Clinical Trial Safety and Dose Finding
3+3 Design Table: Probability of Escalating at Each True DLT Rate
Exact operating characteristics of the 3+3 design at a single dose, and a calculator for a whole escalation scheme.
- DLT rate 1% to 70%
- Exact probabilities
- Whole-trial calculator
3+3 Design Probabilities
Rows: true DLT rate at the dose · Columns: outcome probabilities for a cohort of 3, the expansion to 6, the overall decision and the expected number of patients treated.
| First cohort | Expansion | Overall decision | ||||
|---|---|---|---|---|---|---|
| True DLT rate | 0 of 3 | 1 of 3 | 1 of 3, then 0 of 3 | Escalate | Stop | Expected patients |
| 0.01 | 0.9703 | 0.0294 | 0.0285 | 0.9988 | 0.0012 | 3.09 |
| 0.02 | 0.9412 | 0.0576 | 0.0542 | 0.9954 | 0.0046 | 3.17 |
| 0.05 | 0.8574 | 0.1354 | 0.1161 | 0.9734 | 0.0266 | 3.41 |
| 0.10 | 0.7290 | 0.2430 | 0.1771 | 0.9061 | 0.0939 | 3.73 |
| 0.15 | 0.6141 | 0.3251 | 0.1997 | 0.8138 | 0.1862 | 3.98 |
| 0.20 | 0.5120 | 0.3840 | 0.1966 | 0.7086 | 0.2914 | 4.15 |
| 0.25 | 0.4219 | 0.4219 | 0.1780 | 0.5999 | 0.4001 | 4.27 |
| 0.30 | 0.3430 | 0.4410 | 0.1513 | 0.4943 | 0.5057 | 4.32 |
| 0.33 | 0.3008 | 0.4444 | 0.1337 | 0.4344 | 0.5656 | 4.33 |
| 0.35 | 0.2746 | 0.4436 | 0.1218 | 0.3965 | 0.6035 | 4.33 |
| 0.40 | 0.2160 | 0.4320 | 0.0933 | 0.3093 | 0.6907 | 4.30 |
| 0.45 | 0.1664 | 0.4084 | 0.0679 | 0.2343 | 0.7657 | 4.23 |
| 0.50 | 0.1250 | 0.3750 | 0.0469 | 0.1719 | 0.8281 | 4.12 |
| 0.60 | 0.0640 | 0.2880 | 0.0184 | 0.0824 | 0.9176 | 3.86 |
| 0.70 | 0.0270 | 0.1890 | 0.0051 | 0.0321 | 0.9679 | 3.57 |
Tip: click any value to highlight its row and column.
How to read this table
- Treat 3 patients. With 0 DLTs, escalate. With 1 DLT, add 3 more and escalate only if none of them has a DLT (1 of 6 overall). With 2 or more DLTs, stop.
- Find the row for the true DLT rate at the dose.
- "Escalate" is the chance the design moves past the dose; "Stop" is the chance it declares the dose too toxic.
Worked example
If a dose has a true DLT rate of 30%, the 3+3 design still escalates past it with probability 0.4943 and stops there with probability 0.5057. So a dose at the top of the usual 20–33% target range is passed almost half the time, which is one reason model-based designs such as the CRM and BOIN are preferred.
Click the underlined links to highlight the value in the table.
Calculator
For values between the rows and columns of the table. Results update as you type.
Formula and how the values were computed
Probabilities follow directly from the binomial distribution: escalate = (1 − p)³ + 3p(1 − p)² × (1 − p)³, and the expected number treated is 3 + 3 × 3p(1 − p)². The calculator multiplies these across dose levels, choosing as MTD the dose below the one where escalation stops.
Frequently asked questions
Which version of the 3+3 does this use?
The common version: escalate on 0/3 or 1/6 DLTs and stop on 2 or more. Some protocols also require 6 patients at the declared MTD; the table describes the escalation decision at a single dose, which is the same in both.
What target DLT rate does the 3+3 imply?
It does not have a fixed target, but it tends to choose doses with DLT rates around 20% or lower and has a high chance of stopping early.