## 1. 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, prior-sensitivity, 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 large-sample 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 small-sample 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.

## 2. Bayesian inference

In statistics, we address a research question by a parameter, which is often denoted by *θ*. We 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*, and we denote it by *f*(*θ*). It is a non-negative function, which satisfies *θ* (i.e., parameter space). We then model data *θ*. The likelihood function, denoted by *θ* after observing data

The function *θ* given data

where *k* is the normalizing constant which makes *f*(*θ*) and the likelihood function

### 2.1. 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 *a* and *b* must be specified by the researcher independent from data. Let *y*_{i} = 1 if the _{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 *π* is as follows

where

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 **Figure 1**. The knowledge about *π* becomes more certain (less variance) after observing the data.

### 2.2. 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 **Figure 2** presents two dose-response curves. The solid curve is generated by *β*_{0} increases, the background risk *π*_{0} is interpreted as the probability of tumor development at dose *x* = 0. The dose-response curve increases when *β*_{1} > 0, and it decreases when *β*_{1} < 0. The rate of change in the dose-response curve is determined by|*β*_{1}|.

To express prior knowledge about

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

where

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 *β*_{0} = −2 and *β*_{1} = 5, we have ED_{50} = .4 as describe in the figure with the dotted curve. In the case of *β*_{0} = −1 and *β*_{2} = 2, we have ED_{50} = .5 as described in the figure with the solid curve.

In 1997, International Agency for Research on Cancer classified 2,3,7,8-Tetrachlorodibenzo-p-dioxin (known as TCDD) as a carcinogen for humans based on various empirical evidence [8]. In 1978, Kociba et al. presented the data on male Sprague-Dawley rats at four experimental doses 0, 1, 10 and 100 nanograms per kilogram per day (ng/kg/day) [9]. In the control dose group, nine of 86 rats developed tumor (known as hepatocellular carcinoma); three of 50 rats developed the tumor at dose 1; 18 of 50 rats developed the tumor at dose 10; and 34 of 48 rats developed the tumor at dose 100 [6]. Without loss of generosity, we let *x*_{i} = 0 for *i* = 1,…, 86; *x*_{i} = 1 for *i* = 87,…136; *x*_{i} = 10 for *i* = 137,…, 186; and *x*_{i} = 100 for *i* = 187,…, 234. The given information is sufficient to calculate *β*_{1} > 0, given the observed sample of size *n* = 234, we can generate random numbers of

where *β*_{1} > 0 and **Figure 3**. By transforming (*β*_{0}, *β*_{1}) to _{50} as shown in the right panel. The posterior mean of ED_{50} is

## 3. 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 *M*_{k} for *k* =1,…, *K*. Suppose *θ* is a function of *θ* must be common across all models. Let *M*_{k}. By the Law of Total Probability, the posterior density function of *θ* is as follows

In Eq. (12), the posterior density function *M*_{k}, and the posterior model probability *M*_{k} after observing data, which is given by

In Eq. (13), the prior model probability *P*(*M*_{k}) is determined before observing data such that *M*_{k} requires the integration

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).

### 3.1. 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 *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 *M*_{1} denote the logistic model, and let *M*_{2} denote the quantal-linear model. Assume the uniform prior model probabilities *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 results are very similar to the results reported by Shao and Small [6]. From these model-specific statistics, we can calculate the model-averaged posterior mean

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 **Figure 4**. In the figure, the left panel shows an approximation of *M*_{2} fits the data better than the logistic model *M*_{1} by a Bayes factor of

## 4. 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 conflicting perspectives in Phase I clinical trials and a compromising method based on Bayesian decision theory.

Assume a dose-response relationship follows a logistic model

where *x* is a dose in the logarithmic scale (base *e*) and *π*_{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 *π*_{x} → 0 as *n* patients (i.e., allocated doses) and *y*_{i} = 1 indicates an adverse event and *y*_{i} = 0 otherwise. Let *x*_{n+1}. Based on Bayesian Decision Theory, we want to find *x*_{n+1} which minimizes the posterior mean of

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.

