Why Small Rule Errors Matter
The conventional 3+3 design is simple to state, but its decisions depend on keeping several counts and actions distinct. A cohort may expand because one participant experiences a dose-limiting toxicity (DLT); it does not escalate at that point. A dose may qualify as the maximum tolerated dose (MTD) even if only three evaluable participants were treated there. And replacing an unevaluable participant is not the same as adding three more participants to a cohort.
These distinctions matter because the algorithm is a sequence of protocol-defined decisions, not a general rule to “move up when the dose seems safe” or “move down when toxicity occurs.” This tutorial uses the conventional algorithm with ordered dose levels and completed DLT observation windows. A study protocol must define the DLT, observation window, evaluability, and any replacement rules before the algorithm is applied.
“Evaluable” here means evaluable for the specified DLT assessment under the protocol. The rules are applied to participants whose relevant DLT outcomes are available and classifiable; pending or unevaluable cases cannot simply be counted as having no DLT. The conventional sequence also presumes that decisions are made after the relevant outcomes are known, rather than escalating while the cohort’s DLT assessments remain unresolved.
Follow the Decision Sequence, Not an Intuition
The important branches can be summarized by the observed count at the current dose. The expansion branch is triggered by exactly 1 DLT among the first 3 evaluable participants. Once six have been assessed at that dose, the decision uses the total across all six—not the count among only the last three.
At the current dose, wait for three evaluable DLT outcomes. If the count is 0, move up one dose level. If it is 1, enroll three more at the same dose. If it is 2 or 3, stop escalation at this dose.
After the additional three participants are evaluable, combine their outcomes with the first three. A total of 0 or 1 DLT among 6 permits escalation; a total of 2 or more stops escalation at that dose.
When escalation stops at a dose because its stopping criterion is met, the operational MTD is commonly the highest tested dose that meets the protocol’s tolerability rule, provided the next higher tested dose meets the stopping criterion.
The last step does not require every dose below the stopping dose to have been expanded to six. Under the algorithm, a dose with 0 DLTs among 3 qualifies under the common tolerability rule of no more than 1 DLT among the evaluable participants treated there, up to six. Expansion is only required when the first three include exactly 1 DLT.
Worked Decisions
Worked Example: Zero of Three Before a Toxic Dose
In a hypothetical trial, the first dose is 10 mg and the next dose is 20 mg. At 10 mg, 0 of 3 evaluable participants experience a DLT. At 20 mg, 2 of the first 3 evaluable participants experience a DLT. Apply the conventional rule and identify the operational MTD.
At 10 mg, the observed proportion is \(0/3=0\%\). The rule for 0 DLTs among 3 is to escalate, so the trial proceeds to 20 mg. No additional participants are required at 10 mg.
At 20 mg, the observed proportion is \(2/3=66.7\%\). Because 2 or more of the first 3 experience DLTs, escalation stops at 20 mg; there is no expansion at that dose.
Under the stated conventional MTD rule, 10 mg is the highest tested dose that meets the tolerability criterion: it has 0 DLTs among 3, which is no more than 1 among the evaluable participants treated there. The next higher tested dose, 20 mg, meets the stopping criterion with 2 DLTs among 3. Thus the operational MTD is 10 mg. Requiring six participants at 10 mg would incorrectly alter the algorithm.
Worked Example: Expansion and the Total of Six
In a hypothetical trial, 5 mg has 0 DLTs among 3 evaluable participants. At 10 mg, 1 of the first 3 has a DLT. After three additional evaluable participants are treated at 10 mg, none of those additional three has a DLT. At 20 mg, the first 3 include 1 DLT, and after expansion the total at 20 mg is 2 DLTs among 6. What happens at each dose, and what is the MTD?
At 5 mg, \(0/3=0\%\), so escalation to 10 mg is allowed. At 10 mg, \(1/3=33.3\%\), so the correct action is to add three participants at 10 mg, not to escalate immediately. The total there is \(1+0=1\) DLT among 6, or \(1/6=16.7\%\), so escalation to 20 mg is allowed.
At 20 mg, 1 DLT among the first 3 triggers expansion. The final count is 2 among 6, or \(2/6=33.3\%\). Since the total is at least 2, escalation stops at 20 mg. The 10 mg dose meets the common tolerability rule with 1 DLT among 6, and the next higher dose meets the stopping criterion. Therefore, 10 mg is the operational MTD.
The key calculation at 20 mg is the cumulative total, \(1+1=2\), not just the number of DLTs in the second set of three. Looking only at the additional participants would miss the stopping rule.
Worked Example: Replacement Is Not Expansion
In a hypothetical protocol, a participant who does not complete the required DLT observation window is classified as unevaluable and replaced according to a prespecified rule. At 8 mg, the first three enrolled participants include one unevaluable participant and two evaluable participants, neither of whom has a DLT. A replacement participant completes the assessment without a DLT. What is the decision at 8 mg?
