Bayesian Model Averaging and Compromising in Dose- Response Studies

Dose-response models are applied to animal-based cancer risk assessments and humanbased clinical trials usually with small samples. For sparse data, we rely on a parametric model for efficiency, but posterior inference can be sensitive to an assumed model. In addition, when we utilize prior information, multiple experts may have different prior knowledge about the parameter of interest. When we make sequential decisions to allocate experimental units in an experiment, an outcome may depend on decision rules, and each decision rule has its own perspective. In this chapter, we address the three practical issues in small-sample dose-response studies: (i) model-sensitivity, (ii) disagreement in prior knowledge and (iii) conflicting perspective in decision rules.


Introduction
Dose-response modeling is often used to learn about the effect of an agent on a particular outcome with respect to dose.It is widely applied to animal-based cancer risk assessments and human-based clinical trials.A sample size is typically small; so many statistical issues can arise from a limited amount of data.The issues include the impact of a misspecified model, priorsensitivity, and conflicting ethical perspectives in clinical trials.In this chapter, we focus on cases when an outcome variable of interest is binary (a predefined event happened or not) when an experimental unit is exposed to a dose.Main ideas are preserved for cases when an outcome variable is continuous or discrete.
There are two different approaches to statistical inference.One approach is called frequentist inference.In this framework, we often rely on the sampling distribution of a statistic and largesample theories.Another approach is called Bayesian inference.It is founded on Bayes' Theorem, and it allows researchers to express prior knowledge independent of data.In a smallsample study, Bayesian inference can be more useful than frequentist inference because we can incorporate both researcher's prior knowledge and observed data to make inference for the parameter of interest.Bayesian ideas are briefly introduced for dose-response modeling with a binary outcome in Section 2.
In a small-sample study, we often rely on a parametric model to gain statistical efficiency (i.e., less variance in parameter estimation), but our inference can be severely biased by the use of a wrong model.To account for model uncertainty, it is reasonable to specify multiple models and make inference based on "averaged-inference."In this regard, Bayesian model averaging (BMA) is a useful method to gain robustness [1].The BMA method has a wide range of application, and we focus its application to animal-based cancer risk assessments in Section 3.
In clinical trials, study participants are real patients, and therefore, we need to carefully consider ethics.There are conflicting perspectives of individual-and population-level ethics in early phase clinical trials.Individual-level ethics focuses on the benefit of trial participants, whereas population-level focuses on the benefit of future patients, which may require some level of sacrifice from trial participants.We compare the two conflicting perspectives in clinical trials based on Bayesian decision theory, and we discuss a compromising method in Section 4 [2,3].
A sample size for an early phase (Phase I) clinical trial is often less than 30 subjects.Dose allocations for first few patients and statistical inference for future patients heavily depend on researcher's prior knowledge in sparse data.When multiple researchers have different prior knowledge about a parameter of interest, one compromising approach is to combine their prior elicitations and average them (i.e., consensus prior) [4,5].When we average the prior elicitations, there are two different approaches to determine the weight of each prior elicitation, weights determined before observing data and after observing data.We discuss operating characteristics of the two different weighting methods in the context of Phase I clinical trials in Section 5.

Bayesian inference
In statistics, we address a research question by a parameter, which is often denoted by θ.W e begin Bayesian inference by modeling the prior knowledge about θ.A function, which models the prior knowledge about θ, is called the prior density function of θ,andwedenote it by f(θ).It is a non-negative function, which satisfies ð Ω f ðθÞ dθ ¼ 1, where Ω is the set of all possible values of θ (i.e., parameter space).We then model data y !¼ðy 1 , …,y n Þ given θ.The likelihood function, denoted by f ð y !jθÞ, quantifies the likelihood of observing a particular sample y !¼ðy 1 , …,y n Þ under an assumed probability model.By Bayes' Theorem, we update our knowledge about θ after observing data y ! as The function f ðθjy !Þ is called the posterior density function of θ given data y ! .Since we treat observed data y !¼ðy 1 , …,y n Þ as fixed numbers, we often express Eq. ( 1) as follows where k is the normalizing constant which makes based on the prior density function f(θ) and the likelihood function f ð y !jθÞ without considering the denominator f ðy !Þ¼ ð f ðyjθÞ f ðθÞ dθ in Eq. ( 1) which is called the marginal likelihood.

