Ecology, 74(8),
1993, pp. 2215-2230
?
1993
by the Ecological
Society of
America
PUTTING
THINGS
IN
EVEN BETTER
ORDER:
THE
ADVANTAGES
OF
CANONICAL
CORRESPONDENCE
ANALYSIS'
MICHAEL
W.
PALMER
Department of
Botany,
Oklahoma State
University,
Stillwater, Oklahoma
74078 USA
Abstract.
Canonical
Correspondence
Analysis (CCA) is
quickly
becoming the most
widely used
gradient
analysis
technique in
ecology. The
CCA algorithm
is based
upon
Correspondence
Analysis
(CA),
an
indirect gradient
analysis
(ordination)
technique. CA
and
a
related
ordination
technique,
Detrended
Correspondence
Analysis, have been crit-
icized for
a
number of
reasons. To
test whether
CCA
suffers from the
same
defects,
I
simulated
data
sets with
properties that
usually cause
problems
for DCA. Results indicate
that
CCA
performs quite well
with
skewed species
distributions,
with
quantitative
noise
in
species
abundance data,
with
samples taken
from unusual
sampling
designs, with
highly
intercorrelated
environmental
variables, and with
situations
where not all
of
the factors
determining
species composition are known.
CCA
is
immune to most of the
problems
of
DCA.
Key words:
Canonical
Correspondence
Analysis;
Detrended
Correspondence
Analysis;
Gradient
Analysis;
ordination;
simulation.
INTRODUCTION
The
most
common
kind of
data
set
in
community
ecology
undoubtedly
consists
of
the
abundance or im-
portance
of
taxa
(usually
species)
indexed
by
sampling
units
(e.g.,
quadrats,
releves,
stands,
traps,
etc.).
Typ-
ically,
these
data
are
organized in
a
matrix
with
species
as
rows,
sampling units
as
columns, and
abundance
(or
merely
presence/absence)
as
the
entries.
Since
such
data
matrices are
multidimensional,
and
since
the
hu-
man
mind is
limited
in
its
capacity to
visualize
more
than a few
dimensions,
ecologists are
forced to
find
ways
to
extract
the
most
important
dimensions of
the
data
set.
Fortunately,
most
species
by
sampling-unit data ma-
trices contain
much
redundant
information
(for
ex-
ample,
different
species can
respond
to the
same en-
vironmental
gradients),
and
hence
there
are
typically
very
few
important
dimensions
(Gauch
1982a, b).
There are
two
basic
conceptual
models
for
analyzing
species
by
sampling-unit
matrices. One
model is
that
in
which
sampling-units
(hereafter
referred to
as
sites,
although
the
reader must
keep
in
mind
that
sampling-
units
can be
things
other than
sites,
such as
pitfall
traps,
transects,
or
seine
samples)
are
arranged
into
(often
hierarchical)
groups or
community
types,
and
is known
as
classification. The
other
conceptual
model is
that
in
which
sites
and/or
species
can
be
arranged
along
en-
vironmental
gradients,
and
is
known
as
ordination. This
paper
focuses on
ordination.
Ordination is
increasingly
used for
gradient
analysis,
or
the
study
of
species distributions
along gradients.
'
Manuscript
received
3
November
1992;
accepted
8
Feb-
ruary
1993.
Perhaps
the
most
widely
used
ordination
technique is
Detrended
Correspondence
Analysis
(DCA; Hill
and
Gauch
1980),
which
is an
indirect
gradient
analysis
technique.
In
indirect
gradient
analysis,
environmental
gradients are
not
studied
directly but
are
inferred
from
species
composition
data.
Indirect
gradient
analysis
is not
circular
reasoning,
but
rather
a
quite
logical
way to
uncover
factors
de-
termining
community
structure. It
is
performed
reg-
ularly
and
intuitively
by
experienced field
naturalists.
For
example, an
experienced
ornithologist
can look
at
bird
counts from
several
sites, and
can
(with
some
error) place the sites
along
a
gradient from
wet to
dry,
or north
to
south, or
high
elevation to low
elevation
even
if
data on
these
factors
were
absent.
This is
be-
cause there is
pattern
(and
redundancy)
intrinsic
to the
data.
It
is
fairly simple
to
detect such
pattern
in
small
data
sets,
even for
someone
unfamiliar with
the par-
ticular sites
and
species.
It
is,
however, quite
difficult
to
intuitively
order
large,
complex
data
sets without
the
help
of
multivariate ordination
techniques.
