# How to test for significant differences among monthly samples with small and unbalanced data

Here's the thing: I have the concentration of a certain compound measured in each individual. Individuals were sampled on a monthly basis, with varied lucky, some months I caught more than ten, sometimes I caught only two.

Here is a plot showing the data I am dealing with.

What I want to know is if and which months have average levels of the compound that are significantly higher or lower (peaks mostly). Also I want to know if changes in variance among samples are different from the expected given the different sample sizes.

My first thought was simple one-way ANOVA and then a post-hoc, but my data do not support the assumptions of normality nor homoskedasticity.
After various readings including non parametric tests, permutation test and temporal series, I realized that here are many possible analysis and assumptions to consider so at the end I get a bit lost. What would be the best way to analyze my data to answer my questions above?

I have already tried a one way permANOVA and wrote some code in R to perform permutational T-tests contrasting the mean for each sampling date with that of the pooled data, but I am not sure if that is a proper test.

Any advice is welcomed.

## 1 Answer

If you are interested in comparing means, the textbook non parametric alternative for ANOVA is Kruskal-Wallis test. It will tell you if there are significant differences among groups.

If Kruskal-Wallis test shows differences among groups, you can perform comparisons using t-test or Mann–Whitney U test, but you need to make corrections for multiple tests.

However, if you are interested in comparing variances, you can also do pairwise comparisons (with any available test like Levene's or Bartlett's) and make corrections for multiple comparisons.

• Thank you for your time Pere! I have considered K-W but I readed some things about possible pitfalls of this test in cases like mine here. As example, it seems that if the distributions of the groups under comparison are different, the K-W test may reject the null hypothesis of equal medians even though the medians are the same... An example can be found here. biostathandbook.com/kruskalwallis.html So I still guess that permutational test seem to be the best option here... Maybe someone else can bring light to this. – Nahuel Feb 22 '17 at 12:40