Why Add a Third Cohort?
The conventional 3+3 design uses cohorts of three and, in some circumstances, expands a dose level to six participants. A particular boundary matters: if two of the six experience a dose-limiting toxicity (DLT), the conventional rule stops escalation at that dose. A 3+3+3 variant adds another cohort of three in this situation. The additional observations can change the decision, although they do not turn the rule into a precise estimate of the dose-specific toxicity rate.
There is not one universal rule set called “3+3+3.” The version taught here is defined explicitly so that every boundary is unambiguous: a third cohort is added only when exactly two DLTs have occurred among the first six participants at a dose. The final decision uses the total among all nine. A trial protocol must specify its own rules before enrollment; a design should not be modified informally while outcomes accumulate.
The Cohort and Decision Rules
Assume the trial has an ordered dose ladder, as defined in earlier material on constructing a dose-escalation ladder. Let \(d_i\) denote the current dose. A DLT is an outcome classified as dose-limiting under the trial’s prespecified definition and observation window. At each cohort review, count DLTs among participants treated at \(d_i\); do not combine counts from different dose levels.
The rule assumes that each participant’s DLT status for the review is available before the next cohort decision, the dose and DLT definition remain fixed during that decision, and the cohort has been assessed consistently. The simple probability calculation later also uses independent participant outcomes with a common DLT probability at a fixed dose. These are working assumptions, not guarantees. In particular, late or missing outcomes can make a completed-cohort rule impossible to apply until the protocol’s evaluation requirements are met.
| Participants assessed at the dose | Total DLT count at that dose | 3+3+3 decision |
|---|---|---|
| First 3 | 0 | Escalate to the next dose. |
| First 3 | 1 | Enroll 3 more at the same dose. |
| First 3 | 2 or 3 | Stop escalation at this dose; do not enroll another cohort here. |
| First 6, after expansion | 0 or 1 | Escalate to the next dose. |
| First 6, after expansion | 2 | Enroll the third cohort of 3 at the same dose. |
| First 6, after expansion | 3 or more | Stop escalation at this dose. |
| All 9, after a third cohort | 0, 1, or 2 | Escalate to the next dose. |
| All 9, after a third cohort | 3 or more | Stop escalation at this dose. |
“Escalate” means move to the next dose level in the prespecified ladder, if one remains. “Stop escalation” means do not move to a higher dose under this rule. In the usual ladder-based interpretation, the dose immediately below a dose that triggers stopping may be nominated as the maximum tolerated dose (MTD) candidate, if such a dose exists. That nomination is a consequence of the protocol rule; it is not proof that the dose’s true DLT probability equals a target or that the dose is suitable for every later purpose.
Worked Examples
Worked Example: No DLTs in the First Cohort
At a hypothetical dose \(d_1\), suppose the first three participants have outcomes with zero DLTs. The count is \(0/3\), so the rule says to escalate. The trial enrolls three participants at \(d_2\); suppose one has a DLT. The count at \(d_2\) is \(1/3\), so the trial enrolls three more there rather than escalating immediately.
Suppose none of the second cohort of three at \(d_2\) has a DLT. The total at that dose is \(1+0=1\) DLT among \(3+3=6\) participants, or \(1/6\). Since one is no more than one, the rule escalates to the next dose. Counts at \(d_1\) and \(d_2\) are evaluated separately; the one DLT at \(d_2\) is not added to the zero at \(d_1\).
Worked Example: The Third Cohort Changes the Decision
At hypothetical dose \(d_1\), assume \(0/3\) participants have DLTs, so the trial escalates to \(d_2\). At \(d_2\), one of the first three participants has a DLT. The rule expands the cohort. One of the next three also has a DLT, giving
Exactly two DLTs among six trigger the third cohort in this variant. Suppose the final three participants at \(d_2\) have no DLTs. The final count is \(1+1+0=2\) DLTs among \(3+3+3=9\), or \(2/9\). Because two is no more than two, the variant escalates from \(d_2\) to the next dose level.
Under the conventional 3+3 rule, escalation would stop at \(d_2\) when the count reached \(2/6\); the trial would not enroll the third cohort there. Thus this is a decision pattern where the variants differ. The 3+3+3 rule has used three additional observations at \(d_2\), and their outcomes determine whether escalation proceeds.
