## Peres-Neto P.R., Legendre P., Dray S., Borcard D. (2006).
*Variation partitioning of species data matrices: estimation
and comparison of fractions*.
Ecology, 87:2614--2625.

Establishing relationships between species distributions
and environmental characteristics is a major goal in the
search for forces driving species distributions. Canonical
ordinations such as redundancy analysis and canonical
correspondence analysis are invaluable tools for modeling
communities through environmental predictors. They provide
the means for conducting direct explanatory analysis in
which the association among species can be studied
according to their common and unique relationships with the
environmental variables and other sets of predictors of
interest, such as spatial variables. Variation partitioning
can then be used to test and determine the likelihood of
these sets of predictors in explaining patterns in
community structure. Although variation partitioning in
canonical analysis is routinely used in ecological
analysis, no effort has been reported in the literature to
consider appropriate estimators so that comparisons between
fractions or, eventually, between different canonical
models are meaningful. In this paper, we show that
variation partitioning as currently applied in canonical
analysis is biased. We present appropriate unbiased
estimators. In addition, we outline a statistical test to
compare fractions in canonical analysis. The question
addressed by the test is whether two fractions of variation
are significantly different from each other. Such
assessment provides an important step toward attaining an
understanding of the factors patterning community
structure. The test is shown to have correct Type I error
rates and good power for both redundancy analysis and
canonical correspondence analysis.

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