1 Introduction

In the field of drug development, there is high interest in conducting clinical trials using designs that can enable the incorporation of external information, such as prior or historical information, with trial data to save sample sizes, improve the power, and expedite the trial process. Several studies have focused on developing designs that incorporate external information for phase II or III trials, for example, meta-analytic power prior–based multiple historical sources [1], hierarchical shrinkage method for basket trials [2–4], calibrated power prior for biosimilar trials [5], and Bayesian designs for confirmatory trials [6, 7]. All the above research designs have focused on phase II trials and beyond. Very few studies have discussed how to incorporate external information for phase I trials, though it is known that phase I trials are crucial because all appropriate evaluations of promising new agents in phase II or III trials have to rely on well-conducted phase I trials. On the other hand, in some settings wherein historical information is available (e.g., pediatric clinical trials), the suggested starting dose is 80% of the dose recommended for adults. Due to ethical constraints and a typically small number of patients in pediatric trials, it is essential to know how to formally incorporate prior knowledge from adult trials. Petit et al. [8] proposed a method to extrapolate pharmacokinetic information from the adult population to the pediatric population in dose-finding trials. Their method focused on phase I/II trials that jointly modeled toxicity and efficacy by using the continual reassessment model (CRM). In contrast to that study, our study considers how to use historical information to inform the prior elicitation for phase I trials only. Our proposed method is based on the Bayesian optimal interval design (BOIN) framework [9, 10].

The BOIN design’s escalation/de-escalation decisions are based on two boundaries. Given the DLT target, the two boundaries are fixed (derived by minimizing the overall decision error rate). However, in some situations, we might need to have an unbalanced control of misallocation of patients to under-toxic and over-toxic dose levels. By having accumulative information, we could have a better understanding of the toxicity rate for each dose level tried; fixed boundaries cannot reflect these dynamics. The second goal of this study is to propose flexible boundaries that can change during the trial process. This design is termed as adaptive BOIN (aBOIN).

The rest of the paper gives a brief introduction to the BOIN design, followed by a methodology proposed to incorporate external information based on the BOIN design framework. Next, an approach for extending the BOIN with fixed boundaries to the aBOIN design with non-fixed boundaries is proposed. Empirical findings are shown by comprehensive simulations with derivation of the theoretical properties. The paper ends with a final discussion.

2 Brief introduction to the BOIN design

The BOIN design proposed by Liu and Yuan in 2015 [9] is simple to implement and is similar to the 3+3 design, but is much more flexible, and its operating characteristics are superior to those of more complex model-based methods. An R package (BOIN), a stand-alone graphical user interface–based software, and Shiny app (www.trialdesign.org) have been developed, which are freely accessible to users.

The BOIN design can be summarized as follows:

1. (a). Patients in the first cohort are treated with the lowest or a pre-specified dose level.
2. (b). Let be the observed toxicity rate at the current dose. To assign a dose to the next cohort of patients,
   
   • if , we escalate the dose level to j + 1,
   • if , we de-escalate the dose level to j ‒ 1, or
   • otherwise, i.e. , we retain the same dose level, j.
   To ensure that dose levels of treatment always remain within the pre-specified dose range, the dose escalation or de-escalation rule needs to be adjusted for the lowest or highest levels of j; for example, if j = 1 and or j = J and , the dose remains at the same level, j.
3. (c). This process continues until the maximum sample size is reached or the trial is terminated because of excessive toxicity, as described next.

The selection of interval boundaries λ₁ and λ₂ is critical, because two parameters essentially determine the operating characteristics of the design. The BOIN design is optimal in the sense that it selects λ₁ and λ₂ to minimize incorrect decisions of dose escalation and de-escalation during the trial.

By using p_j to denote the true toxicity probability of dose level j for j = 1, …, J, three point hypotheses are formulated: where ϕ₁ denotes the highest toxicity probability that is deemed sub-therapeutic (i.e., below the MTD) such that dose escalation is required, and ϕ₂ denotes the lowest toxicity probability that is deemed overly toxic, such that dose de-escalation is required.

Under the Bayesian paradigm, each hypothesis was assigned an equal prior probability, denoted as , k = 0, 1, 2. The probability of making an incorrect decision (the decision error rate) is minimized when (1) (2)

Details can be found in [9, 10].

