Seventy possible experiments
The probability comes from the design, not a magical formula.
Only seventy possible assignments
Imagine eight participants were randomly assigned, four to each treatment. Their hypothetical outcomes are listed below. Under the sharp null hypothesis that treatment changes no participant's outcome, we can enumerate every one of the 70 equally likely assignments.
View all numerical outcomes
The observed hypothetical assignment is group A = [6, 7, 8, 9] and group B = [2, 3, 4, 5]. The total set of observed values is held fixed under the sharp null.
| Mean A − mean B | Number of allocations | In p-value? |
|---|
This calculation has an exact design-based interpretation only if assignment was uniformly random and the sharp null is the model being tested. In observational studies, simply permuting labels generally does not justify causal inference. The data here are fictional.