Example
Suppose we observe n = 20 rats for 2 years.Let π be the parameter of interest, which is interpreted as the probability of developing some type of tumor.Suppose a researcher models the prior knowledge about π using the prior density function It is known as the beta distribution with shape parameters a > 0 and b > 0. We often denote the beta distribution by π $ Betaða, bÞ, and the values of a and b must be specified by the researcher independent from data.Let y !¼ðy 1 , …,y n Þ denote observed data, where y i =1i ft h e i th rat developed tumor and y i = 0 otherwise.Assuming y 1 ,…, y n are independent observations, the likelihood function is as follows where s ¼ X n i¼1 y i is the total number of rats developed tumor.By Eq. ( 2), the posterior density function of π is as follows where k ¼ ΓðaþbþnÞ ΓðaþsÞ ΓðbþnÀsÞ is the normalizing constant, which makes If the researcher fixed a = 2 and b =3 and observed s = 9 from a sample of size n = 20, the prior density function is f ðπÞ¼k π ð1 À πÞ 2 with k ¼ Γð5Þ Γð2Þ Γð3Þ ¼ 12, and the posterior density function is f ðπj y !Þ¼k π 10 ð1 À πÞ 13 with k ¼ Γð25Þ Γð11Þ Γð14Þ ¼ 27457584.The prior and posterior distributions are shown in Figure 1.The knowledge about π becomes more certain (less variance) after observing the data.

Example
This example is simplified from Shao and Small [6].In dose-response studies, we model π as a function of dose x.There are many link functions between π and x used in practice.In this example, we focus on a link function which is known as a logistic regression model.It is commonly assumed that a dose-response curve increases with respect to dose, so we assume β 1 > 0 (and β 0 can be any real number).
There are two regression parameters in Eq. ( 6), β 0 and β 1 , and we denote them as with an arbitrarily large value of σ [6].When a reliable source of prior information is available, there is a practical method, which is known as the conditional mean prior [7], and it will be discussed in a later section (see Section 4.2).In an experiment, the experimental doses x !¼ðx 1 , …,x n Þ are fixed, and we observe random binary outcomes y !¼ðy 1 , …,y n Þ.Given y !(and fixed x ! ), the likelihood function is as follows Bayesian Inference where s 1 ¼ X n i¼1 y i and s 2 ¼ X n i¼1 x i y i .By incorporating both prior and data, the posterior density function is as follows In an animal-based studies, one parameter of interest is the median effective dose, which is denoted by ED 50 .It is the dose, which satisfies and it can be shown that ED 50 ¼À β 0 β 1 by algebra.In the case of β 0 = À2 and β 1 = 5, we have ED 50 = .4as describe in the figure with the dotted curve.In the case of β 0 = À1 and β 2 =2,we have ED 50 = .5 as described in the figure with the solid curve.

Prior and Posterior Distributions
where

Bayesian model averaging
In a small sample, we borrow the strength of a parametric model to gain efficiency in parameter estimation.However, an assumed model may not describe the true dose-response relationship adequately.The impact of model misspecification is not negligible particularly in a poor experimental design.In such a limited practical situation, Bayesian model averaging (BMA) can be a useful method to account for model uncertainty.It is widely applied in practice, and in this section, we focus on the application to cancer risk assessment for the estimation of a benchmark dose [1,6,10,11].
Let θ denote a parameter of interest.Suppose we have a set of K candidate models denoted by In Eq. ( 12), the posterior density function Bayesian Model Averaging and Compromising in Dose-Response Studies http://dx.doi.org/10.5772/intechopen.68786 In Eq. ( 13), the prior model probability P(M k ) is determined before observing data such that In the BMA method, all K models contribute to inference of θ through the averaged posterior density function in Eq. ( 12), and the weight of contribution is determined by Bayes' Theorem in Eq. ( 13).