3 BOIN design with incorporating external information

Viele et al. [11] said, “Clinical trials rarely, if ever, occur in a vacuum. Generally, large amounts of clinical data are available prior to the start of a study”. Although the phase I trial is considered the first-in-human study for identifying the MTD, there is still possible information that we can use to enhance our understanding of the toxicity profile for experimented drugs, for example, in the aforementioned phase I pediatric trials or rare diseases occurring in limited patient populations. Another point of view is that well-conducted phase I studies can increase the precision of phase II dose recommendation. High failure rates for late-phase studies can be due to flawed phase I studies. Efficiently using the prior or historical information provides an opportunity to improve the phase I study.

By using the BOIN design framework, prior information can be incorporated naturally via only modifying (1) and (2) by using the following formula: (3) (4)

Comparing (3) and (4) to (1) and (2), prior parameters are incorporated in the above formulas by using prior probabilities of the 3-point hypotheses: π_0j, π_1j, and π_2j.

By notations, assuming there are J dose levels, for each dose level j, j = 1, ⋯, J, we have three prior probability vectors for π_0,j, π_1,j and π_2,j, j = 1, ⋯, J, associated with three point hypotheses H_0j, H_1j, H_2j, j = 1, ⋯, d, which are defined in BOIN introduction. All the prior probabilities can be presented explicitly by the following Table 1:

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Prior probabilities of each dose for three point hypotheses.

https://doi.org/10.1371/journal.pone.0237254.t001

On the basis of data in Table 1, we propose an approach to elicit values for these cells prior to the trial study. If there is strong confidence that dose D_j is closest to the target DLT rate, we assign a larger probability to π_0,j (e.g., 0.6) and then assign π_1,j, π_2,j to be equally half of the rest of the probability (e.g., 0.4/2 = 0.2). Here, we believe a priori with 60% confidence that dose D_j would be the MTD and 20% confidence that this dose would be under-dosing or over-dosing. We define odds_j to be .

By eliciting values of the remaining cells in Table 1, we pre-specify a probability vector for H₀, that is, (π_0,1, ⋯, π_0,J). We emphasize here that pre-specification of the probability vector for H₀ is feasible. For example, if we have strong evidence that one dose is near to the MTD, as in pediatric trials, because MTDs in children and adults correlate strongly and 80% of the adult dose is recommended as the starting dose for children, the investigator can effortlessly select with high confidence the dose that can be the MTD and also other doses. If there is weak prior knowledge, equally likely probabilities can be assigned to this vector.

When eliciting values for two probability vectors of H₁ and H₂, the two vectors need to be in decreasing and increasing orders, respectively. This is because H₁ refers to the under-dosing hypothesis; therefore, probabilities of believing in H₁ would decrease when dose level increases and vice versa for the probability vector for H₂. For example, for a trial with five dose levels, if we assign the probability vector to H₀ to be π_0,1 = 0.05, π_0,2 = 0.15, π_0,3 = 0.6, π_0,4 = 0.15, π_0,5 = 0.05, then π_0,1 = 0.05 it means we have very low confidence that the first dose is the MTD. Since the first dose is the lowest dose among the five doses, the π_1,1 should be the highest among (π_0,1, π_1,1, π_2,1), since it is the safest dose level; that is, we have high confidence that the first dose will lie in the interval defined by the hypothesis H₁, which corresponds to the under-dose interval. As an example, let us assign values (π_0,1 = 0.05, π_1,1 = 0.85, π_2,1 = 0.10) to them by considering the constraint π_0,j + π_1,j + π_2,j = 1, ∀, j. For the second dose, since π_0,1 = 0.15, this again means that we have little confidence that this dose is the MTD and, similarly, π_1,2 should still be the highest dose among (π_0,2, π_1,2, π_2,2). However, the probability of π_1,2 to be in H₁ should now be lower than that for π_1,1, since dose 2 has a higher toxic rate than dose 1. For example, if we assign the probabilities as (π_0,2 = 0.20, π_1,2 = 0.60, π_2,2 = 0.20), there should be a decreasing trend in the probability vector of H₁ and an increasing trend in the probability vector of H₂. Similarly, a decreasing trend will be observed for a vector probability of H₂.

Given the above premise, we propose the following algorithm that can automatically implement the assignment of horizontal probability vectors in Table 1. However, in reality, there will be infinite alternatives to elicit three probability vectors by satisfying the above increasing or decreasing monotone constraints. Our proposed alternative shows just one of the possible cases.

