider a random sample  biostat line 6.1=(Y1,…,Yn ), where Yi ~ N(μ,σ2) with unknown μ and known σ2. Under this exponential family,the sample mean μ̂=Y̅ is sufficient and unbiased for μ. Fork=1,…,K, the conjugate priors μ | Qk ~ N(μk*k2) leads to the posterior μ|biostat line 6.1 ,Qk ~ N(μ̂kk-1), where τkk-2+nσ-2 denotes theposterior precision (i.e. inverse of the posterior variance)and the Bayes estimator μ̂k is in the form of (1-ηkk*kY̅ withηk=nσ-2τk-1. For self-consistency, we require the same prior
precision τk=τ, and the self-consistent estimator of μ is  biostat eqstn 6.6 , where ak=P(Qk ).

 

Each individual marginal likelihood function can be written as

 

biostat eqstn 6.7

where vkk22 n-1. To this end, the posterior contribution ofQk to the weighted posterior ɡ2 (μ |biostat line 6.1 ) is

P(Qk| biostat line 6.1)∝P(Qk) ϕ(y̅; μk*,vk) ,

where ϕ(∙) denotes a normal density function.Then, the posterior weighted estimator of μ is biostat eqstn 6.5μ̂**= , with ak *=P(Qk(y̅; μk *,vk ). In practice, some expert may be more experienced to receive a relatively greater prior weight P(Qk ). However, for a given P(Qk), a strong prior opinion expressed via a small value of σk 2 additionally increases the posterior weight ak *. This is a typical behavior in the posterior weighting scheme through conjugacy. To this end, it is something to consider whether we desire to upweight an individual opinion simply because of a relatively strong prior opinion.

Typically with a small sample, the bias of μ̂** is small relative to the bias of μ̂* because μ̂** has the second chance to adjust weights based on observed data. On the other hand, the variance of μ̂** is large relative to the variance of μ̂* because the data-dependent weight ak * is random from sample to sample. As a numerical example for the trade-off between bias and variance, we consider K=2 prior opinions with σk=0.5 (i.e. prior sample size of two) and μk*=(-1)k for k=1,2. Figure 1 showsthe curve of relative root mean square error (RMSE) of μ̂**) to μ̂* when μ varies from -2 to 2 for small sample sizes n=3,6,9,12.

Focusing on the case of P(Q1 )=P(Q2), RMSE exceeds one (i.e. the self-consistent μ̂* performs better) when μ is close to the midpoint of the two prior estimates. On the other hand, RMSE is below one (i.e. the posterior weighted μ̂** performs better) when μ is substantially closer to one of the two prior estimates. The relative performance of μ̂** is more plausible as n increases. Focusing on the case of P(Q2)=9×P(Q1), a similar trend is shown with the zone of superiority of μ̂* shifted the the toward μ2*=1. When μ is near μ2*, the posterior weighted μ̂** performs better with respect to RMSE. When μ is against the prior assignment P(Q2)=9×P(Q1) (i.e. μ is close to μ1*), RMSE decreases,and the rate of a decrease is faster with a larger sample size. When σ2 is unknown, the relative operating characteristics of μ̂** to μ̂* is similar to when σ2 is known.

 biostat fig 6.1

Figure 1. RMSE of μ̂** to μ̂ * when μ varies from -2 to 2 for n=3,6,9,12. The figure on the left is the case of P(Q1 )=P(Q2 ), and the figure on the right is the case of P(Q2 )=9×P(Q1 ).

A Logistic Regression Model in Phase I Clinical Trials

Background: Phase I Clinical Trials

A maximum tolerable dose (MTD) is the highest dose of a therapeutic treatment that does not cause unacceptable toxicity in a loose definition. A precise definition will be provided later in this section. A primary objective of most Phase I clinical trials is to study the toxicity of a new drug and to determine the MTD for later investigation and future patients. Whitehead and Williamson discussed various Bayesian decision theoretic approaches for dose-finding studies based on a logistic regression model [9] in addition to the design which focused on information gain [10]. Among various gain functions discussed, we focus on the patient gain using the terminology in the paper of Whitehead and Williamson. For the patient gain, the dose allocation rule is optimal for each member of trial participants. This experimental design is analogous to the continual reassessment method (CRM) proposed by O’Quigley et al. which  as first developed to treat cancer patients in severe conditions [11]. One major concern about this Bayesian adaptive design is prior sensitivity. In particular, estimation of the MTD and the number of adverse events (AEs) per trial are highly sensitive to a prior specification because we often have thirty or less subjects in Phase I clinical trials. Sometimes, multiple researchers may have different opinions about the toxicity of a new experimental drug, and the dose allocations for the first few trial participants heavily depend on their subjective opinions. Therefore, the application of a consensus prior is suitable.

