Math 2.0

Statistics Definition

Fisher's exact test

Builds on Contingency tableHypergeometric distributionP-valueStatistical hypothesis test

Definition

Consider a 2×22 \times 2 contingency table of counts a,ba, b in the first row and c,dc, d in the second row, with total n=a+b+c+dn = a + b + c + d. Fix the row totals r1=a+br_1 = a + b, r2=c+dr_2 = c + d and the column totals k1=a+ck_1 = a + c, k2=b+dk_2 = b + d. Assume the null hypothesis that the row factor and the column factor are independent. Then, conditionally on these totals, aa has the hypergeometric distribution P(a)=(r1a)(r2k1−a)/(nk1).P(a) = \binom{r_1}{a} \binom{r_2}{k_1 - a} \Big/ \binom{n}{k_1}. Fisher's exact test computes the p-value as a sum of P(a′)P(a') over tables with the same totals. The sum runs over the tables at least as extreme as the observed one. The one-sided version sums over a′≥aa' \ge a; the usual two-sided version sums over all a′a' with P(a′)≤P(a)P(a') \le P(a).

Remarks

Example: in the lady tasting tea experiment, r1=r2=k1=k2=4r_1 = r_2 = k_1 = k_2 = 4, n=8n = 8, and she classifies all cups correctly, so a=4a = 4. Then P(4)=(44)(40)/(84)=1/70P(4) = \binom{4}{4}\binom{4}{0}/\binom{8}{4} = 1/70, and the one-sided p-value is 1/70≈0.0141/70 \approx 0.014. Non-example: replacing the hypergeometric law by its chi-squared approximation gives Pearson's chi-squared test, whose p-value is only approximate. The test is conservative: since the conditional law is discrete, the probability of a type I error is at most α\alpha and often smaller.

Reading path

To understand this page from the ground up, read these 68 pages in order. The foundations come first.

  1. Binomial coefficient definition
  2. Construction of the reals from Cauchy sequences concept
  3. Contingency table concept
  4. Element definition
  5. Integers object
  6. Ordered field definition
  7. Proposition concept
  8. Random experiment concept
  9. Rational numbers object
  10. Real numbers object
  11. Set definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. The rationals form a field statement
  22. Cartesian product definition
  23. Subset definition
  24. Uniqueness quantifier definition
  25. The rationals form an ordered field statement
  26. Binary relation definition
  27. Universal set definition
  28. Power set definition
  29. Order on the reals definition
  30. Function definition
  31. Complement definition
  32. Interval definition
  33. Codomain definition
  34. Injective function definition
  35. Preimage of a set definition
  36. Sigma-algebra definition
  37. Surjective function definition
  38. Countable set definition
  39. Event definition
  40. Borel sigma-algebra definition
  41. Bijective function definition
  42. Probability axioms concept
  43. Equinumerous sets definition
  44. Probability measure definition
  45. Finite set definition
  46. Probability distribution definition
  47. Probability space definition
  48. Classical definition of probability definition
  49. Conditional probability definition
  50. Random variable definition
  51. Cumulative distribution function definition
  52. Discrete random variable definition
  53. Law of a random variable definition
  54. Independence of two events definition
  55. Random variable with a density definition
  56. Probability mass function definition
  57. Identically distributed random variables definition
  58. Probability density function definition
  59. Mutual independence of events definition
  60. Hypergeometric distribution definition
  61. Continuous uniform distribution definition
  62. Independent sigma-algebras definition
  63. Independence of random variables definition
  64. Independent and identically distributed random variables definition
  65. Statistic definition
  66. Statistical hypothesis test concept
  67. P-value definition
  68. Fisher's exact test definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Fisher%27s_exact_test
About this page
TagYB9390ZE4W
URIm2://YB9390ZE4W
Kinddefinition
WritingW0 stub
MSC62H17, 62F03
DefinesFisher's exact test
UsesContingency table, Hypergeometric distribution, P-value, Statistical hypothesis test

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