To illustrate our proposed methods, we simulated a single trial using the design parameters specified in Section 4.2. The toxicities were generated using a latent uniform approach. Specifically, a random variable with a uniform distribution over interval (0,1) was sampled using the same seed for each patient and denoted as *u*. Supposing that this patient is assigned to dose *d*_{k}, then we set the toxicity outcome of this patient as *T* < 1 if *u* ≤ Pr(*T* < 1|*d*_{k}); 1 ≤ *T* < 1.5 if Pr(*T* < 1|*d*_{k}) < *u* ≤ Pr(*T* < 1.5|*d*_{k}); and *T* ≥ 1.5 if otherwise. This approach implied that given the same dose the toxicity outcome for a patient is the same, and therefore, ensured that the results obtained using the 2 proposed methods in Section 2.2 were comparable.

displays the prior estimates, the dose levels assigned, and the outcomes for the 18 patients in the trial. Since

_{0} = −5.51 and

_{0} = −5.30, both methods recommended dose level 3 hence were permissible. The first patient was treated at dose level 3 with no toxicity, resulting in

_{1} = −3.00 and

_{1} = −2.91. However, since we did not allow for dose skipping, dose level 4 was recommended for the second patient. This was the only occasion in this trial that the no dose-skipping restriction was in effect, indicating that the estimators stabilized quickly after very few observations. In this simulated trial, the recommended MTD and the dose assigned based on both estimators were the same except for patients 12 and 14 who were assigned dose level 3 based on

_{11} = −4.96 and

_{13} = −4.99, respectively, but dose level 4 based on

_{11} = −4.94 and

_{13} = −4.87, respectively. At the end of the trial, dose level 4 was recommended as the MTD by both methods. summarizes graphically the dose assignments and outcomes for the 18 patients using the CRM-MC

_{2} method. The dose was escalated after each of the first 2 patients with no toxicities being observed. The fourth patient experienced a severe toxicity at dose level 5 and the dose was deescalated. The dose was deescalated again after the sixth patient experienced another severe toxicity. As a result, the following 5 patients were treated at dose level 3. With no toxicities being observed, the dose was escalated after the eleventh patient and the method recommended dose level 4 for the remaining 7 patients. One toxicity was observed among these 7 patients. At the end of the trial, toxicities were reported in 3 out of the 18 patients (17%), with 2 (11%) patients having a severe toxicity.

The performance of the 2 proposed methods, CRM-MC_{1} and CRM-MC_{2}, was assessed with further simulations under 6 scenarios that might be encountered in practice and compared with those of the CRM (i.e. ignoring the higher toxicity constraint Pr(*T* ≥ 1.5|*x*) ≤ *p*_{2} = 0.10) and the BT method which is the only alternative in the literature for multiple toxicities. One thousand simulations were performed for each scenario with 18 subjects for each simulation using the 5 doses specified at the end of Section 4.2.

The complete configuration of the true toxicity probabilities for the 6 scenarios is depicted in . Unlike the 10 calibration scenarios used to select the optimal *δ* value in the calibration process, these 6 scenarios were selected in collaboration with clinicians. In the first 4 scenarios, the true probabilities of *T* ≥ 1 correspond to the probabilities of DLT that were originally used in the calibration of the CRM design for the bortezomib trial. In these scenarios, *θ* = *θ*_{1} = *θ*_{2}. In Scenario 5, *θ* = *θ*_{1} < *θ*_{2}. Thus, there is no risk of ignoring the secondary constraint since *θ* = *θ*_{1}. However, in Scenario 6, *θ* = *θ*_{2} < *θ*_{1}. This means that if the secondary constraint is ignored and only the primary constraint is used, the wrongly believed MTD would be *θ*_{1} = *d*_{3}, resulting in an inflated chance of severe toxicity since Pr(*T* ≥ 1.5|*d*_{3}) = 0.23 > 0.10 = *p*_{2}. This is a scenario in which the higher toxicity constraint should not be ignored.

| **Table 2.**True probabilities of the 2 types of toxicities under 6 scenarios used in the simulation study |

