# Nonparametric 2-way ANOVA for data with unequal n's due to possibly informative missingness

So my problem revolves around trying to find the right test for my data in R. I've been doing an experiment that is measuring sublethal effects in a toxic environment, and as a result, even though I started with the same numbers for all groups, my sample numbers have shifted due to mortality, leaving me with groups that have different numbers of data points. I've been trying to do nonparametric tests because the data distribution isn't normal, but can't do Friedman's due to the differences in number of data points. I essentially want something like a nonparametric two way ANOVA.

If you want more specifics I have 3 species that I work with: Chione, Venerupis, and Musculista. I exposed all these species to three different copper exposure levels: 0 ppb, 25 ppb, or 50 ppb. For this particular problem, I'm comparing their mucus production. So, independent variables are copper exposure and species, dependent variable is mucus production. Due to the deaths, they all have different numbers of survivors and therefore different numbers of data points. I want to be able to compare them, and would normally use a ANOVA, but they aren't distributed normally.

The mucus data is already transformed because it was a ratio to begin with (mucus/dry mass ratio), and was transformed via arcsine. The mucus collection procedure is fatal, so it's one value per mollusk. The differences in the data sets arise from the number of mollusks that were able to be tested in each group. For example, the 50 ppb groups high the highest mortality rates, so I was only able to collect mucus from a handful in comparison to the 0 ppb groups.

• Mucus sounds positive, it might be possible to transform it to normality via something like the log. Do you have repeated measures? Even if you had normal data, it isn't clear that an ANOVA would be appropriate. How many measures do you have per mollusk? Feb 26, 2016 at 23:04
• The mucus data is already transformed because it was a ratio to begin with (mucus/dry mass ratio), and was transformed via arcsine. The mucus collection procedure is fatal, so it's one value per mollusk. The differences in the data sets arise from the number of mollusks that were able to be tested in each group. For example, the 50 ppb groups high the highest mortality rates, so I was only able to collect mucus from a handful in comparison to the 0 ppb groups Feb 26, 2016 at 23:10
• This is a very interesting question. In essence, you have informative missingness (NMAR data). BTW, I might question the arcsine transformation (cf, The arcsine is asinine (pdf)), note that a number of other possibilities exist. Did you sacrifice the animals at a series of timepoints to track the progression of the mucus proportion? Feb 26, 2016 at 23:20
• Animals were all sacrificed at the end of the respective experiments to try and maximize the amount of data for the experiment (mucus wasn't the only thing I was testing for, just the source of this stats issue) Feb 26, 2016 at 23:32
• What I mean is, did you sacrifice, eg, 1/4 of each group at week 1, anther 1/4 at week 2, ..., the last 1/4 of each group at week 4, to see how the mucus production was changing over time as a result of the exposure to the copper solution? Feb 26, 2016 at 23:57