Math 2.0

Statistics Definition

Family-wise error rate

Builds on Statistical hypothesis testType I and type II errors

Definition

Let H1,…,HmH_1, \dots, H_m be null hypotheses about the law PP of the data, tested by a procedure that rejects a random subset of them. Let I0(P)⊆{1,…,m}I_0(P) \subseteq \set{1, \dots, m} be the set of indices ii such that HiH_i is true under PP. Let VV be the number of rejected HiH_i with i∈I0(P)i \in I_0(P), the number of type I errors. The family-wise error rate is FWER=P(V≥1)\mathrm{FWER} = P(V \ge 1). The procedure controls the FWER strongly at level α\alpha if P(V≥1)≤αP(V \ge 1) \le \alpha for every PP in the model. It controls the FWER weakly at level α\alpha if this holds for every PP under which all HiH_i are true.

Remarks

Strong control implies weak control. Example: if mm tests of exact level α\alpha are independent and all null hypotheses are true, then FWER=1−(1−α)m\mathrm{FWER} = 1 - (1 - \alpha)^m. So testing each hypothesis at level α\alpha does not control the FWER at level α\alpha for m≥2m \ge 2; this is the non-example. The Bonferroni correction tests each hypothesis at level α/m\alpha/m and gives strong control.

Reading path

To understand this page from the ground up, read these 51 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Ordered pair definition
  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Order on the reals definition
  27. Power set definition
  28. Complement definition
  29. Function definition
  30. Interval definition
  31. Preimage of a set definition
  32. Sigma-algebra definition
  33. Event definition
  34. Borel sigma-algebra definition
  35. Probability axioms concept
  36. Probability measure definition
  37. Probability distribution definition
  38. Probability space definition
  39. Conditional probability definition
  40. Random variable definition
  41. Independence of two events definition
  42. Law of a random variable definition
  43. Mutual independence of events definition
  44. Identically distributed random variables definition
  45. Independent sigma-algebras definition
  46. Independence of random variables definition
  47. Independent and identically distributed random variables definition
  48. Statistic definition
  49. Statistical hypothesis test concept
  50. Type I and type II errors definition
  51. Family-wise error rate definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Family-wise_error_rate
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WritingW0 stub
MSC62J15
DefinesFamily-wise error rate
UsesStatistical hypothesis test, Type I and type II errors

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