first, I would agree with Nirmal that permutation tests are an excellent approach (and should be more widely used by ecologists than is currently the practice).
second, make sure that it is the residuals that you are examining for normality, not the dependent variable itself. and I agree with Stephen that heteroscedasticity is more of a problem than normality. finally, given a dependent variable that is "Gosner stage" (and I am extremely ignorant of what that is), then maybe ANOVA is not appropriate, even if residuals are normal. Instead, ordered logistic regression might be most appropriate. I imagine that a larva goes from stage 1 to stage 2 to stage 3... What you know is that stage 3 is more developed than stage 2 and stage 2 more developed than stage 1. But you may not want to assume that stage 2 is exactly intermediate between stages 1 and 3, which is the assumption using ANOVA (or similar linear model). It will be easy to test main effects and their interaction. good luck, Nadav Nur PRBO Conservation Science 3820 Cypress Drive, #11 Petaluma, CA 94954 e-mail: [EMAIL PROTECTED] -----Original Message----- From: Nirmal Bhagabati [mailto:[EMAIL PROTECTED] Sent: Saturday, March 10, 2007 6:06 AM To: [email protected] Subject: Re: [ECOLOG-L] Dealing with non-normal, ordinal data for 2-way ANOVA with interactions You might also consider permutation-test based ANOVA, which eliminates any need for normality. See the book "Randomization, Bootstrap and Monte Carlo Methods in Biology" By Bryan F. J. Manly, which has a section on this. A two-factorial model is implemented in the program MeV (www.tm4.org), but this is intended for DNA microarray analysis, and the output format might not be optimal for your purposes. Nirmal Bhagabati, Univ. of Maryland. On 3/7/07, Stephen B. Cox <[EMAIL PROTECTED]> wrote: > Well - opinions vary on this topic, but, a couple of things to consider in > 2-way factorial ANOVA with a non-normal response. > > 1) ANOVA is robust with respect to deviations from normality, especially > with decent sample sizes. (Good ole Central Limit Theorem comes in handy!) > So, what is your sample size in each cell of the analysis? You may be > worrying over a non-issue :) > > 2) you can always just rank-transform the data and run the 2-way ANOVA on > ranks. This may have some problems... (see Seaman et al. 1994 TREE 9: > 261-263). > > I am generally of the opinion that folks tend to worry a bit too much about > normality in an ANOVA context (and Mixed models can deal with > heterscedasticity which is more of a problem)... but others disagree. It > would probably be worth your while to actually examine the distributions of > residuals in each cell and get a better idea of what they do look like. > > On 3/7/07, Ryan Earley <[EMAIL PROTECTED]> wrote: > > > > Help with stubbornly non-normal data.... > > > > We have a data set with 2 independent variables and 1 dependent (Gosner > > stage for amphibian larvae). We have tried every creative way to transform > > the data and end up with significant deviation from normality each time. > > What we'd like to ultimately do is test both main effects and their > > interaction (which effectively eliminates the use of two Kruskal-Wallis > > tests or Friedman's two-way ANOVA). We would be indebted to anyone who > > might > > have a suggestion on how to proceed statistically. Thanks for your help > > in > > advance. > > > > Best, > > Ryan L. Earley & Foung Vang > > Cal State Fresno > > >