To compare our results with those of the BT method, we excluded the variable

*W* from the definition of

*T* in (4.6). Thus, in the context of the bortezomib trial,

*T* = 0.19

*I*(

*Y*_{1} = 1) + 0.64

*I*(

*Y*_{1} = 2) + 1.03

*I*(

*Y*_{1} = 3) + 2.53

*I*(

*Y*_{1} = 4) + 0.17

*I*(

*Y*_{2} = 1,2) + 0.40

*I*(

*Y*_{2} = 3) + 0.85

*I*(

*Y*_{2} = 4). This was equivalent to the definition of TTB used by

Bekele and Thall (2004) with the coefficients being the toxicity weights. We obtained the joint probability distributions of the 2 toxicities from the 2 marginals under independence. The marginal probabilities for the grades of neuropathy and low platelets were chosen such that they corresponded to the prespecified Pr(

*T* ≥ 1.5|

*d*_{k}) and Pr(1 ≤

*T* < 1.5|

*d*_{k}). For example, for dose level 1 under Scenario 1, Pr(

*T* ≥ 1.5|

*d*_{1}) = Pr(

*Y*_{1} = 4) +

*P**r*(

*Y*_{1} = 3,

*Y*_{2} = 4) = 0.014 and Pr(1 ≤

*T* < 1.5|

*d*_{1}) =

*P**r*(

*Y*_{1} = 3,

*Y*_{2} < 4) +

*P**r*(

*Y*_{1} = 2,

*Y*_{2} ≥ 3) +

*P**r*(

*Y*_{1} = 1,

*Y*_{2} = 4) = 0.035.

For the BT method, the prior means of the intercept parameters for the 2 toxicity types were 0.5 and the prior means of the slope parameters were 1.0. The prior precision for all regression parameters was 1.0. The doses are 1, 2, 3, 4, and 5. The target TTB score was set at 0.72. The method did not allow for dose skipping.

displays the percentage of recommending a particular dose (% Recommended) and the percentages of toxicities (% *T* ≥ 1 and % *T* ≥ 1.5) for each of the 4 methods. For the BT method, E(TTB) = 0.19*π*_{1,1} + 0.64*π*_{1,2} + 1.03*π*_{1,3} + 2.53*π*_{1,4} + 0.17*π*_{2,1or2} + 0.40*π*_{2,3} + 0.85*π*_{2,4}, where *π*_{j,c} is the marginal probability of toxicity for a grade *c* toxicity of type *j*. When *θ*_{1} = *θ*_{2} (Scenarios 1–4), the PCS using either CRM-MC_{1} or CRM-MC_{2} method is similar to the one using CRM. Compared to the BT method, CRM-MC_{1} and CRM-MC_{2} have similar accuracy and are occasionally superior (Scenario 3). CRM-MC_{1}, CRM-MC_{2}, and BT recommend dose levels above the MTD less frequently than the CRM. Compared with CRM-MC_{2}, the CRM-MC_{1} method recommends a dose above the MTD less frequently and has lower percentages of toxicities across the 6 scenarios but yields a lower PCS when the true MTD is high.

| **Table 3.**Operating characteristics of the CRM, BT, CRM-MC_{1}, and CRM-MC_{2} methods |

When *θ*_{1} < *θ*_{2} (Scenario 5), CRM-MC_{1} and CRM-MC_{2} methods perform similar to the CRM in terms of percentage of dose recommended and percentage of toxicities. However, when *θ*_{1} > *θ*_{2} (Scenario 6), the CRM is more likely to recommend a toxic dose as it does not take into account the secondary constraint. Both CRM-MC_{1} and CRM-MC_{2} methods recommend the correct dose more than 50% of the time with much lower frequencies of recommending a toxic dose. Both CRM-MC_{1} and CRM-MC_{2} methods outperform the BT method in terms of the PCS, the frequency of recommending a dose above the MTD and the percentage of toxicities. However, the overdosing of the BT method is an artifact since the target is misspecified, and in both Scenarios 5 and 6, the target TTB of 0.72 falls in between dose levels. For example, under Scenario 5, the target TTB value of 0.72 lies somewhere between the expected TTB of dose level 3 (0.59) and dose level 4 (0.90). Therefore, it is not unexpected that the BT method recommends dose level 4 with a somewhat high probability. In this sense, the simulation comparison is unfair. On the other hand, it reveals the limitations of the target TTB both in terms of interpretation and elicitation. The objective of the BT method is to find the dose associated with a target TTB that is not directly related to a probability of toxicity.