I have a question about what statistical tests I can do in R that would be most suited to analyse my enzymatic assay results.

My data

Dependent variable: Absorbance (its’ a colorimetric assay)

Factor 1: Protein (type of protein, wt and several mutants)

Factor 2: Substrate (several substrates)

For each assay, (e.g. Prot1 with Substr1) I have 3 – 6 data points (repeats), and it is not feasible to obtain more.

An example of my data would be something like fig1.

What I want to do:

1. Test if the different mutations have a different effect on the enzymes’ activity with the different substrates (basically test the dependence of Absorbance on the interaction between Protein and Substrate: Abs~Protein*Substrate)

Visually, if I plot my data on a bar chart (mean +/-SD), it appears to be the case, but I need to verify that what I see is significant.

Normally, I would do this with a two-way ANOVA, however:

my data is not normally distributed (according to Q-Q plot and skewness test, I have over-dispersed residuals (Laplace distribution), without any skew);

the variance is not homogenous (standardized residuals vs fitted values plot shows heteroskedasticity)

What sort of model could I use instead? Is there a way I can transform my data to allow a parametric test (the only transformations I found were against skewness, which I do not have)?

2. Test for each substrate, which mutations make the activity differ significantly from the wt (e.g. whether activity with Substr2 is significantly different for Prot2 and 3 from that for Prot1)

To avoid doing multiple pair-wise comparisons, I would normally do a pairwise.t.test with a Holm family-wise error rate correction.

For non-normally distributed data of non-equal variance, I saw it is recommended to do a Pairwise Wilcoxon rank sum test.

If I group my data by substrate, with one of the substrates (e.g. Substr2) it was normally distributed and of equal variance. I did the pairwise T test and it gave results consistent with what is seen on the graph. However, when I tried a Pairwise Wilcoxon rank sum test (holm correction) it showed no significant difference between any of the Proteins, which makes no sense (e.g. that there is no significant difference between Prot1 and 2, although one has an activity with the substrate and the other doesn’t). So it looks like the non-parametric test may not be powerful enough/ at all useful.

For the other substrates, either the distribution is not normal (again over-dispersed, Laplace distribution, with no skew), or the distribution is normal, but the variance is not equal (heteroskedasticity).

Just to note, one-way ANOVA or Kruskal-Wallis rank sum tests (where applicable) for Absorbance~Protein for each substrate individually, showed there is a significant variation in Absorbance depending on Protein.

What sort of pairwise comparison tests can I do in these cases? Or, again, how can I transform my data to be able to use a pairwise.t.test?

Thank you in advance!

I really appreciate any advice!