Math 2.0

Statistics Definition

Kruskal–Wallis test

Builds on Statistical hypothesis testIndependent and identically distributed random variablesChi-squared distributionConvergence in distribution

Definition

Let k≥2k \ge 2, and for g=1,…,kg = 1, \dots, k let Xg1,…,XgngX_{g1}, \dots, X_{g n_g} be i.i.d. real random variables with a continuous distribution function FgF_g, the kk samples being independent. Let N=n1+⋯+nkN = n_1 + \dots + n_k. Rank all NN observations together from 11 to NN, let RgR_g be the sum of the ranks in sample gg, and let Rˉg=Rg/ng\bar R_g = R_g / n_g. The Kruskal–Wallis statistic is H=12N(N+1)∑g=1kng(Rˉg−N+12)2=12N(N+1)∑g=1kRg2ng−3(N+1).H = \frac{12}{N(N+1)} \sum_{g=1}^{k} n_g \Big( \bar R_g - \frac{N + 1}{2} \Big)^2 = \frac{12}{N(N+1)} \sum_{g=1}^{k} \frac{R_g^2}{n_g} - 3(N + 1). The Kruskal–Wallis test of H0 ⁣:F1=⋯=FkH_0 \colon F_1 = \dots = F_k rejects H0H_0 when HH is large.

Remarks

Under H0H_0 the law of HH depends only on n1,…,nkn_1, \dots, n_k. As all ng→∞n_g \to \infty, HH converges in distribution to the chi-squared distribution with k−1k - 1 degrees of freedom, which gives approximate critical values. The number (N+1)/2(N+1)/2 is the mean of the ranks 1,…,N1, \dots, N.

Example: k=2k = 2, samples (1,2)(1, 2) and (3,4)(3, 4), so N=4N = 4, R1=3R_1 = 3, R2=7R_2 = 7. Then H=1220(92+492)−15=17.4−15=2.4H = \frac{12}{20}\big(\frac{9}{2} + \frac{49}{2}\big) - 15 = 17.4 - 15 = 2.4. The first form gives the same: 1220(2(1.5−2.5)2+2(3.5−2.5)2)=0.6⋅4=2.4\frac{12}{20}\big(2(1.5 - 2.5)^2 + 2(3.5 - 2.5)^2\big) = 0.6 \cdot 4 = 2.4.

Non-example: repeated measurements on the same subjects under kk conditions are not independent samples. The clause on independence fails; the Friedman test is used for such blocked data.

Reading path

To understand this page from the ground up, read these 80 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. Indicator function definition
  32. Injective function definition
  33. Preimage of a set definition
  34. Sigma-algebra definition
  35. Countable set definition
  36. Event definition
  37. Borel sigma-algebra definition
  38. Probability axioms concept
  39. Probability measure definition
  40. Dirac measure definition
  41. Probability distribution definition
  42. Probability space definition
  43. Conditional probability definition
  44. Random variable definition
  45. Discrete random variable definition
  46. Cumulative distribution function definition
  47. Independence of two events definition
  48. Law of a random variable definition
  49. Random variable with a density definition
  50. Convergence in distribution definition
  51. Joint probability distribution definition
  52. Independence of two random variables definition
  53. Mutual independence of events definition
  54. Expected value definition
  55. Probability density function definition
  56. Identically distributed random variables definition
  57. Marginal distribution definition
  58. Variance definition
  59. Independent sigma-algebras definition
  60. Linearity of expectation statement
  61. Density of the normal distribution definition
  62. Exponential distribution definition
  63. Independence means the joint law is the product of the marginals statement
  64. Independence of random variables definition
  65. Covariance definition
  66. Normal distribution definition
  67. Gamma distribution definition
  68. Expectation of a product of independent random variables statement
  69. Variance of a sum statement
  70. Independent and identically distributed random variables definition
  71. Independent random variables are uncorrelated statement
  72. Statistic definition
  73. Variance of a sum of independent random variables statement
  74. Statistical hypothesis test concept
  75. Estimator definition
  76. Sample mean definition
  77. Bias of an estimator definition
  78. Sample variance definition
  79. Chi-squared distribution definition
  80. Kruskal–Wallis test definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Kruskal%E2%80%93Wallis_test
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MSC62G10
DefinesKruskal–Wallis test
UsesStatistical hypothesis test, Independent and identically distributed random variables, Chi-squared distribution, Convergence in distribution

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