Fisher Randomization Inference
Fisher randomization inference tests a causal hypothesis by comparing the observed statistic with the values it would take under assignments allowed by the experimental design.
Sharp null hypothesis
The sharp null states that treatment has no effect on any unit:
\[\boxed{H_0:Y_i(1)=Y_i(0)\quad\forall i}.\]It is sharp because it reveals every missing potential outcome under the null. Each unit’s outcome would remain the same under every allowed assignment.
The sharp null is stronger than the average null
\[H_0:ATE=0,\]which permits positive and negative individual effects to cancel and does not impute all missing potential outcomes.
Test statistic
A common statistic is
\[T(W,Y)=\bar Y_1-\bar Y_0.\]Other statistics may be chosen before inspecting treatment effects, including rank statistics or studentized differences.
Randomization distribution
For a two-sided test, the randomization $p$-value is
\[p= P\left( |T(W,Y)|\ge |T^{obs}| \mid H_0 \right),\]where the probability is taken over the known assignment mechanism.
Procedure
- State the sharp null and choose $T$.
- Hold the observed outcomes fixed under the null.
- Enumerate all allowed assignments, or sample many assignments from the actual mechanism.
- Recalculate $T$ for each assignment.
- Count the proportion at least as extreme as $T^{obs}$.
For Monte Carlo randomization tests, a common finite-simulation estimate is
\[\hat p=\frac{1+\#\{b:|T_b|\ge|T^{obs}|\}}{B+1},\]which includes the observed assignment and avoids a reported value of zero.
Fisher versus Neyman
| Question | Fisher randomization inference | Neyman repeated-sampling inference |
|---|---|---|
| Typical null or target | Sharp null for every unit | Average treatment effect |
| Main output | Randomization $p$-value | Estimate, standard error, confidence interval |
| Randomness | Assignment mechanism | Assignment mechanism |
| Missing outcomes | Imputed under sharp null | Their variance contribution is bounded conservatively |
The approaches answer related but different questions; one result does not mechanically determine the other.