2-ways anova for non-normal data

Our group took samples of contamination in sediments from around 20 non-related rivers, during 4 months.

I want to know if contaminant (micrograms per liter) is related to river and/or time, thus my first option would be 2-ways ANOVA. The factors are "river" (values 1 to 20) and "month" (values 1-4), and the response variable is contaminant content.

The problem is I have checked that the contaminant variable is non-normal and I didn't find a useful data transformation to avoid this.

So, which would be the best choice in this case? I have read that Kruskal-Wallis can be used with one factor and another one as blocking, but I'm not sure if it's my case (maybe because I don't fully understand factor blocking, my bad).

Any other non parametric 2-factos test I can use in this case? Thanks a lot

• – A. Donda Jul 19 '18 at 16:35

1. You have to check normality of the response variable in each combination of factors; in your case 20 rivers x 4 months = 80 combinations. You cannot just look at a distribution of the reponse variable overall accross all treatments and expect it to be normal (it is not clear from your question if you did that).

2. You don't have to have very nicely normal distributions in each combination. If you don't, you could still proceed with an ANOVA at your discretion, as it is robust to some deviations from normality.

3. The best way to check for normality is to inspect distributions visually by eye. It is more reliable and recommended over formal tests.

4. A non-parametric alternative to 2-way ANOVA exists and is called the Scheirer–Ray–Hare test. It was developed in 1970s, but has since been strongly criticized (sorry, I don't remember a reference for this).

5. If the sample size is balanced across the 4 months, then you could omit month as a factor and analyse the data with river as the sole factor using either 1-way ANOVA or Kruskall-Wallis test.

• With respect to item 1, one might go further and look at normality of residuals around cell means combined over all the 80 cells in the design, pooling the information, rather than looking in each of the 80 cells. That provides a nice pooling of information among samples and might point to particular outliers in terms of rivers/months. If there's only one measurement per cell, however, neither your nor my suggestion can be carried out. A log transformation often helps with these types of data, too, if measurement errors are proportional to the value of the contaminant being measured. – EdM Jul 20 '18 at 15:45

While regular permutation tests cannot be extended beyond one factor, there are so-called "synchronized permutation" tests. Here' a reference for them in the 2-way setting:

Basso D., Pesarin F., Salmaso L., Solari A. (2009) Synchronized Permutation Tests in Two-way ANOVA. In: Permutation Tests for Stochastic Ordering and ANOVA. Lecture Notes in Statistics, vol 194. Springer, New York, NY.