1. Step 1. Assign each dose a probability for H₀, that is, a prior probability vector of (π_0,1, ⋯, π_0,J), to best “guess” which of these J doses to be the MTD.
   This step is not so challenging if clinicians have strong confidence on which dose is closest to the MTD target. For example, in pediatric trials, we can often choose the MTD for adult patients or a starting MTD dose for pediatric patients. In this step, clinicians can also choose a set of skeletons for the CRM.
2. Step 2. If the dose j is believed to be close to the MTD, then let , that is, odds_j = odds(π_1,j, π_2,j) = 1 to assign probabilities to π_1,j and π_2,j given π_0,j in Step 1. Also, let the lowest dose have odds₁ = odds(π_1,1, π_2,1) = 10 and the highest dose have . We can have probabilities for π_1,1, π_2,1 and π_1,J, π_2,J. If the lowest or highest dose levels are believed to be the MTD, then the odds for it is set to be 1.
3. Step 3. Use extrapolation method (see details in Appendix) to elicit prior probabilities for the rest of two vectors (π_1,1, ⋯, π_1,J) and (π_2,1, ⋯, π_2,J) can be easily derived.

The above algorithm is easy to use since it only requires the investigator to provide probability guesses for H₀s for each investigated dose level. All the other remaining probabilities in Table 1 can be automatically computed, which substantially reduces the burden on investigators and improves the “guess” precision. See details of the algorithm and a numerical example to show the algorithm in the Appendix.

4 aBOIN design with adaptive boundaries

This section discusses the extension of the BOIN to aBOIN design with adaptive boundaries. For interval-based designs, the first step is to specify an indifference interval defined by two fixed boundaries to differentiate under-dose from over-toxic dose levels. Based on these boundaries, decision rules of dose assignment are developed. (See Introduction for the BOIN design.) The BOIN design is also categorized as a model-assisted design from the perspective of the modeling approach and how accumulative data are used [10]. The BOIN design derives two underlinefixed boundaries to make the dose escalation/de-escalation decision from its theories, denoted by λ₁ and λ₂, which are indirectly linked to the under- and over-dose hypotheses introduced earlier. If we denote the MTD toxicity rate as ϕ and use the authors’ recommendation of ϕ₁ = 0.6ϕ and ϕ₂ = 1.4ϕ, the two boundaries can be written as a function of ϕ₁ and ϕ₂.

The BOIN design also has useful theoretical properties, such as minimizing the decision-making error, long-term memory coherence, and convergence to the MTD dose. In this section, we first demonstrate that the proposed aBOIN design also inherits theoretical properties from the BOIN design and then conduct simulation studies to see whether this extension could improve the original BOIN design.

5 Simulation studies

In this section, we explore the operating characteristics of the proposed aBOIN design with and without incorporating prior information by comparing it to the original BOIN design. The aims of the simulation study are twofold: (i) to explore the behavior of the aBOIN design that incorporates prior information compared with that of the original BOIN design and the aBOIN design that does not incorporate prior information, and (ii) explore the operating characteristics of the original BOIN and aBOIN designs.

Simulation setting

We consider trials with five dose levels and a maximum sample size of 30 patients, with a cohort size of three patients. Twenty different scenarios (one half with dose-limiting toxicity (DLT) rates of 20%, and the other half with DLT rates of 30%) with various locations and DLT rates are shown in Table 2. We will use them to examine properties of the proposed designs.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Ten true toxicity scenarios with the target DLT rate of 20% and 30%.

https://doi.org/10.1371/journal.pone.0237254.t002

For each scenario, we simulated 10,000 trials. We implemented the BOIN design using the R package BOIN with its default design parameters. For the aBOIN design, we specified the accelerating factors as g₁ = 0.4, g₂ = 0.9, which were derived by trial and error, and we only activated the adaptive shrinking mechanism in at least six patients who had been treated (referred to as the lead-in period). As introduced in [10], four metrics to measure the performance of a design have been considered: (1) the percentage of correct selection (PCS) of the true MTD in 10,000 simulated trials; (2) the average number of patients allocated to the MTD across 10,000 simulated trials; (3) the risk of overdosing, defined as the percentage of simulated trials in which a large percentage (e.g., more than 60% or 80%) of patients are treated at doses above the MTD (i.e., how likely it is that the design treats more than 60% or 80% of patients at doses above the MTD); and (iv) the risk of under-dosing, which is defined as the percentage of simulated trials in which more than 80% of patients are treated at doses below the MTD (potential sub-therapeutic doses). For comparing the aBOIN design with or without prior information, we focus on the PCS of the true MTD by comparing the proposed design to the original BOIN design.