DCA
has
many
desirable
properties as
an
indirect
gradient
analysis
technique.
Unlike
Principal Com-
ponents
Analysis
(PCA) and
Correspondence Analysis
(CA),
DCA
does not
produce
the
arch or
horseshoe
effect,
a
spurious
second axis which
is a curvilinear
function of the first axis
(Gauch
1
982a,
Pielou
1984,
ter Braak
1985,
1987b,
Digby
and
Kempton
1987).
Unlike
Bray-Curtis
(Polar)
Ordination
(Bray
and
Cur-
tis
1957,
Beals
1984),
DCA
does
not
rely
on
the
se-
lection of
arbitrary
endpoints.
Unlike
Nonmetric Mul-
tidimensional
Scaling
(NMDS,
Kruskal
1964a, b)
and
its
variants (Sibson
1972, Minchin
1987a, Faith and
Norris
1989,
Belbin 1
991),
the
number of
dimensions
2216
MICHAEL W.
PALMER
Ecology,
Vol.
74,
No.
8
of
ordination
space
does not
need
to
be
specified in
advance.
DCA,
along
with
Correspondence
Analysis,
Canon-
ical
Correspondence
Analysis (see
The
Correspondence
Analysis
family:
Canonical
Correspondence
Analysis),
and a few
others,
is
a
weighted
averaging ordination
technique. The
main
advantages of
weighted
averaging
ordinations
include the
simultaneous
ordering
of sites
and species
(this
property
is shared
by
a
few other
techniques,
Escoufier
1987), rapid
computation
(rela-
tive to
NMDS),
and
very
good
performance when
spe-
cies have
nonlinear
and unimodal
relationships
to en-
vironmental
gradients, which
produces
severe
problems
for PCA
(Gauch
1982a,
Pielou
1984,
ter
Braak
1985,
1986,
1987a-d, ter
Braak
and
Barendregt
1986,
ter
Braak
and
Looman
1986, ter
Braak
and
Prentice
1988).
Despite its
advantages,
DCA
has come
under
in-
creasing
criticism
(Beals
1984,
Austin
1985,
Allen
1987,
Ezcurra
1987, Minchin
1987a,
Oksanen
1987,
1988,
Wartenberg et
al.
1987, van
Groenewoud
1992).
Al-
though
some criticisms
have been
successfully
rebutted
(Peet
et
al.
1988), a
number of
problems still remain
with
DCA: the
detrending
algorithm
is
inelegant and
arbitrary,
it
sometimes
performs
poorly
with
skewed
species
distributions,
it
may
occasionally
be
unstable,
it
occasionally
does
not
handle
complex
sampling de-
signs
very well, it
may
compress
one
end of
a
gradient
into
a
"tongue"
(Minchin
1987a,
Okland
1990),
and
it
will
destroy
any true
arch
that
actually exists
in
data.
Recently,
a
new
ordination
technique,
Canonical
Correspondence
Analysis
(CCA)
has
come into wide-
spread
use
(e.g.
Stevenson
et al.
1989,
Whittaker
1989,
Wiegleb
et
al.
1989, Allen and Peet
1990,
Borgegdrd
1990,
Carleton
1990,
John
and Dale
1990, Odland et
al.
1990,
Palmer
1990,
Prentice and
Cramer
1990,
Py-
sek
and
Lep?
1991,
Retuerto
and Carballeira
1991).
The
mathematics
and models
behind CCA
and its vari-
ants have
been
most
thoroughly
developed by
ter Braak
(1985,
1986,
1987a-d,
1988),
although
others have
contributed
to our
understanding
of CCA
under other
names
(Sabatier
et
al.
1989,
Lebreton et al.
1991).
A
thorough
bibliography (165
references
between
1986
and
1991)
of
CCA and
related methods
has been com-
piled
by
Birks
and Austin
(1992).
Unlike
DCA,
CCA
is
a
direct
gradient
analysis
tech-
nique,
and
represents a
special
case of
multivariate
regression.
Direct
gradient
analysis
differs from
indi-
rect
gradient
analysis
in
that
species
composition is
directly
and
immediately related
to
measured
envi-
ronmental
variables.
Before
describing
CCA
in
more
detail,
it
is
necessary
to outline
the
essential
features
of the
Correspondence
Analysis
family
of
ordination
methods.
The
Correspondence
Analysis
family
Correspondence
Analysis.