Example
This example is continued from the example in Section 2.2.Recall π x is interpreted as the probability of a toxic event (tumor development) at dose x.In many cancer risk assessments, a parameter of interest is θ γ at a fixed risk level γ, which is defined as follows or equivalently π θ γ ¼ π 0 þð1 À π 0 Þ γ.In words, θ γ is a dose corresponding to a fixed increase in the risk level.In frequentist framework, Crump defined a benchmark dose as a lower confidence limit for θ γ [12].In Bayesian framework, an analogous definition would be a lower credible bound (i.e., a fixed low percentile of the posterior distribution of θ γ ).The definition is widely applied to the public health protection [13].
In practice, γ is fixed between 0.01 and 0.1.Often, the estimation of θ γ is highly sensitive to an assumed dose-response model because we have a lack of information at low doses.Shao and Small fixed γ = 0.1 and applied BMA with K = 2 models, logistic model and quantal-linear model [6].In the quantal-linear model, the probability of tumor development is modeled by with the restrictions 0 < β 0 < 1 and β 1 > 0 under the monotonic assumption.The logistic model was given in Eq. ( 6) of Section 2.2. Let However, we are not able to calculate the 5th percentile of the model-averaged posterior distribution based on the given statistics.In fact, we need to approximate the posterior distribution f ðθ 0: ¼ :951 :049 ¼ 19:4.The 5th percentile of the model-averaged posterior distribution is approximately 5.97, and it is a BMA-BMD based on the BMA method proposed by Raftery et al. [1] and the BMD estimation method suggested by Crump [12].

Application of Bayesian decision theory to Phase I trials
In a Phase I cancer trial, the main objectives are to study the safety of a new chemotherapy and to determine an appropriate dose for future patients.Since trial participants are cancer patients, dose allocations require ethical considerations.Whitehead and Williams discussed several Bayesian approaches to dose allocations [14].One decision rule is devised from the perspective of trial participants (individual-level ethics), and another decision rule is devised from the perspective of future patients (population-level ethics).However, a decision rule, which is devised from the population-level ethics, is not widely accepted in current practice [15].Instead, there are some proposed decision rules, which compromise between the individual-and population-level perspectives [3,16].In this section, we discuss the two Assume a dose-response relationship follows a logistic model where x is a dose in the logarithmic scale (base e)a n dπ x is the probability of observing an adverse event due to the toxicity of a new chemotherapy at dose x.The logarithmic transformation on the dose is to satisfy denote a series of decisions for n patients (i.e., allocated doses) and y !n ¼ðy 1 , …,y n Þ denote a series of observed responses, where y i = 1 indicates an adverse event and y i = 0 otherwise.Let Lðβ !,x nþ1 Þ denote a loss by allocating the next patient at x n+1 .Based on Bayesian Decision Theory, we want to find x n+1 which minimizes the posterior mean of Lðβ !,x nþ1 Þ.I fw el e tA denote an action space, a set of all possible dose allocations for the next patients, the decision rule can be written as follows: A choice of L has a substantial impact on the operating characteristics of a Phase I trial including (i) the degree of under-and over-dosing in trial, (ii) the observed number of adverse events at the end of a trial, and (iii) the quality of estimation at the end of a trial.

Parameter of interest: maximum tolerable dose
Let N denote an available sample size for a Phase I clinical trial.A typical sample size is N ≤ 30.Let γ denote a target risk level, the probability of an adverse event.In a cancer study, a typical target risk level γ is fixed between .15 and .35depending on the severity of an adverse event.
Then, the dose corresponding to γ is called a maximum tolerable dose (MTD) at level γ, and we denote it by θ γ in the logarithmic scale.Under the logistic model in Eq. ( 18), it is defined as follows At the end of a trial (observing N responses), we estimate θ γ by the posterior mean θγ,N ¼ Eðθ γ jy !N Þ for future patients.

Prior density function: conditional mean priors
A consequence of sequential decisions heavily depends on a prior density function f ðβ !Þ.I n particular, the first decision x 1 must be made based on prior knowledge only because empirical evidence is not observed yet.In addition, the later decisions x 2 , x 3 ,… and the final inference of Bayesian Inference θ γ are substantially affected by f ðβ !Þ as a Phase I study is typically based on a small sample.In this regard, we want to carefully utilize researchers' prior knowledge about β !, but it may be difficult to express their prior knowledge directly through f ðβ !Þ.I nt h i ss e c t i o n ,w ed i s c u s sa method of eliciting prior knowledge, which is more tractable than prior elicitation directly on β ! .Suppose a researcher selects two arbitrarily doses, say x À1 < x 0 .Then, the researcher may express their prior knowledge by two independent beta distributions Using the Jacobian transformation from It is known as conditional mean priors under the logistic model [7].

