Sample Size for Kruskal-Wallis Test

your post is confusing.  What I think you are saying is the following:

You have 13 groups and each group has 8 samples.

The data is non-normal

You checked the data using box-plots and didn’t see any outliers

You chose the Kruskal-Wallis test because you had more than 2 groups to compare.

You ran this test and found a significant p-value.

If the above is correct then there are a few things to consider:

1. You don’t detect outliers using a boxplot. The term outlier with respect to a boxplot is not the same thing as a data point whose location relative to the main distribution is suspect.

2. ANOVA is robust to non-normality – run ANOVA and see if you get the same thing – namely a significant p-value.  If you do get a significant p-value then the two tests are telling you the same thing.

The issue you are left with – regardless of the chosen test is this – which group or perhaps groups are significantly different from the rest. In ANOVA you can run a comparison of everything against everything and use the Tukey-Kramer adjustment to correct for multiple comparisons.  In the case of the K-W you would use Dunn’s test for the multiple comparisons to determine the specific group differences.

If you don’t have access to Dunn’s test you will have to run a sequence of Wilcoxon-Mann-Whitney tests and use a Bonferroni correction for the multiple comparisons.

To answer your question:

There are two equations one for alpha and one for beta.

What you need to define is what constitutes a critical difference between any two populations (max value, min value)

What you have is the sample size per population (8) and for the conditions you have specified you have -1.645 (for the alpha = .05) and +1.282 ( for beta = .1).

the alpha equation is  (Yc – Max value)/(2/sqrt(n)) = -1.645

the beta equation is (Yc – Min value)/(2/sqrt(n)) = 1.282

subtract the two equations and solve for n.

Obviously you can turn these equations around – plug in n = 8 and solve for the beta value and look it up on a cumulative standardized normal distribution function table and see what you have.

A point of order:  As I mentioned earlier ANOVA is robust with respect to non-normality. Question: with 8 samples how did you determine the samples were really non-normal?  If you used one of the many mathematical tests for non-normality – don’t.  Those tests are so sensitive to non-normality that you can get significant non-normality declarations from them when using data that has been generated by a random number generator with an underlying normal distribution.  What you want to do is plot the values on probability paper and look at the plot – use the “fat pencil test” – a visual check to see how well the data points form a straight line.

If you are interested in seeing just how odd various sized random samples from a normal distribution look I would recommend you get a copy of Daniel and Wood’s book Fitting Equations to Data through inter-library loan and look at the plots they provide on pages 34-43 (in second edition – page numbers may differ for later editions)