-CCA and
DCA
are
both
variants of
Correspondence
Analysis
(CA).
The
CA
algorithm
can
either
be
expressed
in
terms of
an
ei-
genanalysis
or as a
"reciprocal
averaging"
approach
(reciprocal
averaging
is
actually
a
form
of
eigenanaly-
sis).
The
mechanics
of
reciprocal
averaging have been
described
in
detail
elsewhere
(e.g.,
Hatheway
1971, Hill
1974,
Pielou
1984, ter
Braak
1985,
1987b,
Digby
and
Kempton
1987);
I
will
give
only
a
quick
overview.
The
reciprocal
averaging
approach is
computation-
ally
simple
(Fig.
1A):
arbitrary numbers
are
assigned
to each
site
(any
nonzero
numbers
are
acceptable;
the
particular
numbers
chosen
do
not
influence
the
final
outcome). These
numbers are
site
scores.
Species
scores
are
assigned
to
species
as the
weighted
average of the
site
scores,
where
the
weight is the
abundance
of the
species
in
each
site.
(This is
where
the
data
enter
into
the
algorithm.) At this
stage
species
scores
must be re-
standardized,
or
else scores will
eventually
tend
to-
wards
zero.
Pielou
(1984)
suggests
standardizing from
o
to
100, ter
Braak
and
Prentice
(1988)
suggest sub-
traction
of
the
mean
and
division
by
the
standard de-
viation;
any linear
resealing
will
work.
New
site
scores
are
assigned
as
the
weighted
average
of
the
species
scores
of all
species
that
occur
in
the
site.
Again, the
weights
are
species
abundances.
The new
site
scores
are
(optionally)
re-standardized. The
algorithm
con-
tinues
reciprocally
averaging (and
re-standardizing)
sites
and
species,
until
there is no
noticeable
change
in
spe-
cies
and site
scores
from
one
iteration to
the
next. The
result
is the
first
CA
axis
solution.
Given a
data
set,
an
identical
solution will
result from
any set of
initial
arbitrary numbers.
Computation of
the
second CA
axis
is
more com-
plicated,
but
is
essentially
the
same
as
described
above
except
that
the
linear
effects
of
the
first axis are
factored
out. Third
and
higher
axes can
also be
readily calcu-
lated.
The
reciprocal
averaging
algorithm has
been
consid-
ered
by
some to
be
"circular,"
"mysterious,"
"an
art
form,"
or
"wizardry."
In
reality,
it
is
merely
an
algo-
rithm for
eigenanalysis,
one
of
the
central
techniques
of
matrix
algebra
(Pielou
1984,
Digby and
Kempton
1987).
The
solution
obtained
by
correspondence
analysis
has
desirable
mathematical
properties.
The
first axis
consists of
the
ordering
of
species
and
sites
that
pro-
duces
the
maximum
possible
correlation
between
site
and
species scores
(Gauch
1982a, Pielou
1984).
Second
and
higher axes
also
have
maximal
site-species cor-
relation
subject to
the
constraint
that
axes
are
orthog-
onal.
Eigenvalues
associated
with
each
axis
equal
the
correlation
coefficient
between
species
scores
and site
scores
(Gauch
1982a, Pielou
1984).
Thus an
eigenvalue
close to
1
will
represent
a
high
degree
of
correspon-
dence
between
species
and
sites,
and
an
eigenvalue
close to
zero will
indicate
very little
correspondence.
If
our
fundamental
model of
species
responses
to en-
vironmental
gradients
is
unimodal
(this
is
generally
accepted;
see
Austin
1985, Minchin
1987b),
then
high
December
1993
THE
ADVANTAGES OF
CCA
2217
A
B
Start
Start
Start
Arbitarily
assign
Arbitrarily
assign
Arbitrarily
assign
Site Scores
Site
Scores
LC
Scores
Assign Species
Scores
Assign Species
Scores Aswighted
avere
as
weighted
average as
weighted
average
A
sof LC
Scores
A
of Site
Scores of
Site
Scores
Assign
WA scores
Assign
new
site scores
Assign
new site
scores
as
weighted
averages
as
weighted
averages
as
weighted
averages
of
species
scores
of
species
scores
of
species
scores
Create
LC
site scores
Detrend
Site Scores
as
predicted
values
Detrend
Site Scores
from
multiple
regression
Any change
<
Any
c
aAny
nge
in
scores?
Yes
in
scores?
Ys
i
crs
No
No
No
Ye
Stop
Stop
Stop
FIG.