Posterior density function: conjugacy
For notational convenience, we let y i = a i and n i = a i + b i for i = À1,0.By conjugacy, the posterior density function of β !can be concisely written as follows where s 1 ¼ X n i¼À1 y i and s 2 ¼ X n i¼À1 x i y i .After observing n responses, the decision rule for the next patient is as follows

Loss functions for individual-and population-level ethics
A loss function, which reflects the perspective of individual-level ethics, is as follows: This loss function is analogous to the original continual reassessment method proposed by O'Quigley et al. [17].The square error loss attempts to treat a trial participant at θ γ , and the expected square error loss is minimized by the posterior mean of θ γ .
From the perspective of population-level ethics, Whitehead and Brunier proposed a loss function, which is equal to the asymptotic variance of the maximum likelihood estimator for θ γ [18].The Fisher expected information matrix with a sample of size n + 1 is given by where Then, the loss function (the asymptotic variance) is given by where is the gradient vector, the partial derivatives of θ γ with respect to β 0 and β 1 .Kim and Gillen decomposed the population-level loss function as follows where with the weight defined as w i ¼ τ i ethics, but the compromising process is rather too slow to be implemented in a small-sample Phase I clinical trial [3].

Loss function for compromising the two perspectives
Kim and Gillen proposed to accelerate the compromising process by modifying L P ðβ !,x nþ1 Þ of Eq. (29) as follows where is an accelerating factor [3].It has two implications.First, the compromising process is accelerated toward the individual-level ethics as the trial proceeds (i.e., n increases).Second, the compromising process toward the individual-level ethics is accelerated at a faster rate when an adverse event is observed (i.e., X n i¼1 y i increases).The tuning parameter λ controls the rate of acceleration.It imposes more emphasis on population-level ethics as λ !0 and more emphasis on individual-level ethics as λ !∞.The choice of λ shall depend on the severity level of an adverse event.

Simulation
To study the operating characteristics of L B,λ with respect to λ, we assume the logistic model with β 0 = À3 and β 1 = .8as a true dose-response relationship as shown in Figure 5 in the left panel.The target risk level is fixed at γ = .2,so the true MTD is given by θ .2= 2.02 in the logarithmic scale.We consider three different priors based on the conditional mean priors given in Eq. ( 22).For simplicity, we set a À1 =1 ,b À1 =3 ,a 0 = 3 and b 0 = 1 for all three priors.Then, we let x À1 ¼À4 and x 0 ¼ 4 for Prior 1; x À1 ¼ 0 and x 0 ¼ 8 for Prior 2; and x À1 ¼ 4 and x 0 ¼ 12 for Prior 3.  the up-and-down scheme has a high degree of fluctuation in order to maximize information about θ .2 .When λ = 1, the up-and-down scheme is stabilized after the first few adverse events, and the stabilization occurs quickly when λ = 5 to treat trial participants near an estimated θ .2 .
Let θ:2 ¼ Eðθ :2 jy !N Þ, the posterior estimate of θ .2 at the end of a trial, so π θ:2 implies the true probability of an adverse event at the estimated MTD.We focus on the following criteria: (i) Eðπ θ:2 Þ which we desire to be close to γ ¼ :2 for future patients, (ii) Vðπ θ:2 Þ which we desire to be as low as possible for future patients, (iii) E½ðπ θ:2 À :2Þ 2 which we desire to be as low as possible for future patients, (iv) Eð

Bayesian Inference
Table 1 summarizes simulation results of 10,000 replicates for each prior.For all three priors, we observe similar tendencies.First, Eðπ θ:2 Þ gets closer to θ =. 2a sλ increases.Second, Vðπ θ:2 Þ decreases as λ decreases to zero.The average square distance between π θ:2 and γ ¼ :2 measures a balance between j Eðπ θ:2 ÞÀ:2 j and Vðπ θ:2 Þ, and the superiority depends on priors.Lastly, as λ !0, we have larger Pð3 ≤ X 20 i¼1 Y i ≤ 5Þ and more robust Eð In summary, when we emphasize more on population-level ethics, we have a smaller variance in the estimation for future patients (with a greater absolute bias, potentially due to Jensen's Inequality), and the distribution of X n i¼1 Y i becomes more robust to prior elicitations.When we emphasize more on individual-level ethics, we have a larger variance in the estimation, and the distribution of X n i¼1 Y i becomes more sensitive to prior elicitations.

