# Thread: 2-way ANOVA on non-normal, hetergeneous data

1. ## 2-way ANOVA on non-normal, hetergeneous data

I'm trying to analyze growth data from chicks, and I need to perform the equivalent of a 2-way ANOVA (2x2 design, unequal sample sizes), but my data are heavily skewed with heterogeneous variances, and transformation, which I avoid when possible, isn't helping.

I've searched for alternative, robust or non-parametric 2-way ANOVAs, but the only recommendation I can find is Friedmans (no good, since I don't have repeated-measures data) or rank-transform, which leaves me with data so kurtotic it's funny to look at.

In this forum, I just found this thread while searching for a way to deal with my data. I've had experience using generalized linear models with binomial, poisson and proportional data, but can anyone guide me to how might I use it with continuous data that I don't know what the underlying distribution is?

Thanks from a frustrated grad student for any pointers!

2. ## Re: 2-way ANOVA on non-normal, hetergeneous data

Any other information about the data? For example is it constrained to be positive?

3. ## Re: 2-way ANOVA on non-normal, hetergeneous data

Originally Posted by Dason
Any other information about the data? For example is it constrained to be positive?
Hi Dason,

The growth rate data I've collected are for rate of weight change (g/day) and rate of wing growth (mm/day). For weight values can be negative, and in fact, some values are (for example, when chicks starve, they have negative growth weights as they lose weight). However, for wing growth, yes, I'd say it'd be pretty impossible for wings to shrink, so no negative values there! Other than that, it's continuous: each chick's growth rate was calculated from the slope of it's regression. Would it be helpful to see the raw data itself? I'm not sure what else would be relevant to tell you.

4. ## Re: 2-way ANOVA on non-normal, hetergeneous data

Originally Posted by mogetons
Would it be helpful to see the raw data itself? I'm not sure what else would be relevant to tell you.

It couldn't hurt. If it's a lot of data I wouldn't suggest just putting it in a post. Maybe attach a file instead.

5. ## Re: 2-way ANOVA on non-normal, hetergeneous data

It couldn't hurt. If it's a lot of data I wouldn't suggest just putting it in a post. Maybe attach a file instead.
Roger roger! Here's the file. The other problem with it is very unequal sample sizes :\

6. ## Re: 2-way ANOVA on non-normal, hetergeneous data

I'm bumping this up...does anyone have any suggestions for me? I'm completely stuck. I'm also adding an update:

I'd mentioned above that I tried rank-transforming the data (I'd read on this forum (here) that for a 2x2 design, one can RT the data and perform the 2-way ANOVA on the RT data), but because my histograms and boxplots still look completely non-normal and heterogeneous, does this still hold?

I've also read that for testing interactions in a multifactorial design, rank-transformation is inappropriate because RT is non-linear (from Quinn & Keough's "Experimental Design and Data Analysis for Biologists")...but maybe this is not applicable to a 2x2 design?

I've gone ahead with the 2-way ANOVA on both the untransformed (non-normal, heterogeneous data) and the rank-transformed equivalent. For my wing growth, the results are the same, but for weight growth, the 2 tests give different results. I assumed that if the tests showed the same results, they would likely be true, so for wing, I'm assuming I'm ok. However, I have no idea if that's true! And I have no idea what to think about the weight.

If anyone can point me in the right direction, I'd really appreciate it!

7. ## Re: 2-way ANOVA on non-normal, hetergeneous data

Why don't u use a non-parametric alternative for the two-way ANOVA comprising Freidman's or Sheirer-Ray-Hare tests?

The Freidman's test is not necessarily for repeated data. It can be used for non-repeated but matched data too (just like the repeated-measures and two-way ANOVAs). I think the best option for you is the Sheirer-Ray-Hare test, which is an extension to the Kruskal-Wallis test and works like the two-way ANOVA on non-normally distributed data. AFAIK the only software which can compute it is BIOMstat. There is also a SAS module designed to calculate this test. However, I think it needs the sample to be balanced (all the groups must have the same number of specimens).

8. ## Re: 2-way ANOVA on non-normal, hetergeneous data

Why don't u use a non-parametric alternative for the two-way ANOVA comprising Freidman's or Sheirer-Ray-Hare tests? The Freidman's test is not necessarily for repeated data. It can be used for non-repeated but matched data too (just like the repeated-measures and two-way ANOVAs).
I don't know what you mean by matched data...is this different than repeated-measures? However, I think you're right: the Sheirer-Ray-Hare test looks appropriate. I hadn't heard of it before, nor did I find it in my many searches.

As it turns out for my growth data, the residuals proved to be close enough to normality that I think I'm going with untransformed original data in my 2-way ANOVA. It's frustrating to me that I only just realised that I could ignore the non-normality of my original dataset if my residuals were ok...(many texts seem to harp on checking the original data before you do anything!!), but anyway, there you go.

However, I have another set that I may look into this Sheirer-Ray-Hare test you suggest...unfortunately, my entire data set is severely unbalanced, so it may not work, but we'll see.

9. ## Re: 2-way ANOVA on non-normal, hetergeneous data

Brunner, E., Domhof S. and Langer, F. (2002). Nonparametric Analysis of Longitudinal Data in Factorial Designs. Wiley, New York. (unfortunately out if print).

Brunner, E. and Puri, M.L. (2001). Nonparametric Methods in Factorial Designs. Statistical Papers 42, 1-52.

Noguchi, K., Gel, Y., Brunner, E. , Konietschke, F. (2012). nparLD: An R Software Package for the Nonparametric Analysis of Longitudinal Data in Factorial Experiments. Journal of Statistical Software 50, Iss. 12.

Shah, D.A. and Madden, L.V. (2004). Nonparametric Analysis of Ordinal Data in Designed Factorial Experiments. Phytopathology 94, 33-43. Electronic Appendix: Instructions on the Use of Software and Applications (e-extra).

Kind regards,
Edgar Brunner
ebrunne1@gwdg.de

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