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Allocate the family budget
Let the family target be alpha and let m be the number of planned hypotheses. Give each test a rejection rule with true-null probability at most alpha/m. The union bound adds the false-rejection probabilities for the true nulls; their sum cannot exceed m × alpha/m = alpha. Independence is unnecessary for this inequality.
Use valid p-values consistently
Compare a valid raw p-value with the divided threshold, or the capped adjusted value with alpha.
| Raw p | Adjusted p | Meets adjusted rule? |
|---|---|---|
| 0.001 | 0.02 | Yes |
| 0.004 | 0.08 | No |
| 0.2 | 1 | No |
Do not divide the observed p-value
Dividing p by m would make rejection easier as more tests are added. The correction instead tightens the threshold or increases the adjusted probability. Preserve the stated p ≤ threshold convention and adequate numerical precision.
Check the inputs behind the bound
The bound requires valid individual tests and a properly defined family. It does not repair invalid p-values caused by ignored selection or violated assumptions. It can be conservative, so report the error target and method rather than claiming that the corrected findings must all be true.