1.
Algorithms
for (A)
Correspondence
Analysis,
(B)
Detrended
Correspondence
Analysis,
and (C)
Canonical
Cor-
respondence
Analysis,
diagrammed as
flowcharts. LC
scores are
the
linear
combination
site scores,
and
WA scores
are the
weighted
averaging site
scores.
eigenvalues are
associated
with
long and
strong envi-
ronmental gradients
(Gauch 1982a).
Detrended
Correspondence Analysis. -DCA
is iden-
tical
to CA except that a
detrending step is
added (Fig.
1B).
The detrending
consists of removing
the previ-
ously
described "arch
effect" by various
artifices, such
as
cutting
the first axis
into
segments and
re-setting the
average
of
each segment to zero
(Hill and
Gauch 1980),
or
by
fitting a
polynomial, usually
quadratic, equation
to the
relationship and
subtracting
its
effect
(ter Braak
1987,
Knox 1989). Site
scores may also be
rescaled to
equalize species turnover
along the axes (Hill
and Gauch
1980,
Gauch 1982a).
Such artifices do
eliminate the
major
problems with
CA, but they introduce
inelegan-
cies
that have been justly
criticized
for
their
uncertain
effects
(Minchin 1987a,
Oksanen 1987,
1988,
Warten-
berg et
al.
1987).
Canonical
CorrespondenceAnalysis.
-Like
DCA,
the
most
common algorithm
for CCA involves
the addi-
tion of
steps
to
CA
(Fig.
1
C).
However, the
new
steps
are
added not
to
remove an
undesirable
effect,
but to
take
advantage
of
supplemental
data
in
the
form of
environmental variables.
This
is what makes CCA
a
direct
gradient analysis.
A
multiple
linear
least-squares
regression
is
performed
with
the site scores
(deter-
mined
from
weighted
averages
of
species)
as
the
de-
pendent
variables,
and the
environmental variables as
the
independent
variables.
New
site scores are now
assigned
as the value
predicted
using
the
regression
equation.
Since this
regression
equation
is
formally
a
Linear
Combination
of
variables,
let us
label
the
new
site
scores LC
scores,
in
contrast
to the
site
scores
determined by
Weighted
Averaging
(WA).
Although
the
CCA
solution
is
most
commonly
ob-
tained by a
weighted
averaging
algorithm, the
solution
is
essentially an
eigenanalysis,
and
can
hence be
ob-
tained
by
any
eigenanalysis
algorithm
(ter Braak
1986,
1987c).
Indeed,
Chessel
et al.
(1987)
present
a
more
efficient
eigenanalysis
solution
for
CCA.
Nevertheless,
the
weighted
averaging
algorithm
is
sufficiently
rapid
and
accurate
for
practical
use,
and is
discussed here
because of
its historical
importance
and intuitive
ap-
peal.
The
statistical model
underlying CCA is
that
a
spe-
cies'
abundance or
frequency
is
a
unimodal
function
of
position
along
environmental
gradients. CCA is
an
approximation to
Gaussian
Regression under a
certain
set of
simplifying
assumptions, and is
robust to
vio-
lations of
those
assumptions
(ter
Braak
and
Prentice
1988).
CCA
is
inappropriate
for
extremely
short
gra-
dients,
in
which
species abundance or
frequency
is
a
linear
or
monotonic
function of
gradients
(ter
Braak
1
987b,
ter
Braak and
Prentice
1988).
For
further details
on
the
nature
of
the
statistical
models
underlying
CCA
and other
members of
the
CA
family,
the
reader
is
referred to
Lebreton et
al.
(1990),
Sabatier et
al.
(1989),
ter Braak
(1985,
1986,
1987b-d,
1988),
and ter
Braak
and
Looman
(1986,
1987).
Since
CCA,
by any
algorithm, produces
two sets of
site
scores,
it
is
unclear which
is
the most
appropriate
2218
MICHAEL
W. PALMER
Ecology,
Vol. 74,
No.
8
TABLE 1. Parameters used
in
COMPAS
simulations (unless
otherwise stated). See Minchin (1987b)
for
computational
details.
Two gradients
24 sites on a 6
x
4
regular grid
300 species
Maximum abundance for species lograndomly
distributed
from 1 to
100
Species ranges
on both
gradients
taken from
a normal
dis-
tribution, ,
=
100,
a
=
30
Species modes
from uniform random distribution between
-95 and 195
Alpha and gamma (skewness parameters)
taken from uni-
form random distribution
between 0.5
and
3.5
Quantitative noise
taken
from the normal
distribution, and
proportional to the square root of abundance
to
use
in
an ordination
diagram.
