Partial Ordination
Vegetation responds to many factors, but not all of these factors are
necessarily of interest to the ecologist. Fortunately, CANOCO has the capability
to "factor out" such influences. The result is a partial ordination, which
is directly analogous to partial correlation (see Draper and Smith 1981).
Partial analysis can be performed for both direct and indirect gradient
analysis. The variables to be "factored out" are termed covariables. Examples
of covariables follow.
Differences among observers
In a large study, there is often more than one person responsible for collecting
the data. Unfortunately, there will inevitably be some subtle variations
in how different observers estimate cover, identify species, etc. If observers
were represented by dummy variables, such variation can be factored out.
This capability of CANOCO is usseful, but should not be an excuse for sloppy
collection of data.
Phenological variation
Collection of data takes time. In a study involving many plots, it is possible
that many weeks separate the first and last plots. The vegetation may change
substantially during this time period. Unless phenological change is an
objective of the study, it is desirable to use the time of the year as
a covariable, so that the true site-to-site gradients may be more obvious.
Similarly, the year of sampling (coded as a dummy variable) might be a
desirable covariable if understanding year-to-year variation is not an
objective of the study.
Among-plot variation
If an observer is interested only in temporal trends within repeatedly
sampled plots, then variation among plots is worth "factoring out". This
can be done by creating a dummy variable for each plot, and by using these
dummy variables as covariables.
Block effects
Suppose you have an experiment which is replicated between several sites
or blocks (for example, in a "split plot" design). You can code your
sites or blocks as dummy variables, so that you can specifically focus
on experimental effects.
Uninteresting gradient
Many gradients are so well known that it is not worth focusing on them
in an analysis. For example, an ecologist studying the effects of grazing
practices on vegetation in a mountainous region might wish to use elevation
as a covariable.
Spatial dependence
One of the basic assumptions of statistics is that different observations
are independent of each other. However, vegetation data often suffer from
spatial dependence (Palmer 1988, 1990, Legendre and Fortin 1989), which
can lead to incorrect statistical inference. Use of spatial coordinates
as covariables can help factor out some aspects of spatial dependence (Borcard
et al 1992, Økland and Eilertsen 1994). I urge caution in using
spatial coordinates as covariables, because not all spatial dependence
is a linear (or polynomial) function of location, and because excessive
use of polynomial terms in spatial covariables might lead to the arch effect
(for some of the reasons, see variance explained
and variation partitioning). If data are collected in a transect or
grid, it is possible to factor out spatial dependence using a special permutation
test (ter Braak 1990, ter Braak and Wiertz 1994).
Everything
In an exploratory analysis, an investigator is often interested in revealing
all of the important environmental factors which determine species distributions.
If all of the environmental variables in a study are inputted as covariables,
and an indirect gradient analysis is performed (e.g. a partial Detrended
Correspondence Analysis or pDCA), then the result indicates the residual
variation in species composition. The investigator can then use his or
her knowledge and intuition while examining the location of species and
samples in ordination space. The distribution of different kinds of species
in different parts of ordination space might suggest that some unmeasured
gradient is important.
CAVEATS
Using covariables does not mean that you are completely factoring out an
effect. For example, suppose that you use elevation as a covariable. It
is possible that you have an important elevation by aspect interaction
which is still "hidden" in the data set. Also, it is possible that elevation
is not best expressed as a linear term. If species composition varies as
a function of the square root of elevation, you will not be removing the
entire "elevation effect". These concerns shouldn't bother us much: usually
a variable will be so highly correlated with any transformation of that
variable, so removing a linear effect will essentially remove the bulk
of any nonlinear effect, and interaction effects (except in extreme cases)
will also be correlated with the covariable. However, getting the "wrong"
transformation dampens what would otherwise be crisp hypotheses. Note that
this "dampening" is not unique to gradient analysis: it is a problem with
ANCOVA, partial correlation, or any other method which uses covariables.
Variation partitioning and stepwise ordination
It is possible to use the same variable as both a covariable and as an
environmental variable in different parts of the same analysis. In
variation partitioning, covariables are useful
for distinguishing the relative contributions of different groups of variables
in explaining species composition. In stepwise
ordinations, variables are included in a regression
model, one at a time. As soon as they are added, they become
covariables. This procedure can be automated in CANOCO.
Significance testing and Permutation Blocks
Permutation tests with covariables are somewhat
more complex than without covariables. Ter Braak and Smilauer (1998)
and Legendre and Legendre (1998) discuss some of these complexities.
However, the tets become much cleaner if the permutations are "conditioned
upon covariables". In almost all cases, this is done when your covariables
are dummy (categorical) variables. When you condition upon covariables,
you are only permuting your data within each category of your covariables.
For example, if your covariables represent the blocks
of a split-plot design, you would only permute data within your blocks.
References Cited
(see also selected references for self-education)
Borcard, D., P. Legendre, and P. Drapeau. 1992. Partialling out the
spatial component of ecological variation. Ecology 73:1045-55.
Draper, N. R., and H. Smith. 1981. Applied Regression Analysis. second
edition. Wiley, New York.
Legendre, P., and M.-J. Fortin. 1989. Spatial pattern and ecological
analysis. Vegetatio 80:107-38.
Legendre, P. and L. Legendre. 1998. Numerical Ecology. 2nd English edition.
Elsevier, Amsterdam. 853 pages.
Palmer, M. W. 1988. Fractal Geometry: a tool for describing spatial
patterns of plant communities. Vegetatio 75:91-102.
Palmer, M. W. 1990. Spatial scale and patterns of species- environment
relationships in hardwood forests of the North Carolina piedmont. Coenoses
5:79-87.
ter Braak, C. J. F. 1990. Update notes: CANOCO version 3.10. Agricultural
Mathematics Group, Wageningen, The Netherlands.
ter Braak, C. J. F., and P. Smilauer. 1998. CANOCO Reference Manual
and User's Guide to Canoco for
Windows: Software for Canonical Community Ordination (version 4). Microcomputer
Power (Ithaca, NY USA)
352 pp.
ter Braak, C. J. F., and J. Wiertz. 1994. On the statistical analysis
of vegetation change: a wetland affected by water extraction and soil acidification.
J. Veg. Sci. 5:361-72.
Økland, R. H., and O. Eilertsen. 1994. Canonical correspondence
analysis with variation partitioning: some comments and an application.
J. Veg. Sci. 5:117-26.
This page was created and is maintained by Michael
Palmer.
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