Pairwise non-parametric test for small sample sizes

Question

I was asked to check a small dataset (3 replicates for 5 treatments, each) for significant outcome differences.

I give an example:

The variance within replicates in the data cannot be assumed equal, so i thought a pairwise t-test is not appropriate.

For another dataset (60 replicates per treatment), i used glht() from the multcomp-package in R, but i was told the sample size of my current data set is too small for it too work.

The question is, are there even tests that are suitable for nonparametric data with large within-replicate-variance and small sample sizes?

If so, is it possible to get a pairwise comparison chart output? I thought that agricolae's kruskal() comes in very handy, but i have no idea if kruskal is very good for my problem.

Any alternatives (if there are any) are much welcome.

Answer 1

For a nonparametric approach for pairwise comparisons, I recommend using a Holm correction to the Wilcoxon Rank Sum (aka Mann-Whitney U) test. Please follow the assumptions for the Wilcoxon Rank Sum test. Reference: https://statistics.laerd.com/spss-tutorials/mann-whitney-u-test-using-spss-statistics.php

The Holm correction (https://en.wikipedia.org/wiki/Holm%E2%80%93Bonferroni_method)

In R, this can be implemented as:

> data("PlantGrowth")
> head(PlantGrowth)
  weight group
1   4.17  ctrl
2   5.58  ctrl
3   5.18  ctrl
4   6.11  ctrl
5   4.50  ctrl
6   4.61  ctrl
> 
> pairwise.wilcox.test(x = PlantGrowth$weight, g = PlantGrowth$group, p.adjust.method = "holm", paired = FALSE)

    Pairwise comparisons using Wilcoxon rank sum test 

data:  PlantGrowth$weight and PlantGrowth$group 

     ctrl  trt1 
trt1 0.199 -    
trt2 0.126 0.027

P value adjustment method: holm 
Warning message:
In wilcox.test.default(xi, xj, paired = paired, ...) :
  cannot compute exact p-value with ties
> 
> ## see: ?wilcox.test
> ## By default (if exact is not specified), an exact p-value is computed if the samples contain less than 50 finite values and there are no ties. 
> ## Otherwise, a normal approximation is used.
> 

Answer 2

You can try nonparametric Mann-Whitney U-test, which is generally applicable for two samples of 8 and 8 elements (some handbooks recommend 20 and 20). It was described in H. B. Mann and D. R. Whitney. 1947. On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics, 18:50-60 (openly available here). Table 1 of the paper contains critical values for samples of the smaller sizes. Minimal sample sizes with significance level $\alpha = 0.05$ (one-tailed) are 3 and 3.