The initial
publica-
tions
on
CCA
do not advise whether to plot
WA
scores
or
LC
scores
(ter
Braak
1986,
1987a-d).
Most
papers
using CCA
fail to state
which
site scores
are
used.
Even
the
manual for the
program
CANODRAW
(Smilauer
1990) designed to plot CCA results does
not state
which
set
of scores is
used, although
a
computer file
accom-
panying
the
program
indicates
that
LC scores
are the
default. The most recent version
of
CANOCO
(the
leading computer program for CCA)
employs LC scores
as
the default, whereas previous
versions utilized
WA
scores
(ter
Braak
1990).
I
suggest
that
ecologists
use
linear combinations
in
most cases,
for reasons to be
discussed below.
There is
yet another variant
of CA known as De-
trended Canonical
Correspondence
Analysis (DCCA,
ter Braak
1986, 1987a).
As
the name implies, DCCA
incorporates
both a
detrending
step and a linear re-
gression step into the reciprocal
averaging algorithm.
I
intend to argue that detrending
is unnecessary for
CCA.
The
anatomy of
CCA diagrams
Like CA and
DCA,
CCA allows the
simultaneous
plotting of species and
site scores as points
in
an
or-
dination
diagram
known
as
a
joint
plot.
CCA
has an
additional benefit:
environmental variables
can
be
rep-
resented by
arrows
along
with
the
species
and site scores
in
a diagram known as a triplot.
If
the
appropriate form
of
scaling
is used
(see
ter Braak
1990),
the
length
of an
arrow indicates
the
importance
of
the environmental
variable, the direction indicates
how
well
the
environ-
ment
is
correlated with the various
species composition
axes,
the
angle
between arrows indicates
correlations
between
variables,
the location
of
site scores relative
to
arrows indicates the environmental characteristics
of the
sites,
and
the
location of
species
scores
relative
to the arrows indicates the environmental
preferences
of
each
species.
Details of the
interpretation
of
CCA
diagrams
are
given
in
ter Braak (1986,
1
987a-d,
1990), and excellent
examples
of
such
diagrams
include
Stevenson et
al.
(1989),
Whittaker (1989),
Wiegleb et al. (1989),
Allen
and Peet
(1990), Borgegird
(1990), Carleton
(1990),
John and
Dale (1990), Odland
et al. (1990), Prentice
and
Cramer (1990), Pysek
and
Lep?
(1991),
and Re-
tuerto and Carballeira
(1991).
Elegance
of CCA
Ter
Braak (1986) reveals
that the CCA
algorithm is
conceptually simple and
algorithmically
elegant, and
nicely
unites two distinct
bodies of statistical
tech-
niques
(i.e., weighted
averaging techniques and
mul-
tivariate
regression
techniques).
There
is
no
reason,
from
simply studying the
algorithms, that CCA
should
not work
(this may
be
why
CCA,
unlike most
other
ordination
techniques,
has not
previously been tested
by
simulation).
Unfortunately, elegance
in
the past has been
decep-
tive.
Extremely elegant
techniques
such as
Principal
Components
Analysis
and Canonical
Correlation
Analysis
perform very poorly
on most ecological
data
(Gauch
and
Wentworth
1976,
Gauch
1982a,
Pielou
1984,
Digby and Kempton
1987, Minchin
1987a, ter
Braak
1987b). It is clear that
elegance alone is
insuf-
ficient
reason
for
accepting
a
multivariate
method.
In
this
paper,
I
examine the
behavior of CCA
with
data
sets whose
properties
are
completely
known-
namely,
simulated data sets.
Furthermore,
I
test
CCA's
performance
with
high
levels of noise.
Since
CCA
is
part
of the
correspondence
analysis family,
I
also test
whether the
newer technique
has inherited any
defects
possessed
by
its relatives.
METHODS
Simulation
of species
distributions
I
simulated species distributions
using
COMPAS,
a
computer
program
written
by
Minchin
(1 987b).
COM-
PAS
simulates species abundance
along gradients
as a
beta
function,
which allows
species
to have
nonsym-
metrical,
or skewed distributions
along
environmental
gradients.
In
this
study
the default
parameters
for
COMPAS
are
used
(Table
1).