Suppose we observe biostat line 6.1 =(Y1,…,Yn ), where Yi ~ Bernoulli(Πxi) with Yi=1 indicating an AE at the treated dose xi in logscale. Using a logistic regression, the likelihood function of = (β01 ) is

biostat eqtn 6.6

by assuming independence among patients. We let β0∈(-∞,∞) and β1∈(0,∞) by assuming a monotonic dose-response relationship. We consider a conditional mean prior [12]. This method of prior elicitation requires the selection of two arbitrarily doses, say x-1<x0 without loss of generosity. Then, we specify two independent Beta priors Πxi ~ Beta(pi,qi ) for i=-1,0. By the Jacobian transformation, we obtain the joint prior density function

biostat eqtn 6.7

hence pi and qi can be thought of as the pseudo numbers of AEs and non-AEs, respectively, at the selected dose xi. To this end, we achieve the conjugacy by the posterior density function

biostat eqtn 6.8

where yi=pi and 1-yi=qi for i=-1,0.

If we have multiple prior opinions, Q={Q1,…,QK}, we may ask each kth investigator independently to specify (xik,pik,qik) for i=-1,0. In addition to the conjugacy, the conditional mean prior is advantageous because it is more interpretable than a direct specification of the joint prior density function of . Hence, it is easy to control the amount of prior information in a consensus prior. By the conjugacy, the weighted posterior density function is

biostat_eqtn_6.9

where yik=yi, nik=1 and cik=1 for i=1,…,n and yik=pik, nik=pik+qik and

for i=-1,0. It may be plausible to keep x0k-x-1k constant or nearly constant across investigators because the distance contributes to the weighted posterior. In addition, the values of (pik,qik) are highly influential when they are not small. For example, if (pi1,qi1)=(1,2) and (pi2,qi2)=(2,4) with the same value of x0k-x-1k, then ci2/ci1=10.

Here we provide the precise definition of the MTD. For a given risk level γ∈(0,1), we define MTD as

biostat eqtn 6.10

under the logistic model. We simplify the notation Dγ(biostat line 6.5 ) by Dγ. Equivalently, it is the dose such that under the model. Assuming both efficacy and toxicity of a treatment is monotonic with respect to dose, Dγ is the target treatment dose for a fixed risk level, and it is the parameter of interest. A typical value of γ is between 0.15 and 0.35 in a cancer study.

Allocating Trial Participants to New Experimental Doses

We modify the patient gain discussed in Whitehead and Williamson [9] to the patient loss by taking the inverse of the gain. For the patient loss, we allocate the (n+1)th patient at

biostat eqtn 6.11

where the kth posterior mean contributes to this decision with the data-dependent weight of ak*=P(Qk|biostat line 6.5 ). From the perspectiveof the Bayesian decision theoretic approach, each patient is allocated at the posterior weighted estimator

biostat_eqtn_6.12

Due to a finite sample and the nonlinear transformation of the regression parameters, it is difficult to construct a self-consistent estimator. In particular, it is difficult to link the class of estimators

 biostat_eqtn_6.13

With the data-independent weight ak=P(Qk) to self-consistency.

Simulation

Simulation Designs

We design simulation studies to investigate the relative operating characteristics of the two consensus estimators D̂γ* and D̂γ** in Phase I clinical trials. For numerical experiments, we set the target risk level at γ=0.2 and assume N=25 patients are available for a Phase I clinical trial. We suppose three investigators have divergent prior opinions. The first investigator specifies Q1 with (x-11, p-11, q-11)=(-4.0,1.2,3.8) and (x01,p01,q01 )=(4.0,3.8,1.2). The prior specification added 1.2 AEs and 3.8 non-AEs at the arbitrarily low dose x-11=-4 (in logscale) and added 3.8 AEs and 1.2 non-AEs at the arbitrarily high dose x01=4 (in log-scale as well). The second investigator specifies Q2 by (x-11, p-11, q-11)=(0.0,1.2,3.8) and (x01,p01,q01 )=(8.0,3.8,1.2), and the third investigator specifies Q3 by (x-12,p-12,q-12)=(4.0,1.2,3.8) and (x02,p02,q02)=(12.0,3.8,1.2). The three prior elicitations are designed to be equally strong by making x0k- x-1k=9 a nd . I n a ddition, we assume the equal prior probability P(Qk )=1/3 for k=1,2,3. The prior mean of Dγ is Dγ,k*=E(Dγ|Qk)=-3.1,0.9,4.9 for k=1,2,3, respectively, so Q1 is relatively conservative and Q3 is relatively anti-conservative. If a trial is proceeded by each investigator separately, the first trial patient is allocated at the three individual prior estimates. The remaining trial patients are allocated at the updated estimates of Dγ, therefore the three priors shall lead to different results at the end of trial with respect to the final estimation of Dγ and the total number of observed AEs, denoted by . On the other hand, if the three investigators make a compromise through a consensus prior, the first patient is allocated at (-3.1+0.9+4.9) / 3=0.9 and the remaining decisions are based on the consensus estimators. If D̂γ * is used, the three investigators contribute the entire trial equally. If D̂γ ** is used, the contribution of each investigator varies depending on accumulated data.
 
