Nonparametric Analysis of Variance (ANOVA)

 

Kruskal-Wallis Nonparametric ANOV

Kruskal-Wallis Nonparametric ANOVA compares between the medians of two or more samples to determine if the samples have come from different populations.

If the distributions prove not to be normal and/or the variances are different then the Kruskal-Wallis should be used to compare the groups.

If a significant difference is found then there is a difference between the highest and lowest median.

Non-parametric multiple comparisons

Non-parametric multiple comparisons (Behrens-Fisher test procedure) will then be used to compare all possible pairs 2 samples.  

If there are 3 samples (eg. g1, g2, g3), then 3 pairs will be compared (g1-g2, g1-g3, g2-g3); if there are 4 samples, then 6 pairs will be compared.

Friedman Rank Sum Test

If the data is un-replicated complete block designs (i.e., there is exactly one observation in y for each combination of levels of groups and blocks), where normality assumption may be violated, Friedman Rank Sum Test will be performed.

Check the box “Randomized block design” and select the block factor.

The null hypothesis for Friedman Rank Sum Test is that apart from an effect of blocks, the location parameter of y is the same in each of the groups.

Below is the sample input window

 

 

 

Below is the sample output of the above:

 

Nonparametric analysis of variance

 

         Kruskal-Wallis rank sum test

 

data:  OZONE by MONTH

Kruskal-Wallis chi-squared = 29.2666, df = 4, p-value = 6.901e-06

 

                                                             

 /////////////////////////////////////////////////////////////

 / npmc executed                                             /

 /                                                           /

 / NOTE:                                                     /

 / -Used Satterthwaite t-approximation (df=42.2997241588321) /

 / -Calculated simultaneous (1-0.05) confidence intervals    /

 / -The one-sided tests 'a-b' reject if group 'a' tends to   /

 /  smaller values than group 'b'                            /

 /////////////////////////////////////////////////////////////

 

 

$`Data-structure`

  group.index class.level nobs

5           1           5   31

6           2           6   30

7           3           7   31

8           4           8   31

9           5           9   30

 

$`Results of the multiple Behrens-Fisher-Test`

   cmp    effect    lower.cl  upper.cl   p.value.1s   p.value.2s

1  1-2 0.8634409 0.726919876 0.9999618 5.864368e-09 4.495918e-09

2  1-3 0.7507804 0.560766565 0.9407943 2.240692e-03 4.411993e-03

3  1-4 0.7320499 0.542366215 0.9217337 4.969701e-03 9.470567e-03

4  1-5 0.5387097 0.320429504 0.7569899 9.271835e-01 9.969969e-01

5  2-3 0.3575269 0.132396127 0.5826576 1.000000e+00 4.085782e-01

6  2-4 0.3510753 0.128523473 0.5736271 1.000000e+00 3.479492e-01

7  2-5 0.1761111 0.008427483 0.3437947 1.000000e+00 1.066767e-05

8  3-4 0.5026015 0.286465254 0.7187377 9.946963e-01 1.000000e+00

9  3-5 0.2231183 0.040772368 0.4054642 1.000000e+00 7.075776e-04

10 4-5 0.2505376 0.065384644 0.4356906 1.000000e+00 3.468519e-03

 

Note:

cmp: comparison of two samples (group.index1-group.index2)

effect: rank difference

lower.cl:  low 95% confidence interval of rank difference

upper.cl:  high 95% confidence interval of rank difference

p.value.1s:  p value of one side test, which tests “a-b” reject if group “a” tends to be smaller than group “b”

p.value.2s:  p value of two side test.

 

Empower also output following box plot to show the distribution for each group.