The two evaluable participants and the replacement give 3 evaluable outcomes with 0 DLTs. The unevaluable participant is not counted as a fourth DLT-free outcome, and the replacement is not an expansion cohort. The evaluable count is \(0/3\), so the conventional rule permits escalation to the next dose.
The participant who was unevaluable remains part of the trial’s safety records; “not evaluable for this rule” does not mean “no toxicity occurred.” Whether a participant can be replaced, and how that person’s information is handled, must follow the protocol. The relevant distinction for this decision is that replacement completes the required evaluable count, whereas expansion adds three participants after 1 DLT among the first 3.
Why the MTD Rule Follows the Algorithm
The selection rule can be justified directly from the decision sequence. Suppose escalation stops at dose \(d_{j+1}\) because that dose reaches the stopping criterion. The algorithm has tested \(d_{j+1}\) as too toxic under its operational rule, so it does not continue upward. The candidate immediately below is \(d_j\); it qualifies if its observed DLT count is within the protocol’s tolerability limit.
This is an operational selection based on the observed cohort outcomes and the specified algorithm. It is not a claim that the selected dose has a particular true DLT probability, nor that it is the only clinically acceptable dose. If the study stops for another reason, or the dose pattern does not provide the usual higher-dose stopping boundary, the protocol’s selection rule determines what can be reported; the ordinary boundary-based description should not be asserted without its conditions.
Common Errors and How to Detect Them
- Escalating after 1 of 3. This skips the required expansion. The correction is to treat three additional evaluable participants at the same dose, then use the total among all six.
- Expanding after 0 of 3. This adds participants when the conventional rule calls for escalation. Do not impose a six-person cohort at every dose; expansion is triggered by exactly 1 DLT among the first 3.
- Using only the second set of three after expansion. The decision after expansion uses the combined count among six. Record the first-three count and the added-three count, then sum them.
- Calling the dose with 2 or more DLTs the MTD. That is the dose at which escalation stops, not usually the selected MTD. The candidate is typically the highest tested dose below it that meets the tolerability rule.
- Requiring six participants at every candidate MTD. A dose with 0 DLTs among 3 can meet the common tolerability rule. Check the protocol’s criterion and the actual evaluable denominator rather than assuming all doses must have six.
- Treating replacement as expansion or as a DLT-free result. Replacement addresses an evaluability rule; it does not automatically add three people, and the unevaluable participant should not be counted as having no DLT.
- Automatically assigning the next cohort to a lower dose. The conventional algorithm stops escalation when its stopping criterion is met; it does not, by itself, prescribe a new cohort at a lower level. Any such action must be specified by the protocol.
A practical diagnostic is to write, for each dose, the number of evaluable participants and the cumulative number of DLTs before recording the action. If a decision cannot be reproduced from those two counts and the rule, check whether outcomes are pending, whether a replacement rule was applied, or whether a protocol-specific modification is being used.
Comparison and Limits
The 3+3 algorithm is a rule-based design: each action follows from a small set of observed counts. That makes its decisions transparent and easy to audit. A common alternative error is an informal “three at every dose” approach, which is not the conventional algorithm because it ignores mandatory expansion after 1 of 3. Conversely, treating six as mandatory at every dose is also a different procedure.
Model-based dose-finding designs use a specified statistical model and may use accumulating dose-toxicity information differently; their dose recommendations are not obtained by applying the 3+3 branches. Even within rule-based designs, protocol modifications can change expansion, evaluability, or stopping rules. Therefore, the written protocol takes precedence over a generic version of the algorithm.
The conventional algorithm is deliberately limited: a small number of observed DLTs controls each decision, and the operational MTD depends on the tested doses and the stopping boundary. It should not be interpreted as a precise estimate of the dose-toxicity curve. Careful application prevents avoidable procedural errors, but it does not remove the design’s statistical limitations.
Check Your Understanding
For each question, apply the stated conventional 3+3 rule. Solutions follow the questions.
- At 12 mg, 0 of the first 3 evaluable participants have a DLT. What is the next action, and must the dose be expanded?
- At 24 mg, 1 of the first 3 has a DLT. The next three include 1 DLT. What is the final count and decision at 24 mg?
- A trial has 0 of 3 DLTs at 6 mg and 2 of 3 at 12 mg. Under the common operational rule described here, which dose is selected as the MTD, and why is a six-person cohort at 6 mg not required?
- After 1 DLT among the first 3 at a dose, an analyst counts only the additional three participants and observes 0 DLTs. What is the mistake, and what count should determine the decision?
Solutions. (1) Escalate to the next dose: \(0/3=0\), so expansion is not required. (2) The total is \(1+1=2\) DLTs among 6, so escalation stops at 24 mg. (3) The selected dose is 6 mg: it meets the tolerability rule with \(0/3\), while 12 mg meets the stopping criterion with \(2/3\). The algorithm does not require expansion after 0 of 3. (4) The analyst ignored the first three. The correct total is 1 DLT among 6, so escalation is allowed.