From Participant Outcomes to Dose Decisions
A 3+3 trial turns observed dose-limiting toxicities (DLTs) into a sequence of cohort decisions. To follow that sequence correctly, it is not enough to remember a rule such as “expand after one DLT.” The investigator must also track which participants have completed the DLT observation window, whether a replacement is required, and which outcomes belong to the current dose. A decision made before the required outcomes are available can send the trial down the wrong dose path.
This tutorial traces participant outcomes under the conventional 3+3 decision rules. It assumes dose levels have been specified in advance and that the protocol defines the DLT and its observation window. Each example is hypothetical. The rules determine a dose transition from the observed counts; by themselves, they do not establish that a dose is safe or effective for a broader patient population.
The Decision Rule Used in a Trace
At a given dose level, the initial decision cohort consists of three evaluable participants. If none has a DLT, the trial proceeds to the next dose level, if one is available. If one has a DLT, three additional evaluable participants are treated at the same dose before the decision is made. The six outcomes are then counted together. With no more than one DLT among six, the rule permits escalation; with two or more among the first three or six, escalation stops at that dose. At the highest planned dose, escalation is impossible regardless of the count.
This is an operational description of the conventional rule, not a statistical model for individual risk. In particular, a cohort decision is made only when the participants needed for that decision have evaluable outcomes. A participant still within the DLT window does not yet contribute a final no-DLT outcome.
For example, \(1/3\) is not a final escalation decision: it triggers expansion. The next decision uses the combined six-person count, not the outcome of the added cohort in isolation. If the initial three have one DLT and the added three have none, the total is \(1+0=1\) DLT among six, so escalation follows. If the added cohort has one DLT, the total is \(1+1=2\) among six, so escalation stops.
How to Keep the Trace Straight
A reliable trace records dose, participant outcomes, evaluability, the current cohort count, and the action taken. A participant-level indicator can make arithmetic transparent: write 1 for a DLT and 0 for no DLT, but only for participants whose outcomes are evaluable. The cohort total is the sum of those indicators. Do not combine participants from different doses, and do not count an unevaluable participant as a zero.
Confirm which participants have completed the required DLT assessment and can be classified.
Record evaluable DLTs among the initial three. If one DLT is observed, retain that count and add three more participants at the same dose.
After expansion, sum DLTs across all six evaluable participants at that dose. Record escalation or stopping from the combined count.
Write the next dose explicitly, or state that escalation stops. If no higher planned dose exists, record that escalation cannot proceed.
Worked Example: One DLT Followed by Expansion
Consider three planned doses, D1, D2, and D3. All participants complete the same protocol-defined DLT window, and the table records their classified outcomes. What sequence of decisions follows?
| Dose | Initial three outcomes | Expansion outcomes, if needed | Decision |
|---|---|---|---|
| D1 | 0, 0, 0 | Not needed | Escalate to D2 |
| D2 | 0, 1, 0 | 0, 0, 0 | Escalate to D3 |
| D3 | 1, 0, 1 | Not needed | Stop escalation at D3 |
At D1, the count is \(0+0+0=0\) DLTs among three, so the rule permits escalation to D2. At D2, the initial count is \(0+1+0=1\) DLT among three. This triggers three additional participants at D2; it does not trigger escalation yet. The combined D2 count is \(0+1+0+0+0+0=1\) DLT among six, so the rule permits escalation to D3.
At D3, the initial count is \(1+0+1=2\) DLTs among three. The stopping branch applies, so no expansion or further escalation is indicated by this rule. The trace ends with escalation stopped at D3. It does not, by itself, specify which dose should ultimately be selected; that is a separate dose-selection question.
Why Expansion Uses the Combined Count
The expansion branch can be justified directly from the decision rule. After one DLT in the initial three, the trial gathers three more outcomes at the same dose. If \(x\) DLTs occur in those additional three, then the total among six is \(1+x\). The escalation branch allows no more than one DLT in six. Since \(x\) can only be 0, 1, 2, or 3, that condition holds exactly when \(x=0\).
Thus, after one DLT in the initial cohort, zero DLTs in the expansion gives one in six and permits escalation; one or more in the expansion gives at least two in six and stops escalation. This arithmetic explains why the expansion outcomes cannot be interpreted separately from the initial three.
