Two-way ANOVA with non-normal data and heterogeneity of variances

[#1]

Dear friends of data analysis!

I'm struggling with the data for my thesis - I want to do two-way anovas, but my data is non-normally distributed and shows no homogeneity of variances.

My data was collected monthly over a one-year period, with 4 data points per month (respectively for the below-mentioned variables) per treatment (of which I have two) = ~ 48 data points per treatment, ~96 in total.

I have different variables I want to test: Biodiversity (Shannon and Simpson Indices), Ethylene production and Primary productivity. 

My original two-way anova model is (e.g.) Shannon ~ Community*Time 

Transformations of the data did not lead to a fulfillment of the assumptions.

Is there a way to do a non-parametric test with JMP, but to not only test one-way (e.g. Shannon~Community, Shannon~Time), but also the interaction term?

What other alternatives can you suggest?

I tried a GLM with JMP, but the distributions (binomial, normal, exponential, poisson) do not seem to be well-suited for my data... I tried Poisson, but then I have problems with the overdispersion (when to select it and when not?)

I hope to find some help here, thanks in advance!

Best,

Franka

[#2]

This is tricky because you don't really escape the equal variances assumption by going non-parametric.  Some non-parametrics don't assume equal variance, some do.  You could collapse the 2 factors into 1 and do a Welch's ANOVA.  This is like testing the 2 main effects and the interaction simultaneously.  Problem is you won't be able to test each factor for significance separately. 

Also, there is not a good followup test in JMP like Tukey HSD to look at all pairwise comparisons that also has a Behrens-Fisher adjustment to account for unequal variances.  There is a test called Games-Howell that is basically a Welch's Tukey HSD, but you'd have to use R or SAS for that analysis. 

JMP does have Steel-Dwass, which I believe is a type of permutation test.  I've never had much luck with it because my group sizes were too small.  With your sample sizes, you would probably be ok using that method.  I'm just not sure how robust it is to unequal variances.

Non-parametric 2-way ANOVA is Friedman's Test, but that is not currently in JMP.  There is a discussion here: https://community.jmp.com/t5/Discussions/Friedman-test-on-JMP/td-p/5974.

Non-normality is not much of an issue for you because you have a large sample size for each treatment.  CLT kicks in by n=30 for most situations, and the real assumption for ANOVA is that the sampling distributions of the means are normally distributed.  Your bigger problem is the unequal variances.

[#3]

It sounds like you're not sure how to account for time, or change over time in your model. 

What are you doing with Month? is it a random effect? how are you accounting for change over time in your model?

The second variable is your treatment (community?)

In fit model, maybe your effects are:  (or instead of month, days since start, or something that makes sense in you experiment.)

community

month & Random

month*community & Random

Then you can do a custom test, or multiple comparisons (red triangle, estimates, multiple comparisons.)  to get the least squares means for each community. 

After the analysis, check your residuals for normality. The dispersion in your experimental responses may be explained by the treatments and random effects?