As shown in Figure 2, we consider the nine scenarios by crossing the three values of the intercept β0=(-6,-3,0) and the three values of the slope β1=(0.5,0.8,1.2) under the logistic model. The indexing number of each scenario is attached in the ascending order of Dγ. From Scenarios 1 to 9, the nine values of Dγ are -2.77, -1.73, -1.16, 1.34, 2.02, 3.23, 3.84, 5.77 and 9.23. A low value of Dγ implies that a new experimental treatment is safe, and a high value of Dγ implies that it is fairly toxic. In each subfigure, the true value of Dγ in each scenario is indicated by the dotted line, and prior guesses Dγ,k* for k=1,2,3 are indicated by the three solid lines.
 
Evaluations
 
We let D̂γ generically denote a Bayes decision for Dγ. Then, the distribution of interest is rather than the distribution of D̂γ itself. In other words, if D̂γ deviates from Dγ, practitioners’ interest is the deviation of from the target risk level γ.
  
We evaluate simulation results in three respects. First, it is important to treat trial participants near the target risk level γ=0.2 (from the perspective of current patients). In this regard, we measure the mean, standard deviation (SD) and root mean square error (RMSE) of πX with respect to 0.2, where X is the random variable denoting a treated dose in a trial. Second, it is also important to have near γ=0.2 at the end of a trial (from the perspective of future patients). In this regard, we measure
 
biostat fig 6.2
 
the mean, SD and RMSE of after observing the outcomes of N=25 patients. Third, practitioners concern about the distribution of , the sum of AEs at the end of a trial. In particular, as we fixed N=25 and γ=0.2, a desirable distribution of
S25 should have a mode near N×γ=5 with small V(S25) and large P(4≤S25≤6). To this end, we summarize the distribution of S25 by the mean, SD and P(4≤S25≤6).
 
Simulation Results
 
The simulation results for using individual Q1, Q2, Q3 and the consensus prior consisting of {Q1,Q2,Q3} are summarized in Table 1. In the table, the consensus prior with the prior weighted scheme (i.e. D̂γ*) is denoted by biostat line 6.6 and the consensus prior with the posterior weighted scheme (i.e. D̂γ**) is denoted by biostat_line_6.7 .
 
Focusing on the distribution of πX (see the left three columns of the table), the use of a single prior opinion can be very sensitive depending on the true dose-response curve. In Scenario 1, 2 and 3, when a new experimental dose has relatively high toxicity, the respective means of πX were 0.711, 0.787 and 0.860 for the anti-conservative prior Q3 which imply that large proportion of trial patients were overdosed in trials. By the use of a consensus prior, the respective means reduced to 0.428, 0.393 and 0.363 for biostat_line_6.7 , and they reduced to 0.268, 0.238 and 0.224 for biostat_line_6.7 , respectively, which were closer to γ=0.2. The SDs of π_X were greater in than biostat_line_6.7 in biostat line 6.6. The RMSEs were lower for . When the true value of Dγ increases from Scenario 4 to Scenario 9, biostat line 6.6 yielded the averages of πX closer to γ=0.2 with larger SDs when compared to biostat line 6.6 . As a consequence, the RMSE of πX was greater in  biostat_line_6.7than in biostat line 6.6.
 
Focusing on the distribution of πX (see the middle three columns of the table), similar trends were found for Q1, Q2, Q3, biostat line 6.6and biostat_line_6.7 with respect to the mean, SD and RMSE. In each scenario, in  biostat_line_6.7 was closer to the target γ=0.2 on average with larger variability when compared to . The resulting RMSEs of biostat_line_6.7 were smaller in Scenarios 1, 2, 3, 8 and 9, the cases when the combined prior opinions were relatively distant from the true  value of Dγ. The gross negative impact of prior misspecification was reduced by weighting each prior opinion based on data.
 
When the combined prior opinions were relatively close to the truth as in Scenarios 4, 5, 6 and 7, the RMSEs of  biostat_line_6.7were smaller due to less variability of Dγ* than the variability ofD̂γ**. This general tendency was similar to the case under the normal model when we compared the self-consistent estimatorμ̂* and the re-weighted estimator μ̂**
 
We turn our focus to the distribution of S25 (see the right three columns of the table). The resulting E(S25) generally decreased as the true Dγ increased for each method of Bayes estimation with rare exceptions. In the nine scenarios, the range of E(S25) was (0.066, 4.835) for Q1, (0.433, 10.674) for biostat line 6.6 Q2, (2.044, 17.778) for Q3, (0.436, 10.674) for biostat_line_6.7and (1.590, 6.686) for . By adaptively weighting each individual’s opinion based on accumulated information, biostat_line_6.7 resulted in the shortest range of E(S25) across the various scenarios, and it yielded S25 closest to N×γ=5 on average among the five considered approaches. Furthermore, the posterior adaptive weighting scheme provided robustness to P(4≤S25≤6).
 
