Table 1.

Designs investigated that are extensions of the 3 + 3 design that allow only escalation.

Design	Assignment rule	Ways to escalate	Approximate range for toxicity rate targeted by the design (Table 4.1, Chapter 4, Ting, 2006 [30]; Storer, 2001 [19])
3 + 3	If 0 out of 3 enrolled patients have a DLT, then escalate to the next dose level and enroll 3 more; if 1 out of 3 patients has a DLT, then add 3 more patients at the same dose level; if 2 or more patients out of 3 or 6 patients experience a DLT, then stop the trial. The MTD is one dose level below.	0/3 = 0% or 1/6 = 16.7% i.e. can escalate if we observe 0 DLTs out of 3 patients, or 1 DLT out of 6 patients	0.17<Γ < 0.26 or 0.2<Γ < 0.25
2 + 4	If 0 out of 2 enrolled patients have a DLT, then escalate to the next dose level and enroll 2 more; if 1 out of 2 patients has a DLT, then add 4 more patients at the same dose level; if 2 or more patients out of 2 or 6 patients experience a DLT, then stop the trial. The MTD is one dose level below.	0/2 = 0% or 1/6 = 16.7% i.e. can escalate if we observe 0 DLTs out of 2 patients, or 1 DLT out of 6 patients	0.17<Γ < 0.26
4 + 4 a	If 0 out of 4 enrolled patients have a DLT, then escalate to the next dose level and enroll 4 more; if 1 or 2 out of 4 patients have a DLT, then add 4 more patients at the same dose level; if 3 or more patients out of 4 or 8 experience a DLT, then stop the trial. The MTD is one dose level below.	0/4 = 0% or 1/8 = 12.5% or 2/8 = 25% i.e. can escalate if we observe 0 DLTs out of 4 patients, or 1 DLT out of 8 patients, or 2 DLTs out of 8 patients	0.25<Γ < 0.31
5 + 5 a	If 0 out of 5 enrolled patients have a DLT, then escalate to the next dose level and enroll 5 more; if 1 or 2 out of 5 patients have a DLT, then add 5 more patients at the same dose level; if 3 or more patients out of 5 or 10 experience a DLT, then stop the trial. The MTD is one dose level below.	0/5 = 0% or 1/10 = 10% or 2/10 = 20% i.e. can escalate if we observe 0 DLTs out of 5 patients, or 1 DLT out of 10 patients, or 2 DLTs out of 10 patients	0.2<Γ < 0.25
3 + 3+3	If 0 out of 3 enrolled patients have a DLT, then escalate to the next dose level and enroll 3 more; if 1 out of 3 patients has a DLT, then add 3 more patients at the same dose level; if 2 out of 6 patients have a DLT then add 3 more patients at the same dose level; if 2 or more patients out of 3 patients experience a DLT or 3 or more out of 6 or 9 patients experience a DLT, then stop the trial. The MTD is one dose level below.	0/3 = 0% or 1/6 = 16.7% or 2/9 = 22.2% i.e. can escalate if we observe 0 DLTs out of 3 patients, or 1 DLT out of 6 patients, or 2 DLTs out of 9 patients
Simple Accelerated Titration Design	Successively assign a single patient at each dose level until the patient has a DLT. Then switch to the 3 + 3 design (i.e. add 2 more patients to the dose level at which a DLT is first seen and then follow the rules of the 3 + 3 design).

The table above provides the rules for the escalation only designs but we also allow de-escalation in the 3 + 3, 2 + 4, 4 + 4 a, and 5 + 5 a designs and follow the algorithm described in the methods section. The designs that also allow de-escalation will target a slightly lower DLT rate than their counterparts that allow only escalation. One method to estimate the approximate target DLT rate of each design that also allows de-escalation is to run simulations for each design using several different dose-toxicity curves and then perform the following calculation: one needs to compute the sum of the product of the true DLT rate at each dose and the probability that that dose is selected as the MTD from simulations for each scenario and then find the average of this value across the various scenarios (dose-toxicity curves). Based on our results for the logistic, log-logistic and linear dose-toxicity curves in Table 3, Table 4, Table 5, we find that the approximate target DLT rate of the 3 + 3 design with de-escalation is 0.17, of the 2 + 4 design with de-escalation is 0.18, of the 4 + 4 a design with de-escalation is 0.21 (which is why we also included the 4 + 4 a design, even though its target DLT rate for the escalation only case is a little higher than 0.2), and of the 5 + 5 a design with de-escalation is 0.17. The 3 + 3+3 design targets an approximate DLT rate of 0.21.