These values result
in
skewed
species distributions,
and
have
been used to
criticize the
performance
of
DCA
(Minchin
1987a).
Sites are
situated
as
a
6
x
4
regular grid
along
two
major
(hypothetical)
environmental
gradients
(Figs.
2
and
3).
The
simulated
data consist
of
the abundance
of
each
species
in
each site. It must
be stressed that
this
design
does not
represent
a
spatial grid,
but
merely
a
grid
in
"ecological space" (sensu
Gauch
1982).
This
sampling
scheme
is
used
in
all
simulations below unless
otherwise
stated.
A
grid
design may
not be
realistic,
but
it
allows
rapid
visual evaluation of the
performance
of a
technique (Gauch
1
982a,
Kenkel
and
Orloci
1986,
Bradfield and Kenkel
1987, Minchin
1987a).
Although
Minchin
(1987a) found differences
in
simulation re-
sults
between sites
placed
in
a
regular grid
and
sites
placed
randomly,
I
tested both
options
and detected
December
1993
THE ADVANTAGES
OF CCA
2219
Ark
ro?
Environmental
Gradient
I
FIG. 2.
The simulated
distribution
of
the 26
most abundant
species
along
the two
simulated
environmental
gradients.
The
small rectangles
indicate
placement
of sites along
the
gradients.
The vertical
axis
indicates
abundance
(e.g., biomass)
of
species.
Since
it is virtually
impossible
to illustrate the abundance
of many species simultaneously
as
a
function
of
two
gradients,
transectss"
are taken
across the
first environmental
gradient
at five
different
levels
of the second
environmental
gradient
(this
is similar
to the
method
employed
by van
Groenewoud
1992).
The
same species
in different
transects
are
indicated by
different line
styles.
no
substantial
differences;
hence only
the
former
are
presented
here.
Ordination
CCA
and DCA
were
performed
using the
computer
program
CANOCO
version 2.1
(ter
Braak
1
987a) with
all
the
program
defaults. One
of the
major
choices
made in
DCA is
whether
to
detrend by
segments or by
polynomials
(ter Braak
1987a, Knox
1989,
Okland
1990).
Detrending is by
polynomials
in
this
paper. When
detrending
the
simulated data
by
segments
(not pre-
sented
here)
the
configurations
of
the
DCA
diagrams
were
usually
similar;
however,
when
the
two tech-
niques
produced
dissimilar
results the
performance of
both
techniques
was
consistently poor.
An
ideal
ordination
technique
on the
simulated data
should
result
in
a
grid identical
to that
illustrated
in
Fig. 3. If
CCA
works
optimally, there
should be an
arrow
representing
gradient
1
pointing to
the
right, and
an
arrow
representing
gradient
2
pointing
perpendic-
ular
to it. Of
course, no
ordination
technique
will
per-
form
perfectly
if
there is an
extremely high
level
of
noise
in
the
data.
However, a
robust
and powerful
technique
should
give
results similar to
those
in
Fig.
3
in
spite
of high
noise.
If
the
only
environmental
gradients
input into CAN-
OCO
were
gradient
1
and
gradient
2,
we are
practically
guaranteed
near-perfect
results.
This
is
because
a
reg-
ular
grid
will
result as
a
linear
combination of two
perpendicular
gradients.
Unfortunately,
we
rarely
know
a
priori
what
the
most
important
gradients
are.
If
we
did, there
would
be little
purpose
in
performing
mul-
tivariate
gradient
analysis at
all. We
are usually
more
interested in
determining
which
environmental
vari-
ables
represent real
gradients
and
which
variables are
unimportant to
species
composition.
In
order
to rep-
resent such
"unimportant"
variables, I
input
four vari-
ables
in
which
the
values
were taken
from a
uniform
random
distribution,
and which
had no
systematic re-
lationships
with
simulated
species
abundance
data.
Thus the
environmental data
consist
of two
gradients
and
four random
variables,
labelled
gl, g2, rl,
r2, r3,
and
r4.
100
cmJ
E
80
. .
.
.
.
0)
60
cO
60
. .
.
.
.
E
40
.
.
C
2
>i 20
. .
.
.
.*
20
-20
0
20 40
60
80
100
120
Environmental
Gradient
1
FIG. 3.
The
location
of simulated sites
along
two simu-
lated
environmental
gradients.
This
sampling design
is used
for
the DCA
and
CCA
analyses
described
below,
unless oth-
erwise
stated.