5.1 Simulation 1: Adaptive BOIN design with incorporating prior information

To incorporate prior information, we first specify a probability vector for H₀ row in Table 1. In our simulations, we assign one set of probability vectors for all 20 scenarios in Table 2, because this enables us to check whether the performance of the aBOIN design with prior information is robust or not through various locations of MTDs in Table 2. To be specific, we assign H₀ with probabilities (π_0,1, ⋯, π_0,5) = (0.2, 0.45, 0.7, 0.45, 0.2) for all scenarios, and the other two probability vectors are derived by using the procedure introduced above to be (π_1,1, ⋯, π_1,5) = (0.72, 0.44, 0.15, 0.12, 0.08) and (π_2,1, ⋯, π_2,5) = (0.08, 0.11, 0.15, 0.43, 0.72). Considering these specific settings, this means that we have 70% confidence that dose level 3 could be the MTD and 45% confidence that dose level 2 or 4 could be the MTD.

Simulation results of the PCS (%) for the BOIN and the aBOIN design with or without incorporating prior information are shown in Table 3. For example, with a DLT rate of 20% for the PCS (%) metric, there are 10 scenarios. The first row with “BOIN” refers to the PCS (%) of the original BOIN design. For instance, for scenarios 1, 3, and 8, the corresponding PCSs (%) are 37.15%, 16.1%, and 45.41% with MTD locations at dose levels 1, 3, and 3, respectively. The second row with aBOIN¹ refers to PCS (%) of the aBOIN design without incorporating prior information. Similarly, for scenarios 1, 3, and 8, the corresponding PCSs (%) are 32.22%, 16.75%, and 43.82% with MTD locations at dose levels 1, 3, and 8, respectively. The third row with aBOIN² refers to the PCS (%) of the aBOIN design incorporating prior information. The corresponding PCS (%) for scenarios 1, 3, and 8 are 26.03%, 22.61%, and 47.65%, respectively. Because the prior has placed high confidence on dose level 3 being the MTD, and scenarios 3 and 8 are scenarios with MTD locations at dose level 3, for scenarios 3 and 8, aBOIN² has highest PCSs (%) among the three designs. For the remaining scenarios, the PCS (%) of aBOIN² is comparable to or lower than that of the BOIN and aBOIN¹ designs. For example, for scenario 3 of DLT with a 20% toxicity rate, the PCS (%) of the aBOIN² is 22.61%, whereas those of the BOIN and aBOIN¹ are 16.1% and 16.75%, respectively, because for scenario 3 with the MTD located at dose level 3, the prior guess also has the strongest confidence at dose level 3. However, if we check scenario 1 with the MTD located at dose level 1, we put only 20% confidence into dose level 1 and find that the aBOIN² design has the worst performance in terms of the PCS% (26.03%), whereas BOIN and aBOIN¹ have a higher PCS% (37.15% and 32.22%, respectively). At a DLT rate of 30%, similar patterns can be observed.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Percentage of correctly selection percentage of MTD for ten scenarios in Table 1.

https://doi.org/10.1371/journal.pone.0237254.t003

From these results, we infer that when the prior guess for the MTD location is close to the truth, the aBOIN version incorporating prior information performs the best in terms of the PCS metric; in other scenarios, its results vary widely and can sometimes even be very inaccurate. Given these observations, we recommend that in actual practice, the aBOIN incorporating prior information should be used only when the investigator has strong confidence or there is prior or historical information on which dose is or approximate to the MTD.

5.2 Simulation 2: Adaptive BOIN design without incorporating prior information

In this subsection, we investigate the aBOIN¹ design, that is, the aBOIN design without incorporating prior information. However, in this subsection, we still call this version aBOIN for brevity. We closely examine not only the PCS% but also the other three metrics, percentages of patients allocated to a true MTD during the trial (MTD%), and the mean number of observed DLTs throughout the trial (# of DLTs). Results are shown in Table 3 and Figs 1 and 2.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Operating characteristics of ten scenarios on the left panel (DLT 20%) of Table 2 by two competing methods BOIN and aBOIN.

https://doi.org/10.1371/journal.pone.0237254.g001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Operating characteristics of ten scenarios on the left panel (DLT 30%) of Table 2 by two competing methods BOIN and aBOIN.

https://doi.org/10.1371/journal.pone.0237254.g002

Results.