Worked Example: A Replacement Must Be Evaluated Before the Decision
At hypothetical dose D2, three participants begin treatment. Participant A completes the DLT window without a DLT, and participant B has a DLT. Participant C withdraws before the window is complete, so the protocol classifies C as unevaluable and requires a replacement. The replacement, participant R, completes the window without a DLT. What is the initial cohort count, and what happens next?
The three evaluable outcomes for the initial decision are A = 0, B = 1, and R = 0. Participant C is excluded from the DLT count; C is neither a DLT nor a no-DLT outcome. The replacement has completed the assessment, so the required three evaluable outcomes are now available.
The correct action is to enroll three additional participants at D2 and wait for their DLT assessments. It would be incorrect to make the decision using only A and B, or to call C a no-DLT outcome. It would also be incorrect to say that three evaluable outcomes are available and then wait for another replacement before deciding: in this scenario R is the required replacement and supplies the third evaluable outcome.
Suppose the three additional participants later have outcomes 0, 1, and 0. The combined count is \(0+1+0+0+1+0=2\) DLTs among six. Therefore escalation stops at D2. The replacement is part of the initial three evaluable outcomes, while the three added participants form the expansion cohort.
Worked Example: Two DLTs in the Expansion Cohort
At hypothetical dose D4, the initial three outcomes are 0, 1, and 0. Three more participants are enrolled at D4 and have outcomes 1, 0, and 1. Determine the total and the next action.
The initial count is \(0+1+0=1\) DLT among three, so expansion at D4 is required. After the added cohort completes assessment, count all six outcomes in their participant-level order:
Because three is at least two, the rule stops escalation at D4. Notice that the expansion cohort alone has two DLTs, but the conclusion is not based on an unlabelled count from that cohort. The full dose-level total is three DLTs among six. Writing out the six indicators helps prevent accidental miscounting or omission of a participant.
Diagnostics, Alternatives, and Limits
A trace should be auditable: another reader ought to be able to reconstruct each action from the recorded outcomes. Before moving to the next dose, check the following:
- Assessment status: Are all outcomes required for the decision complete? If not, defer the decision.
- Denominator: Is the count based on three or six evaluable participants at this dose, as appropriate?
- Arithmetic: Do the participant-level indicators sum to the stated total?
- Dose separation: Are counts from different dose levels kept separate?
- Transition: Does the recorded action match the count, and is a higher planned dose actually available?
A common mistake is to treat a withdrawal without a recorded DLT as a zero and decide from the resulting count. That reasoning confuses absence of an observed event with an outcome that could not be assessed. The correction is to follow the protocol’s evaluability and replacement rules, then decide only when the required number of evaluable outcomes is present. Another frequent arithmetic error is to add cohort totals that do not match the listed participant outcomes. Re-summing the individual indicators is a simple check.
The conventional 3+3 is a transparent, rule-based approach: for a given sequence of completed outcomes, its next action is fixed. Model-based dose-finding methods instead use a specified statistical model and an allocation or decision rule to relate outcomes across dose levels. These approaches answer related but distinct operational questions and require their own assumptions and implementation. The simple trace here does not estimate a toxicity curve, quantify uncertainty in a dose-specific risk, or account for outcomes still pending. Its decision path can also depend strongly on the small cohorts and the discrete thresholds in the rule.
The algorithm assumes that the protocol’s DLT definition, observation window, evaluability criteria, and handling of withdrawals are clear and applied consistently. If toxicity is assessed too early, if outcomes are delayed beyond the window, or if classification differs across participants, the recorded trace may not represent the intended rule. Do not substitute a time-to-event adjustment or an alternative replacement convention unless the protocol specifies it.
Check Your Understanding
For each problem, identify the count and action. Solutions are given after the questions.
- At a dose, the initial three evaluable outcomes are 0, 0, 0. What is the action?
- The initial outcomes are 1, 0, 0, and the three added outcomes are 0, 0, 0. How many DLTs are there among six, and what follows?
- The initial outcomes are 0, 1, 0. One additional participant has completed assessment, but two are still pending. Can the six-person decision be made?
- At a dose, the initial outcomes are 1, 1, 0. Should the cohort be expanded?
Solutions. (1) There are \(0+0+0=0\) DLTs among three, so escalate if a higher planned dose exists. (2) The total is \(1+0+0=1\) DLT among six, so escalation is permitted if a higher dose exists. (3) No. The expansion cohort requires three additional evaluable outcomes; wait until all three are classified, then count them together with the initial three. (4) No expansion is indicated: \(1+1+0=2\) DLTs among the initial three, so stop escalation at that dose.