The minimum value of 'p' for the KRUSKALWALLIS function can be resolved through two possible methods:
1) The KRUSKALWALLIS function calculates the value of 'p' based on ranks. So the function statistic, and therefore the p-value, is necessarily discrete. For small sample sizes, the p-value will not be able to get very small before it is zero. This is not a numerical precision issue usually present with digital computers; it is a property of the function itself.
2) The p-value is an upper tail of a cumulative distributed function (CDF). The CDF value is approximately a chi-squared distribution and is computed as 1 - CHI2CDF (...). The smallest possible value of ‘p’, before zero, is eps/2 where EPS is the floating-point relative accuracy of a number. Here is an example where it reaches that value:
rand('twister',7);
N = 1000;
x = 1:N;
k = 80;
g = [repmat(1,1,k) randsample(1:2,1000-2*k,true) repmat(2,1,k)];
[p table stats] = kruskalwallis(x,g,'off') = kruskalwallis(x,g,'off');
p
The value of 'p' would be:
p =
1.1102e-16
This is a numerical precision issue which digital computers have. Given the third output of KRUSKALWALLIS, stats, it is possible to take the value of the function and the chi-square state.
For example, if the value of the variables 'df' and 'Chi-sq' are taken to be as:
'df'=3
'Chi-sq'=126
The value of the upper tail is computed directly as:
x = 126;
v = 3;
p = gammainc(x/2,v/2,'upper')
This would give the following output:
p =
3.9353e-27
An important thing to note would be that chi-squared distribution is only an approximation and therefore will never be exact. Secondly, this is a hypothesis test, and there are very few hypothesis-testing applications where one would need to know the difference between a p-value of 1e-6 and 1e-27.