Worked Example: The Third Cohort Confirms a Stopping Boundary
At a hypothetical dose \(d_2\), suppose the first three participants produce one DLT, and the next three produce one DLT. The count is \(2/6\), so the 3+3+3 rule enrolls a third cohort. Suppose one of those three also experiences a DLT. The total is
Three is greater than two, so the rule stops escalation at \(d_2\). It does not proceed to a higher dose. In the usual interpretation, if \(d_1\) is the immediately lower dose and remains eligible under the protocol, \(d_1\) may be nominated as the MTD candidate. This outcome also would have stopped escalation under conventional 3+3, which stops at \(2/6\) before the third cohort is enrolled.
Why the Added Cohort Can Change the Rule’s Decision
The added cohort resolves one specific boundary case by making the decision depend on the total of nine outcomes rather than stopping at six when the count is exactly two. For example, after \(2/6\), zero DLTs in the third cohort give \(2/9\) and escalation; one or more give at least \(3/9\) and stopping. The extra cohort therefore distinguishes these possible observed totals. It does not guarantee that the resulting decision is correct about the underlying toxicity risk.
A short probability calculation makes the dependence on the extra cohort explicit. Suppose, for illustration, that at a fixed dose each participant independently experiences a DLT with probability \(p\). The probability of exactly two DLTs among the first six is the binomial probability below. Given that count, the probability that the third cohort has zero DLTs is \((1-p)^3\), so the probability of this particular path to escalation from a \(2/6\) count is
The factor \(15p^2(1-p)^4\) accounts for exactly two DLTs among the first six; the final factor accounts for zero among the next three. This is a path probability under the stated model, not a probability that the dose is safe, and it does not by itself summarize the design’s overall behavior.
Diagnostics, Comparisons, and Limitations
Before making a decision, check that the count is for the current dose only, the cohort size matches the review point, and every outcome has been classified using the same prespecified DLT definition and assessment window. A count reported as “two DLTs” is not enough on its own: the distinction between \(2/3\), \(2/6\), and \(2/9\) is essential. Also check that the decision follows the correct branch. In this version, exactly \(2/6\) triggers expansion; \(3/6\) stops immediately.
The chief difference from conventional 3+3 is narrow but consequential. Both rules escalate at \(0/3\), expand after \(1/3\), and escalate after \(0/6\) or \(1/6\). Both stop at \(2/3\) or more and at \(3/6\) or more. They differ at exactly \(2/6\): conventional 3+3 stops, while this variant enrolls three more and then escalates for \(0\), \(1\), or \(2\) total DLTs among nine, stopping for \(3\) or more. The added enrollment is concentrated in this boundary case rather than applied at every dose.
The variant remains a rule-based dose-escalation procedure. Its count thresholds do not estimate a dose-response curve, account for uncertainty through a model, or ensure that the selected dose has a DLT probability near a particular target. The result can also be sensitive to the particular participants enrolled: a different set of outcomes can send the same trial down a different path. As with any fixed-cohort rule, its interpretation depends on consistent DLT assessment and faithful application of the protocol.
Check Your Understanding
Use the specific 3+3+3 rules in this tutorial. Each question is followed by its solution.
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At a dose, the first three participants have no DLTs. What is the next action?
Solution: Escalate to the next dose, because the count is \(0/3\). -
At a dose, the first three include one DLT, and the next three include no DLTs. What is the decision after six participants?
Solution: The total is \(1/6\). Escalate, because the count is no more than one. -
At a dose, the first six participants have two DLTs. What does this variant do, and how does conventional 3+3 differ?
Solution: This variant enrolls a third cohort of three at the same dose. Conventional 3+3 stops escalation at that dose at \(2/6\). -
After a \(2/6\) count, the third cohort has two DLTs. What is the final count and decision?
Solution: The total is \(2+2=4\) DLTs among \(6+3=9\) participants. Since \(4\geq3\), stop escalation at that dose. -
Why must DLT counts at separate dose levels not be added together when applying this rule?
Solution: Each decision boundary is defined for participants treated at one particular dose. Combining dose levels would not give the count required by the rule and could produce the wrong escalation decision.