[R-sig-eco] Proper Contrasts for an Ordered ANOVA

[Joseph Simonis]

Hi Everyone,

     I'm analyzing some data from an experiment and could use a quick 
bit of clarification (and perhaps advice?) on ANOVA contrasts.

     Briefly, the experiment was testing the 'top down effects' of 
predation by an aquatic insect on a freshwater food chain: insect eats 
zooplankton, zooplankton eat algae.  But the insect has a few different 
instars, and laboratory experiments I've done show that, not 
surprisingly, later instars are "stronger" predators (consume more 
zooplankton).  So, I wanted to see if the change in predation capability 
with instar would translate into a change in the strength of the trophic 
cascade.

     To answer the question, I set up an experiment with four treatment 
levels: no insects (trmnt 1), young juveniles (2), old juveniles (3), 
and adults (4).  And the specific question, in relation to the 
experiment is, does zooplankton density decrease and algae density 
increase going from treatment 1 -> 4.

     So, the treatments (and my question) are ordinal, but aren't 
numeric or technically 'evenly spaced'.  And so I'm wondering if 
polynomial contrasts are valid or not in this case, and then what it 
means for the interpretation of the coefficients.  It seems like (from 
Pinheiro and Bates, pg 46) that the fact that the treatment is ordered 
means I should do polynomial, and then from Venables and Ripley (pg 156; 
ed 3) that the non-evenly spaced and non-numeric part means I can't 
interpret the coefficients in the model as coefficients in a polynomial 
regression, which makes sense to me.  The treatment here is just ordered 
factors.

     So then, how do I interpret the coefficients generated for the 
polynomial contrast?  Is it mostly just in terms of the sign, rather 
than the actual numerical value?  Again, I'm really most interested in 
the trend (decreasing vs. increasing), so that gets me the qualitative 
answer I want, but I wasn't sure if there was anything more I could pull 
out of the estimates for the coefficients.

     And, feel free to let me know if there's a better way to set up the 
contrasts for the question I have--I imagine Helmert contrasts provide 
another way to do this?  (And a posting or two by Dalgaard on some 
websites made me think it might be a better way to deal with ordinal 
treatments?)  But I'm not quite sure I have my head wrapped around 
exactly what Helmert contrasts would mean, either...

     Thanks in advance!

--Joe

-- 
Joseph L. Simonis

Ph.D. Candidate
Cornell University
Department of Ecology&  Evolutionary Biology
E447 Corson Hall
Ithaca, NY 14853 USA

E-mail:  jls468 at cornell.edu