For a DLT rate of 20%, Table 3 and Fig 1 show that the PCS (%) of the aBOIN design for scenarios such as 1, 2, 6, 7, 8, and 10 is comparable to or lower than that for the BOIN design. For the remaining scenarios, performance of the aBOIN design by PCS (%) as a metric is comparable with that of the original BOIN design. Findings are similar for the criterion “# of Patients at MTD.” However, for the criterion “Risk of Overdosing 60 (%),” in almost across all scenarios, the risks associated with the aBOIN design are higher than for the BOIN design. Nevertheless, for the criterion “Risk of Underdosing 60 (%),” the BOIN design performs poorer than the aBOIN design. For a DLT rate of 30%, Table 3 and Fig 1 show that the BOIN and aBOIN designs have comparable performances for the criteria “PCS (%)” and “# of Patients at MTD.” However, at a DLT rate of 20%, for the criterion “Risk of Overdosing 60(%),” overall the aBOIN design is associated with lower risks than for the BOIN design across all scenarios but higher risks than that for the BOIN design for the criterion “Risk of Underdosing 60(%).”

We also examined the convergence rate with the PCS (%) metric for asymptomatic properties for both designs. We present partial results for the first four scenarios for DLT rates of 20% and 30%. Fig 3 shows that for DLT rates of 20%, in all explored scenarios the curve of the aBOIN design is eventually above that of the original BOIN design. This indicates that as the sample size increases, the aBOIN design performs better than the BOIN design in terms of the PCS metric. Although we know that phase I trials usually have small sizes (approximately 30 patients), asymptotic findings show that the idea of shrinking boundaries comes into effect only when the sample size is larger, and therefore this idea has little practical use although it is asymptotically or theoretically meaningful. In summary, for the finite sample, the aBOIN and BOIN designs have comparable performances with respect to the four criteria. For large samples, the aBOIN design performs better than the original BOIN design, but it has little practical use due to limited sample size in practice, and the corresponding simulated results can be seen in Fig 4.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Percentage of correctly selection percentage of MTD of the first four scenarios on the left panel (DLT rate 20%) in Table 2 by using the two competing methods BOIN and aBOIN with a large sample.

https://doi.org/10.1371/journal.pone.0237254.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Correctly selected percentage of the maximum tolerated dose for the first four scenarios in the right panel at a DLT rate of 30% in Table 2 by using original BOIN and aBOIN designs in a large sample.

https://doi.org/10.1371/journal.pone.0237254.g004

Note that if investigators have vague or less confidence about prior experience or information, we still suggest that they use the BOIN design without prior information.

6 Discussion

We have developed two extensions of the BOIN design. The first one develops an accessible approach to allow the incorporation of prior or historical information in the phase I trial. The second extension proposes adaptive shrinking boundaries (aBOIN design), whereas the original BOIN design has fixed boundaries. The aBOIN design uses accelerating factors to control the shrinking speed rates of lower and upper boundaries. Theoretical properties were derived for the aBOIN design.

Performances of the proposed methods were discussed by simulations. When setting up the location for the MTD a priori that was close to the MTD, the aBOIN design incorporating prior information showed better performance than the original BOIN design. However, if the prior deviated from the truth, performance of the aBOIN design was inferior to that of the BOIN design. This is understandable, since there were very few sample sizes and therefore it was hard to dominate the estimation procedure for deciding the dose. Therefore, we caution practitioners to use prior information in real trials unless there is strong confidence. The second extension of the proposed aBOIN design was examined numerically by using a finite sample and a large sample. For finite sample sizes, performances were similar when comparing the aBOIN without incorporating prior information to the BOIN design. Although the proposed aBOIN design outperforms in asymptotic properties, it has limited use in actual phase I trials due to the small sample size. In summary, the original BOIN design can be improved only if very informative historical information is available.

Appendix 1: Proof of theoretical properties

Proof. Coherence. Since λ_1j < ϕ and λ_2j > ϕ, we can easily obtain the coherence:

Thus, the aBOIN is long-term memory coherent.

Proof By the definition of ϕ₁ and ϕ₂, we can get λ₁ → ϕ, λ₂ → ϕ, as n_j tends to ∞. By the L’hopital’s rule, we get (9)

The proof of λ₂ → ϕ as n_j → ∞ is similar as above.

That is, both λ₁ and λ₂ shrink toward the MTD target ϕ.

