[R-sig-eco] evaluating multiple responses using chi square

[David Warton]

Hi Steve,
Yes you are right in what you say, and it looks like you have identified the problem already - you can sum chi-square random variables to obtain a chi-square variable whose df is the sum of the df's of component variables, but only if they are mutually independent.

Community datasets, with abundances or presence-absences from multiple taxa collected at the same place, are commonly referred to as multivariate precisely because the multiple responses are typically dependent, and hence statistics calculated for separate response variables are also dependent.  You can still get somewhere with the theory though - the sum of dependent chi-squares could be re-expressed as a weighted sum of chi-squares - but the weightings couldn't be estimated reliably unless you had lots of information in the data from replicate observations, which we tend not to.  This is the reason we went for resampling.  (The main alternative, which we are currently looking at, is covariance modelling as a strategy to estimate and account for correlation in a parsimonious way.)

All the best
David

 
David Warton
Professor and Australian Research Council Future Fellow
School of Mathematics and Statistics and the Evolution & Ecology Research Centre
The University of New South Wales NSW 2052 AUSTRALIA
phone (61)(2) 9385-7031
fax (61)(2) 9385-7123
 
http://www.eco-stats.unsw.edu.au/ecostats15.html




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Date: Wed, 10 Jun 2015 09:57:37 -0500
From: Steve Brewer <jbrewer at olemiss.edu>
To: <r-sig-ecology at r-project.org>
Subject: [R-sig-eco] evaluating multiple responses using chi square
Message-ID: <D19DBA91.384E6%jbrewer at olemiss.edu>
Content-Type: text/plain; charset="UTF-8"

Dear Listserv community,

I realize that this is more a statistical theory question, rather an R application question, but I hope those familiar the theory underlying manyglm and manylm could help me.

In evaluating the overall response of a community to a treatment, I'm aware that one class of approaches involves doing univariate analyses for each species (e.g., ANOVA, t-test, chi square, logistic modeling, etc) and then "summing" the results across all species and evaluating statistical significance with a randomization procedure.

My question is has anyone considered using a chi square test instead of randomization to obtain the significance value? The p value for any test statistic (F, t) can be converted to a chi square value with 1 df. Because chi square values are additive (assuming independence), it makes sense to me that you could simply add up the chi square values for all species and evaluate the significance of the resulting sum assuming a df equal to the number of tests (species). Presumably, one could use different tests for different species, depending on whichever is most appropriate (e.g., anova for common species that differ in abundance between treatments or chi square or a logistic model for species that differ in terms of frequency of occurrence between treatments). If one were confident that the univariate assumptions held for each species' test, other than the assumption of independence of responses among species, I'm wondering what if anything is wrong with such an approach for obtaining a significance value. Perhaps something similar is being done when the log likelihood value is calculated?
If so, what are the similarities or differences?

Thanks, and I apologize if this question is too basic or has already been answered.

Steve




J. Stephen Brewer
Professor
Department of Biology
PO Box 1848
 University of Mississippi
University, Mississippi 38677-1848
 Brewer web page - http://home.olemiss.edu/~jbrewer/ FAX - 662-915-5144 Phone - 662-202-5877



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