Non-parametric alternative to repeated-measures ANOVA with a between-subjects factor?

[PHO_M]

Hello,

I'm trying to do a repeated-measures ANOVA but my data is likely violating the multivariate normality assumption.

My question is:
Does water contamination differ by year when comparing one group of beach waters vs. another group of beach waters?

My repeated-measure is year (I'm sampling from the same waters for all years) and my between-subjects variable is location (2 levels: group 1 beach waters, group 2 beach waters).

I examined multivariate normality in SPSS by looking at kurtosis and skewness, as well as by examining Q-Q plots and running a Shapiro-Wilk's test on the studentized residuals of the repeated-measures ANOVA. Kurtosis was pretty bad (more than +/- 1) and the p-values for the Shapiro-Wilk's test were very low as well (often lower than 0.002). I had also tried transformations, but they were not helping either.

If I didn't have a between-subjects factor then I could easily run a Friedman's ANOVA. But (un)lucky for me, you can't do one of those using Friedman's ANOVA. Does anyone know something else I could do?

Many thanks

[victorxstc]

I think that test is a two-way repeated-measures ANOVA, and some previous posts discuss its non-parametric alternatives:

Please click here and here. Note that each one links to other sources.

[PHO_M]

Thanks victorxstc, but I'm a little confused. I've done SRH tests before, but I don't see how one would apply here, as I have a repeated-measures ANOVA, and the SRH test is for regular ANOVAs only. Is there more information about SRH tests that I'm missing? As for the Friedman's ANOVA, because I don't know of any statistical package that can handle the between-subjects effect, I just can't do one.

[victorxstc]

Thanks victorxstc, but I'm a little confused. I've done SRH tests before, but I don't see how one would apply here, as I have a repeated-measures ANOVA, and the SRH test is for regular ANOVAs only. Is there more information about SRH tests that I'm missing? As for the Friedman's ANOVA, because I don't know of any statistical package that can handle the between-subjects effect, I just can't do one.

I think Friedman can be used for two-way repeated-measures ANOVA too (also check this one, and this, and this). But I have not run it personally, so I really don't know any tricks to do it easily.

But as a suggestion, if the above links were difficult to perform, perhaps you can run a Friedman for the within-subject evaluation, and then do the between-subject assessment using simple pairwise tests such as a Mann-Whitney U with Bonferroni correction for multiple comparison. It is the easiest way coming to my mind, agreeing that it will not incorporate the effect of between-subject differences in the whole framework, and also cannot assess the interaction.

Also you can transform your data, and run a normal parametric test then.

Hello,
I examined multivariate normality in SPSS by looking at kurtosis and skewness, as well as by examining Q-Q plots and running a Shapiro-Wilk's test on the studentized residuals of the repeated-measures ANOVA. Kurtosis was pretty bad (more than +/- 1) and the p-values for the Shapiro-Wilk's test were very low as well (often lower than 0.002). I had also tried transformations, but they were not helping either.

Citing Dason who told me this (I can't remember the thread), it is only the Q-Q plot which really matters, and normal distribution of data is not an assumption. So there might be still hope! If your Q-Q plot shows a linear pattern, your data can be analyzed using a two-way repeated measures ANOVA, of course if the other requirement is met, which that one is the sphericity.

[PHO_M]

Thanks victorxstc. I don't want to try too many fancy transformations, as I'm working with geometric means data. I know that you can use log and squareroot tranformations with geometric means data, both of which I've tried to no avail.
I disagree that normal distribution of data is not an assumption. Multivariate normality is an assumption, and I've read this in several textbooks. Anyway, my Q-Q plots are not that great (kind of a snake-like coil around the diagonal line).

One of your links for the Friedman's ANOVA might be good. I'll have to grab that book from my campus library and check it out. The one-way Friedman's ANOVA + Mann-Whitney test sounds better than nothing too, so I'll look into it.