Consensus prior
In Bayesian inference, researchers are able to utilize information, which is independent of observed data.It allows researchers to incorporate any form of information, such as one'sexperience and existing literature, which may be particularly useful in a small-sample study.On the other hand, we concern subjectivity and prior sensitivity in sparse data.Furthermore, it is possible to have disagreement among multiple researchers' prior elicitations about a parameter θ.
Suppose there are K researchers with their own prior density functions, say f ðθjQ k Þ for k ¼ 1;…,K, and they have the same likelihood function f ð y !jθÞ.Each prior elicitation leads to a unique Bayes estimator is the posterior density function of θ given data y ! and the k th prior elicitation Q k .For posterior estimation, one reasonable approach to compromise is a weighted average In this section, we discuss two different weighting methods.The first method is to fix w k before observing data (referred to as prior weighting scheme).The second method is to determine w k ðy !Þ after observing data y !so that w k ð y !Þ increases when the k th prior elicitation Q k is better supported by the observed data y !(referred to as posterior weighting scheme) [5].
For a prior weighting scheme, we denote w k ¼ PðQ k Þ which quantifies the credibility of the k th prior elicitation.For a posterior weighting scheme, we consider where f ðy !jQ k Þ¼ ð f ð y !jθÞ f ðθjQ k Þ dθ is the marginal likelihood from the k th prior elicitation.
This formulation is similar to the BMA method discussed in Section 3. It can be shown that θk is the Bayes estimator (the posterior mean of θ) when a consensus prior . Samaniego discussed self-consistency when compromised inference is used through the prior weighting scheme be the prior expectation, the mean of the prior density function f ðθÞ.Let θ denote a sufficient statistic, which serves as an unbiased estimator for θ.When we satisfy Eðθj θ ¼ θ Ã Þ¼θ Ã ,i ti s called self-consistency [4].
Self-consistency can be achieved under simple models.For example, let Y !¼ðY 1 , …,Y n Þ be a random sample, where Y i $ BernoulliðθÞ, and assume θ $ Betaða, bÞ for prior.It can be shown Bayesian Inference that the maximum likelihood estimator θ ¼ 1 n X n i¼1 Y i is a sufficient statistic and an unbiased estimator for θ.The posterior mean is a weighted average between θ * and θ as follows where c ¼ aþb aþbþn .If we observe θ ¼ θ Ã , we can achieve the self-consistency because Eðθj θ ¼ θ Ã Þ¼θ Ã .In words, when prior estimate and maximum likelihood estimate are identical, the posterior estimate must be consistent with the prior estimate and the maximum likelihood estimate.The self-consistency can be also achieved in the prior weighting scheme under certain conditions as illustrated in the following example.

Binomial experiment
Let Y i $ BernoulliðπÞ for i ¼ 1;…,n and assume Y 1 , …,Y n are independent.Suppose the k th researcher specifies the prior distribution πjQ k $ Betaða k ,b k Þ for k ¼ 1;…,K.For the prior weighting scheme, let w k ¼ PðQ k Þ, the prior probability for the k th prior elicitation (fixed before observing data).Since EðπjQ k Þ¼ a k a k þb k and the expectation E (Á) is a linear operator, the average of "consensus prior" is EðπÞ¼ Let EðπÞ¼π Ã and suppose the K researchers observed the consistent result π ¼ 1 n X n i¼1 Y i ¼ π Ã .The individual-specific Bayes estimator is as follows for the k th researcher, where c k ¼ a k þb k a k þb k þn .The compromised Bayes estimator is as follows If we allow individual-specific prior elicitation a k and b k with the restriction a k +b k =mfor all K researchers (i.e., the same strength of prior elicitation), the value c k ¼ m mþn is constant over all researcher.By letting the constant c k = c, so the self-consistency is satisfied.For the posterior weighting scheme given data y !¼ðy 1 , …,y n Þ, the marginal likelihood from the k th prior elicitation is as follows where s ¼ X n i¼1 y i is an observed sufficient statistic.Then, the posterior weighting scheme becomes If we desire an equal strength from each researcher's prior elicitation, we may fix a k þ b k ¼ m and w k ¼ 1 K .In the posterior weighting scheme, it is difficult to achieve the self-consistency.Whether self-consistency is satisfied, the practical concern is the quality of estimation such as bias, variance and mean square error.Assuming K = 2 researchers have disagreeing prior knowledge and a sample of size n = 10, let us consider three cases.Suppose two researchers express relatively mild disagreement as ða !Þ πk to the prior weighting scheme X 3 k¼1 w k πk .When a relative MSE is smaller than one, it implies a smaller MSE for the posterior weighting scheme.As the true value of π is well between the two prior guesses EðπjQ 1 Þ¼:25 and EðπjQ 2 Þ¼:75, the posterior weighting scheme shows a greater MSE due to greater variance.When the true value of π deviates away from either prior guess, the posterior weighting schemes show a smaller MSE due to smaller bias.The tendency is stronger when the two disagreeing prior elicitations are stronger (i.e., stronger prior disagreement).The bottom line is a clear bias-variance tradeoff  Bayesian Inference when we compare the two weighting schemes.
X 3 k¼1 w k ð y !Þ πk is able to reduce bias when there is strong discrepancy between "consensus prior" and data, but it has larger variance than X 3 k¼1 w k πk because w k ð y !Þ depends on random data.