Proof. λ₁ < ϕ and λ₂ > ϕ). Prove λ₁ < ϕ.

Since we have proved that λ₁ converges to ϕ(>0), if we prove λ₁ is an increasing function of ϕ₁, then we can prove λ₁ < ϕ. Let and The (using inequality for all x > −1)

(since 0 < ϕ, ϕ₁ < 1)

Thus, , that is, λ₁ = f(ϕ₁) is an increasing function of ϕ₁ with limit at ϕ.

Hence, we have λ₁ < ϕ.

Similarly, we can prove λ₂ > ϕ.

Proof. Convergency. Denote the event . We only need to show that when n_j is large enough, pr(A) = 1. Since lim_{nj → ∞}λ₁/ϕ₁ = 1 and lim_{nj → ∞}λ₂/ϕ₂ = 1, for g₁, g₂ < 1 by the CLT, we can get

Then, the proof provided by Oron, Azriel, and Hoff (2011) can be directly used to obtain the result.

Appendix 2: Algorithm of generating priors in Table 1

Assuming J dose levels, we firstly elicit a prior vector probability for hypothesis H₀, that is, guessing which dose would possibly be the MTD, denoted as π_0,1, ⋯.π_0,j, ⋯, π_0,J. We can also assume odds of H₁ to H₂ at dose level 1, since at the lowest dose, it would have high confidence that this first dose would be under-dosing than over-dosing, thus, we let can be any large number, for instance, in our algorithm, and, vice versa, the odds of H₁ to H₂ at dose level J, would be a small number, for instance, we let . If dose level j is assigned highest probability, that is, dose j is believed to be closet to the MTD prior to the study and we assume there has equal chance to be under- or over-dose at this dose level, that is, .

Based on the above odds₁ and odds_J, we use the definition of the odds to evaluate the prior probabilities for H₁ at dose levels 1 and J as: (10) then, the prior probabilities for H₂ at dose levels 1 and J are: (11)

Now, we have the prior probabilities of the first row for H₀ and first and last (J-th) columns in Table 1. Based on these information, we then use an interpolation technique to assign probabilities for the rest cells in Table 1.

To be specific, for computing probabilities of H₁ for dose levels from 2 to j − 1, the following linear interpolation formula is used: (12)

For computing probabilities of H₁ for dose levels from j + 1 to J, the following linear interpolation formula is used: (13)

Thus, based on the above steps, we assign probabilities for the first (H₀) and second rows (H₁) in Table 1. Probabilities for third row (H₂) are: (14)

We provide a numerical example for showing the above procedure. Assuming there are 5 dose levels and, without losing generality, assuming the 3rd dose level is closest to the MTD prior to the study. For example, the prior probability vector is set to be (π_0,1, ⋯, π_0,5) = (0.2,0.45,0.7,0.45,0.2), that is, this is the 1st row in Table 1. To be specific, we think that the dose level 3 may be close to the MTD with 70% confidence, and then dose level 2 and 4 with 45% confidence to be the MTD while the first dose level and last dose level have the minimal confidence to be the MTD with 20% for each. Since the odds (defined by the algorithm) for the dose level 3 is odds₃₌ 1, so we have π_1,3 = π_2,3 = 0.15 since π_0,3 = 0.7 now. By the algorithm, we also know that odds for the first and last dose levels are as odds₁ = 10 and .

By using the above formula (10) and (11), we can have

Then, we can have π_2,1 = 1 − π_1,1 − π_0,1 = 1 − 0.72 − 0.2 = 0.08 and π_2,5 = 1 − π_1,5 − π_0,5 = 1 − 0.08 − 0.2 = 0.72.

Thus, by using the formula (13) and (14), we have thus, for the second row (H₁) in Table 1, we have assigned prior probabilities as (π_1,1, π_1,2, π_1,3, π_1,4, π_1,5) = (0.72, 0.44, 0.15, 0.12, 0.08).

Then, for the third row (H₂), we have (π_2,1, π_2,2, π_2,3, π_2,4, π_2,5) = 1 − (π_0,1 + π_1,1, π_0,2 + π_1,2, π_0,3 + π_1,3, π_0,4 + π_1,4, π_0,5 + π_1,5) = (0.08, 0.11, 0.15, 0.43, 0.72).

Supporting information

Acknowledgments

The authors really appreciate Dr. Vani Shanker from St. Jude Children’s Research Hospital for scientific editing of this manuscript.