### 4.1. 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 .35 depending 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

### 4.2. Prior density function: conditional mean priors

A consequence of sequential decisions heavily depends on a prior density function *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 *θ*_{γ} are substantially affected by

Suppose a researcher selects two arbitrarily doses, say *x*_{−1} < *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].

### 4.3. Posterior density function: conjugacy

For notational convenience, we let *y*_{i} = *a*_{i} and *n*_{i} = *a*_{i} + *b*_{i} for *i* = −1,0. By conjugacy, the posterior density function of

where *n* responses, the decision rule for the next patient is as follows

### 4.4. 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

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

(29) |

where

with the weight defined as

### 4.5. Loss function for compromising the two perspectives

Kim and Gillen proposed to accelerate the compromising process by modifying

(31) |

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., *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.

### 4.6. Simulation

To study the operating characteristics of *L*_{B,}_{λ} with respect to *λ*, we assume the logistic model with *β*_{0} = −3 and *β*_{1} = .8 as 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*_{−1} = 1, *b*_{−1} = 3, *a*_{0} = 3 and *b*_{0} = 1 for all three priors. Then, we let **Figure 5** in the right panel shows an approximated *f*(*θ*_{.2}) for each prior. Prior 1 significantly underestimates the true

Let *N* = 20 be a fixed sample size. Let *Y*_{i} = 1 denote an adverse event observed from the *i*th patient (*Y*_{i} = 0 otherwise), so *N* = 20 to the true MTD *θ*_{.2}. **Figure 6** shows three simulated trials under the loss function *L*_{B,}_{λ} with *λ* = 0,1,5. When *λ* = 0, 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} at the end of a trial, so

**Table 1** summarizes simulation results of 10,000 replicates for each prior. For all three priors, we observe similar tendencies. First, *θ* = .2 as *λ* increases. Second, *λ* decreases to zero. The average square distance between *λ* → 0, we have larger

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

## 5. 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’s experience 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

where *θ* given data *k*^{th} prior elicitation *Q*_{k}. For posterior estimation, one reasonable approach to compromise is a weighted average *w*_{k} > 0 for *k* = 1,…,*K* and *w*_{k} before observing data (referred to as prior weighting scheme). The second method is to determine *k*^{th} prior elicitation *Q*_{k} is better supported by the observed data

For a prior weighting scheme, we denote *k*^{th} prior elicitation. For a posterior weighting scheme, we consider

where *k*^{th} prior elicitation. This formulation is similar to the BMA method discussed in Section 3. It can be shown that *θ*) when a consensus prior

Samaniego discussed self-consistency when compromised inference is used through the prior weighting scheme *θ* denote a parameter of interest and

be the prior expectation, the mean of the prior density function *θ*. When we satisfy

Self-consistency can be achieved under simple models. For example, let *θ*. The posterior mean is a weighted average between *θ*^{*} and

where

### 5.1. Binomial experiment

Let *k*^{th} researcher specifies the prior distribution *k*^{th} prior elicitation (fixed before observing data). Since *E* (⋅) is a linear operator, the average of “consensus prior” is

Let *K* researchers observed the consistent result

for the *k*^{th} researcher, where

If we allow individual-specific prior elicitation *a*_{k} and *b*_{k} with the restriction *a*_{k} *+ b*_{k} *= m* for all *K* researchers (i.e., the same strength of prior elicitation), the value *c*_{k} = *c*,

so the self-consistency is satisfied.

For the posterior weighting scheme given data *k*^{th} prior elicitation is as follows

where

If we desire an equal strength from each researcher’s prior elicitation, we may fix

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 **Figure 7** provides the relative bias, variance and mean square error (MSE) for comparing the posterior weighting scheme *π* is well between the two prior guesses *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 when we compare the two weighting schemes.

### 5.2. 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 *Q*_{k} is given by

The prior means were

For simulation study, we consider three simulation scenarios with sample size *N* = 20. In Scenario 1, we assume *β*_{0} = −5 and *β*_{1} = .6, so the true MTD is *β*_{0} = −3 and *β*_{1} = .8 as in Section 4.6, so *β*_{0} = −1 and *β*_{1} = 1.2, so

**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 *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

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

When researchers determine consensus prior elicitations before initiating a trial, the multiplicative term

## 6. 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 small-sample 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 (data-dependent 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.