Applications to Phase I trials under logistic regression model
In this section, we apply the prior weighting scheme and the posterior weighting scheme to Phase I clinical trials under the logistic regression model.We consider the three priors considered in Section 4.6.We denote Prior 1, 2 and 3 by Q 1 , Q 2 and Q 3 , respectively.The three priors had the same hyper-parameters a The prior means were Eðθ :2 jQ 1 Þ¼À1:70, Eðθ :2 jQ 2 Þ¼1:40 and Eðθ :2 jQ 3 Þ¼5:38 for Priors 1, 2 and 3, respectively.
Table 2 provides the simulation results of 10,000 replicates for each scenario under the prior weighting scheme and under the posterior weighting scheme.Since the posterior weighting scheme adaptively updates w k ð y !Þ based on empirical evidence, it can reduce bias, but it has greater variance in the estimation of θ 2 .As a consequence, when the true MTD was close to one extreme prior estimate (Scenarios 1 and 3), the use of the posterior weighting scheme yields a smaller E ðπ θ :2 i¼1 Y i ≤ 5Þ closer to one when compared to the use of the prior weighting scheme.In Scenario 3, the average number of adverse events was 4.6 for the posterior weighting scheme, but it was as high as 7.1 in the prior weighting scheme.On the other hand, when the true MTD was well surrounded by the three prior estimates (Scenario 2), the use of the prior weighting scheme yielded more plausible results.
The simulation results are analogous to the simpler model in Section 5.1.When the true parameter is not well surrounded by prior guesses, the posterior weighting scheme is preferable with respect to mean square error due to smaller bias.When the true parameter is well surrounded by prior guesses, the prior weighting scheme is beneficial with respect to mean square error due to smaller variance.
As a final comment, we shall be careful about the strength of individual prior elicitations when we implement the posterior weighting scheme in Phase I clinical trials.The strength of individual prior elicitations depends on (i) the hyper-parameters a i, k and b i, k , (ii) the prior weight w k ¼ PðQ k Þ as well as (iii) the distance between the two arbitrarily chosen doses x 0;k À x 1;k .I t can be seen through the expression When researchers determine consensus prior elicitations before initiating a trial, the multiplicative term w k ðx 0;k À x À1;k Þ shall be carefully considered together with the hyper-parameters a i, k and b i, k [5].

Concluding remarks
In this chapter, we have discussed Bayesian inference with averaging, balancing, and compromising in sparse data.In the cancer risk assessment, we have observed that low-dose inference can be very sensitive to an assumed parametric model (Section 3.1).In this case, the Bayesian model averaging can be a useful method.It provides robustness by using multiple models and posterior model probabilities to account for model uncertainty.In the application of Bayesian decision theory to Phase I clinical trials, we have observed that the sequential sampling scheme heavily depends on a loss function.A loss function, which is devised from individual-level ethics, focuses on the benefit of trial participants, and a loss function, which is devised from population-level ethics, focuses on the benefit of future patients.It is possible to balance between the two conflicting perspectives, and we can adjust a focusing point by the tuning parameter (Sections 4.5 and 4.6).Finally, the use of a weighted posterior estimate can be a compromising method when two or more researchers have prior disagreement.We have compared the prior and posterior weighting schemes in a small-sample binomial problem (Section 5.1) and in a smallsample Phase I clinical trial (Section 5.2).The prior weighting scheme (data-independent weights) outperforms when prior estimates surround the truth, and the posterior weighting scheme (datadependent weights) outperforms when the truth is not well surrounded by prior estimates.One method does not outperform the other method for all parameter values, so it is important to be aware of their bias-variance tradeoff.

