The 3+3 Decision Table

Why a Decision Table Helps

The conventional 3+3 design makes dose-level decisions using a small set of rules applied to observed dose-limiting toxicities (DLTs). Its simplicity can make the rules seem obvious, but a decision depends on which participants have been observed and when. In particular, the six-participant decision is conditional: a dose is expanded to six only after exactly one DLT among the first three participants. Therefore, the possible cumulative DLT totals at that decision are not every integer from zero through six.

This tutorial makes each branch explicit. The DLT definition and observation window are set by the protocol; the earlier material on defining DLTs and observation windows establishes why a DLT must be assessed over a specified period. Here, the task is to translate the conventional algorithm into a complete table and apply it without confusing the added cohort’s count with the cumulative count.

The Complete Decision Table

The table has two stages. The first decision is made after three participants at a dose have completed the protocol-defined DLT assessment. A count of zero leads to escalation to the next dose, if one is available. A count of one leads to expansion at the current dose. A count of two or three leads to no escalation from that dose; the cohort is not expanded under this rule.

Only the one-DLT branch reaches a six-participant decision. The added three participants are assessed at the same dose. Their DLT count, \(A_3\), is added to the one DLT already observed, so \(D_6=1+A_3\). Consequently, \(D_6\) can be 1, 2, 3, or 4. A cumulative total of one leads to escalation; totals of two through four lead to no escalation from the current dose.

The six-participant rows are conditional on \(D_3=1\), not a general classification of any group of six. Under this procedure, \(D_6=0\) is unreachable after expansion because one DLT is already known to have occurred. Likewise, \(D_6=5\) and \(D_6=6\) are unreachable in this branch: only three additional participants are treated, so the largest possible cumulative count is four. Such totals should not be added as ordinary decision rows for this conditional table.

Why the Six-Participant Totals Are Restricted

The restriction follows directly from how the six-person cohort is formed. Expansion occurs only when the first three participants have exactly one DLT. Among the three additional participants, the DLT count can be any integer from zero through three. Adding that count to the known first-cohort count gives:

The escalation condition at six is \(D_6=1\). Substituting \(D_6=1+A_3\), this is equivalent to \(A_3=0\). If even one additional DLT occurs, then \(A_3\ge1\), so \(D_6\ge2\) and the rule gives no escalation from that dose. This is a deterministic classification of observed outcomes; it is not a probability statement about future toxicity.

Operational Conditions and Checks

A table gives an unambiguous action only when the data entering it are well defined. The conventional rule assumes that the dose level and cohort are identifiable, that each participant’s DLT status is assessed according to the protocol, and that the applicable observation period has been completed before the relevant decision is made. Participants with pending or unevaluable safety follow-up cannot simply be treated as having no DLT; how to handle them must be specified by the protocol.

The DLT count is a count of participants meeting the protocol’s DLT criterion, not a count of all adverse events. If one participant experiences several qualifying events, the participant still contributes one DLT to the participant count. Conversely, whether a particular event qualifies is a clinical adjudication governed by the protocol’s definition. The decision table cannot repair an inconsistent DLT definition or incomplete outcome information.

Before applying a row, check three items:

• Has the correct number of participants at this dose completed the required assessment?
• Is the count for the first three, the additional three, or all six?
• Was expansion actually triggered by exactly one DLT in the first three?

These checks also expose a common data-summary problem. A report might state “one DLT in the six-person cohort” when it means one in the added three, or it might state “one DLT among six” when it means the cumulative total. Those are not interchangeable summaries. Record \(D_3\), \(A_3\), and \(D_6\) separately whenever the expansion branch is used.

How This Rule-Based Approach Differs From Alternatives

The conventional 3+3 table is a deterministic rule: once the observed DLT count and decision stage are known, the action is fixed. It does not require estimating a dose-toxicity curve to apply the table. This makes the immediate cohort decision easy to communicate and audit.

Model-based dose-finding approaches instead use a statistical model to relate dose to toxicity and use accumulated data to guide choices. Their decisions depend on model assumptions and the analysis rule, rather than only on the small set of count categories in this table. Neither description makes a design automatically suitable for every trial. The decision table’s simplicity does not mean it summarizes all uncertainty about toxicity, and model-based methods require their own assumptions and checks. The comparison here is about how decisions are generated, not a claim that one approach is universally preferable.

The table also has operational limits. It describes escalation decisions at a dose, but it does not by itself define every action at the top of a dose ladder, specify how unevaluable participants are replaced, or resolve unusual clinical circumstances. A protocol must state how such cases are handled. At the highest planned dose, “escalate” cannot mean assigning participants to a nonexistent next dose; the planned dose range places a boundary on the rule.

Common Mistake: Treating Six as an Unconditional Cohort

Wrong reasoning: “At six participants the DLT count could be any number from zero to six, so the table should have rows for all seven totals.”

Correction: The design reaches six participants at a dose only after exactly one DLT in the first three. If \(A_3\) is the number among the added three, then \(D_6=1+A_3\), with \(A_3\in\{0,1,2,3\}\). Thus the reachable cumulative totals are 1, 2, 3, and 4. A total of 1 permits escalation; totals 2 through 4 do not. Counts 0, 5, and 6 are unreachable in this branch.