17.5.6.2 Algorithms (Kruskal-Wallis ANOVA)
The procedure below draws on NAG algorithms.
- Rank all observation in ascending order. Average ranks are assigned to tied scores
- Sum up the ranks of observations in each group, to give the ranks sums
/math-6cf4401f49e69d4a93f489104fd34474.png?v=0 "R_i \,\!")
, for /math-4ef08e127bbc112d645d462efa9f3b5d.png?v=0 "i=1,2,\ldots ,k \,\!")
- The Kruskal-Wallis test statistic
/math-a2596afee8670c8ad332ca72d2d1b35d.png?v=0 "H \,\!")
is computed as: /math-070e48b5443fd11c01cd49ccfc8263c6.png?v=0 "H=\frac{12}{N(N+1)}\sum_{i=1}^k\frac{R_i^2}{l_i}-3(N+1) \,\!")
where /math-4b382581f2f870c7449baa3e9c477efb.png?v=0 "N=\sum_{i=1}^kl_i\,\!")
, i.e. /math-dddfbab30f141da749dec815073642cc.png?v=0 "N \,\!")
is the total number of observations. If there are tied scores, is corrected by dividing by /math-b3d4d467162e11667188bdb8fd11355c.png?v=0 "1-\frac{\sum (t^3-t)}{N^3-N}\,\!")
where /math-444ded531df49dd5e9d60e42154722ce.png?v=0 "t \,\!")
is the number of tied scores in a group and the summation is overall tied groups.
The significance level is based on the /math-c15009ca5cbb034d023f145a4ed23b8d.png?v=0 "\chi ^2\,\!")
distribution, with /math-777fa234183c06685e04a5579229be52.png?v=0 "k-1 \,\!")
degree of freedom.
For more details of the algorithm, please refer to nag_kruskal_wallis_test (g08afc)