In summary, the resulting distributions of πX , and S25 from the individual priors Q1, Q2 and Q3 showed high prior sensitivity, and robustness was gained by the use of a consensus prior. When we compared the prior weighting schemebiostat line 6.6 and the posterior weighting scheme biostat_line_6.7, biostat_line_6.7yielded relatively small variability with respect to dose allocation and final estimation of Dγ, biostat_line_6.7and yielded relatively small deviation from the target on average. The distribution of S25 was significantly more robust for biostat_line_6.7 by adjusting the weights of contributions based on updated data.
  
Figure 3 graphically summarize the relative performance ofbiostat_line_6.7 to biostat line 6.6 . The figure on the left shows the relative RMSE of πX (solid) and the relative RMSE of (dotted). A relative RMSE lower than one implies a smaller RMSE for , and the result is analogous to Figure 1. When the consensus prior guess and  the true value of a parameter of interest are fairly close, theprior weighting scheme tends to be better with respect to the  RMSE. When the two quantities are distant (i.e. one extremeprior guess is relatively close to the true value), the posterior weighting scheme tends to be better with respect to the RMSE. In Figure 3, the figure on the right highlights the robustness of 
biostat table 6.1

the posterior weighting scheme with respect to E(S25 ), and the comparison appears to be of practical importance (i.e. safety of dose-finding studies).

biostat fig 6.3

Discussion

In small-sample studies, there are cases when we want our prior knowledge to influence a decision in a robust manner. In a scientific community it is impossible that two or more researchers have the exactly same prior opinion about a parameter of interest, while gathering various sources of information is encouraged. For some scientific topics, experts have various and perhaps strong opinions. To this end, a consensus prior can be useful in a practical sense. We provided the theoretical perspective of a consensus prior and the class of re-weighted Bayes estim ators in comparison to the class of Bayes estimators with prior weights.

The prior weights have a path to self-consistency under simple models, while the re-weighted Bayes estimators have a  ifficulty with respect to self-consistency. However, the posterior weighting scheme allows to adjust the contribution of each investigator’s opinion based on empirical evidence, which is a natural reaction given data. Based on our numerical  llustrations, the prior weighting method yields smaller variability while the posterior weighting method yields smaller bias. If
the true value of parameter is located near the middle of prior guesses, the prior weights seem preferable. In the other case,the posterior weights seem preferable. This may be an intuitive result because when data cannot clearly determine whose prior specification was more plausible, the prior disagreement would be more controversial in posterior. On the other hand, when data favors one prior specification than the other, the degree of disagreement can be decreased in posterior.

When we formulate a consensus prior, it is important to check the balance of prior elicitations. We may not want one person’s
prior opinion to be upweighted simply because it is dogmatic. 


To this end, we may impose some restriction on the magnitudes of hyper-parameters. For example, in a Phase I clinical trial with the logistic model, the amount of prior information

can be measured by the total number of pseudo observations Unless the strengths of prior elicitations are intentionally unbalanced, we may match the hyper- parameters or pre-investigate the impact of different hyper- parameters under a model specified.

A primary focus of this manuscript has not been to urge one Bayes estimator outperforms better than the other Bayes  stimator based on a single measure of the operating characteristic (e.g. RMSE). A choice of estimator shall depend on the context of a problem. For example, in Scenario 7 of Table 1, the posterior weighting scheme allocated patients at the risk level of 0.181 on average with SD of 0.144. On the other hand, the prior weighting scheme allocated patients at the risk level
of 0.103 on average with SD of 0.086 which resulted in the smaller RMSE (0.145 versus 0130). We may want to carefully think about whether precise underdosing throughout a trial or slightly less precise dosing near the target is more preferable. As such, it would be important to understand the operating characteristics of consensus Bayes estimators under a given setting. In sparse data with strongly divergent priors, it is possible to face false minima in optimization, and a smoothed version of the estimator can be an alternative approach using surrogate data [13]. Alternatively, when a conservative decision is needed in early phase clinical trials, a decision can be made with the constraint that the predicted risk does not exceed a fixed threshold, such as a fixed quantile of the posterior distribution [14]. As such, a choice shall depend on the practical situation.

{tab=References}

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{/tabs}

Cite this article: Steven B. Kim. An Alternative Perspective on Consensus Priors with Applications to Phase I Clinical Trials. J J Biostat. 2016, 1(1): 006.

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