Species Richness, Dominance and Diversity Differences

Question

I have collected 70 organisms from 4 different sites; two sites of treatment 1 and two sites of treatment 2. I also have a continuous explanatory variable (average temperature) which is different for each site. How can I test if measures of richness and diversity differ between sites or by temperature?

Richness is the number of species in each sample

Diversity is a weighted average of the proportions of each species present. In this case we are using the Shannon Index which is:

$^1\!D= \exp\left(-\sum_{i=1}^R p_i \ln p_i\right)$

where $p_i$ is the proportion of each species $i$ at that site, and $R$ is the richness of the site.

What sort of model can I use when diversity and richness are my response variables?

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This question had been abandonned by the OP without giving enough info to answer it properly, the above is an attempt to provide an answerable question in the spirit of what was asked.

Original Question:

I have collected ca. 70 species of organism from 4 sites. 2 sites of Treatment 1 and 2 sites of Treatment 2. How do I test using R, whether the richness, dominance, abundance and diversity is different between the two if I have 6 explanatory variables?

Answer 1

I think neither of these responses fit perfectly into any of the standard GLM link functions. Taking a pragmatic approach it is probably sufficient to pick a link function that is broadly doing the right thing.

Your raw data is from a multinomial distribution, and Richness, $R$, is the number of categories with a score of 1 or more. Although this is count data, it isn't quite the same as the normal sort of count data. The most striking difference being that you can never have a count of zero. As pragmatic first attempt, I would consider modeling $(R-1)$ using a poisson link function. The handwaving reasoning here could be this: If you were catching animals at a random rate for a fixed length of time, the number caught would be modelled well by Poisson. You are catching a fixed number of animals, but at a random rate the type of animal changes. If it wasn't for the fact your return to species already seen, this would be quite a good match.

For diversity, things are even less intuitive. However, the diversity is a sort of average of the abundance ($p_i$) of each species, and as such is bounded by $[0,1]$. This suggests that you could probably treat it as a proportion and use a logit link function.

Advanced Alternative

If the pragmatic matching of support and "looks approximately right" approach is not satisfying, it appears there is another way. I don't know much about this, but there is something called the *Multinomial Diversity Model" which is designed to deal with these sorts of problems. It is described in this paper, http://www.ncbi.nlm.nih.gov/pubmed/23185889 but I have no idea how popular it is.

It is, however, implented in R and availble on CRAN as MRM, apparently by the author of the above paper. It includes a spider data set used as an example in the help:

library(MDM)
data(spider6)
fit0 <- mdm(y2p(spider6[,1:6])~1,data=spider6)
fit1 <- mdm(y2p(spider6[,1:6])~Water,data=spider6)
fit2 <- mdm(y2p(spider6[,1:6])~Water+Herbs,data=spider6)
fit3 <- mdm(y2p(spider6[,1:6])~Site,data=spider6,alpha=TRUE)
anova(fit0,fit1,fit2,fit3)