Figure 2
Figure 2 presents two dose-response curves.The solid curve is generated by β !¼ðÀ1; 2Þ, and and the interpretation of θ must be common across all models.Let f ðβ !k jM k Þ and f ðy !jβ !k ,M k Þ denote the prior density function and the likelihood function, respectively, under M k .By the Law of Total Probability, the posterior density function of θ is as follows

PðM 1
j y !Þ¼:049 and PðM 2 j y !Þ¼:951, respectively, as shown in Figure 4.In the figure, the left panel shows an approximation of f ðθ 0:1 jM 1 ,y !Þ, the middle panel shows an approximation of f ðθ 0:1 jM 2 ,y !Þ, and the right panel shows an approximation of the averaged posterior f ðθ 0:1 j y !Þ.The averaged posterior density f ðθ 0:1 jy !Þ is bimodal, but it is very close to f ðθ 0:1 jM 2 ,y !Þ because the quantal-linear model M 2 fits the data better than the logistic model M 1 by a Bayes factor of

Figure 4 .
Figure 4. Posterior distributions of θ 0.1 from the logistic model (left panel), the quantal-linear model (middle panel), and the Bayesian model averaging (right panel).

[ 3 ]
. Eq. (29) has the following important remarks.In fact,L P ðβ !,x nþ1 Þ considers individual-level ethics by including L I ðβ !,x nþ1 Þ¼ðx nþ1 À θ γ Þ 2 in the numerator.By including ðx nþ1 À μ n Þ 2 inthe denominator, where μ n ¼ X n i¼1 w i x i , the population-level loss function reduces a loss by allocating the next patient further away from the weighted average of previously allocated doses (i.e., devised from information gain).In long run, L P ðβ !,x nþ1 Þ is devised from a compromise between individual-and population-level Bayesian Inference

Figure 5
in the right panel shows an approximated f(θ .2 ) for each prior.Prior 1 significantly underestimates the true θ :2 ¼ 2:02 with prior mean Eðθ :2 Þ¼À1:70, Prior 3 overestimates the truth with Eðθ :2 Þ¼5:38, and Prior 2 has a prior estimate relatively close to the truth with Eðθ :2 Þ¼1:40.Let N = 20 be a fixed sample size.Let Y i = 1 denote an adverse event observed from the ith patient (Y i = 0 otherwise), so X N i¼1 Y i denotes the total number of adverse events observed at the end of a trial.The sum X n i¼1 Y i is random from a trial to another trial, and we want to behave like Binomialð20;:2Þ which is the case when we treat N = 20 to the true MTD θ .2 .Figure6shows three simulated trials under the loss function L B,λ with λ = 0,1,5.When λ =0, Bayesian Model Averaging and Compromising in Dose-Response Studies http://dx.doi.org/10.5772/intechopen.68786

Figure 5 .Figure 6 .
Figure 5.The true dose-response relationship π x ¼ e β 0 þβ 1 x1þe β 0 þβ1 x with β 0 = À3 and β 1 = .8(where x is the dose in the logarithmic scale) in the simulation (left panel) and the three prior distributions of θ .2approximated by kernel density (right panel).

1 ,b 1 3 k¼1 w k ð y
scheme X 3 k¼1 w k ð y !Þ πk to the prior weighting scheme

Figure 7 .
Figure 7. Comparing prior and posterior weighting schemes for different degrees of disagreements.

Dose−Response Curves (Logistic Link)
[6] denote the logistic model, and let M 2 denote the quantal-linear model.Assume the uniform prior model probabilities PðM 1 Þ¼PðM 2 Þ¼:5 and flat priors on the regression parameters.By posterior sampling, we can approximate the posterior model probabilities Under M 1 , the posterior mean of θ 0.1 is 20.95 with the 5th percentile 16.74.Under M 2 , the posterior mean is 8.25 with the 5th percentile 5.95.These Bayesian Inference results are very similar to the results reported by Shao and Small[6].From these modelspecific statistics, we can calculate the model-averaged posterior mean ! Þ¼:049 and PðM 2 j y !Þ¼:951.!Þ PðM k j y !Þ¼20:95 ð:049Þþ8:25 ð:951Þ¼8:87 : can be shown that the prior density function of β ! is given by

Table 2 .
Simulation results of 10,000 replicates for the prior and posterior weighting schemes.