Ministerie
van
Landbouw
en
Visserij
Directoraat-Generaal
Landbouw
en
Voedselvoorziening
Directie
Landbouwkundig
Onde:rzoek
GROEP
LANDBOUWWISKUNDE
CANOCO
- a
FORTRAN
program
for
canonical
community
ordination
by
(partia~
fdetrended1
[canonical]
correspondence
analysis,
principal
components
analysis
and redundancy
analysis
(version
2.1).
Cajo
J.F.
Ter Braak
Agricultural
Mathematics Group
Box
100,
6700
AC
Wageningen
The
Netherlands
This
report
is
reprinted
(with
permission,
and with
corrections
and
some
additions)
from
the
technical
report
with
number
87
!TI
A
11
of
the
TNO
Institute
of
Applied Computer
Science,
Statistics
Department Wageningen, which
is
the
former
affiliation
of
the
author.
Technical
report:
LWA-88-02
January
1988
GLW
Postbus
100
6700
AC
Wageningen
Copyright
Agricultural
Mathematics Group, Wageningen, 1988.
No
part
of
this
publication,
apart
from
bibliographic
data
and
brief
quotations
in
critical
reviews,
may
be
reproduced,
re-recorded
or
published
in
any form
including
print,
photocopy,
microfilm,
electronic
or
electromagnetic
record
without
written
permission
from
the
Agricultural
Mathematics Group, P.O.Box 100,
6700
AC
Wageningen,
The
Netherlands.
- i -
CONTENT
OVERVIEW
1 •
INTRODUCTION
1.1
General
objective
1.2
Models, methods and
algorithm
1.3 Terminology
1.~
CANOC0
1
s
efficiency
for
ordination
of
community
data
1.5
Outline
of
the
manual
2.
DATA
INPUT
2.1
Cornell
condensed format
2.2
Full
format
2.3
Presence/absence
data
and nominal
data
for
ordination
2.~
Linking
up
samples
in
different
data
files
3.
TERMINAL
DIALOGUE
3.1
How
to
activate
CANOCO
3.2
Input
and
output
3.3
Ways
to
answer
the
questions
3.~
Questions
to
specify
the
type
of
analysis
and
in-
and
output
files
3.5
Questions
to
omit samples and
to
manipulate
environmental
variables
and
covariables
3.6
Questions
to
specify
transformation
of
species
data
3.7
Questions
to
specify
the
output
3.8
Questions
to
specify
additional
analyses
3.9
Example
~.
OUTPUT
~.1
Samples and
species
in
the
analysis
~.2
Iteration
report,
eigenvalue
and
length
of
gradient
~.3
Correlation
matrix,
means,
standard
deviations
and
inflation
factors
4.~
Percentage
variance
accounted
for
by
firsts
axes
of
species-
environment
biplot
~.5
Species
scores
~.
6 Samples
scores
~.7
Regression/canonical
coefficients,
t-values
and
linear
combinations
of
environmental
variables
~.8
Inter-set
correlations
of
environmental
variables
with
axes
~.9
Biplot
scores
of
environmental
variables
~.10
Centroids
of
environmental
variables
in
the
ordination
diagram
~.11
Monte
Carlo
permutation
test
5.
NONSTANDARD
ANALYSIS
6.
EXAMPLES
6.1
Dune
meadow
data
6.2
Weeds
in
summer
barley
6.3
Gene
frequency
data
7.
MISCELLANEOUS
TOPICS
7.1
Percentage
data/compositional
data
7.2
Nominal
response
data
..
7.3
Multiple
regression,
redundancy
analysis,
principal
components
analysis
and
.canonical
correlation
analysis
7.4
Principal
coordinates
analysis
(PCO)
7. 5
Interchanging
species
and samples; weighted
averaging
ordination
7.6 Weighting samples and
species
7.7
Calibration
by
CANOCO
7.8 Canonical
variates
analysis
(CVA)
8.
ITERATIVE
ORDINATION
ALGORITHM
9.
TECHNICAL
DETAILS
9.1 Dimensioning
-
ii
-
9.
2
Structure
of· the main program
9.3
Scaling
of
the
axes
9.4
Monte
carlo
permutation
test.
9.5
Some
points
concerning
CVA
10.
INSTALLATION
NOTES
11.
ACKNOWLEDGEMENTS
12.
REFERENCES
APPENDIX
A:
Theorem
on
the
eigenvalue
equation
solved
by
CANOCO
APPENDIX
B:
Constrained
principal
coordinates
analysis
APPENDIX
C:
Trace and
short-cut
formulae
(4.17)
and
(4.19)
- 1 -
OVERVIEW
Aim
A
common
problem
in
community ecology and
ecotoxicology
is
to
discover
how
a
multitude
of
species
respond
to
external
factors
such
as
environmental
variables,
pollutants
and management regime, Data
are
collected
on
species
composition
and
the
external
variables
at
a number
of
points
in
space
and
time.
Statistical
methods
available
so
far
to
analyse
such
data
either
assumed
linear
relationships
or
were
restricted
to
regression
analysis
of
the
response
of
each
species
separately.
To
analyse
the
generally
non-linear,
non
monotone
response
of
a community
of
species,
one had
to
resort
to
the
data-analytic
methods
of
ordination
and
cluster
analysis
-
"indirect"
methods
that
are
generally
less
powerful than
the
"direct"
statistical
method
of
regression
analysis.
Recently,
regression
and
ordination
have been
integrated
into
techniques
of
multivariate
direct
gradient
analysis,
called
canonical
(or
constrained)
ordination.
The
use
of
canonical
ordination
greatly
improves
the
power
to
detect
the
specific
effects
one
is
interested
in.
One
of
these
techniques,
canonical
correspondence
analysis,
escapes
the
assumption
of
linearity
and
is
able
to
detect
unimodal
relationships
between
species
and
external
variables.
The
computer program
CANOCO
is
designed
to
make
these
techniques
available
to
ecologists
studying
community
responses.
CANOCO
can
carry
out
most
of
the
multivariate
techniques
described
inTer
Braak (1987)
and Ter Braak and
Prentice
(1988)
using
a
general
iterative
ordination
algorithm.
Researchers
in
other
fields
may
find
CANOCO
useful
as
well,
for
example,
to
analyse
percentage
data/compositional
data,
nominal
data
or
(dis)-
similarity
data
in
relation
to
external
explanatory
variables.
such use
is
explained
in
separate
sections
in
the
manual.
CANOCO
is
particularly
suited
if
the
number
of
response
variables
is
large
compared
to
the
number
of
objects.
Techniques
covered
1.
CANOCO
is
an
extension
of
DECORANA
(Hill,
1979).
CANOCO
formerly
stood
for
canonical
correspondence
analysis
(Ter Braak, 1986a, b) and
included
weighted
averaging,
reciprocal
averaging/[multiple)
correspondence
analysis,
detrended
correspondence
analysis
and
canonical
correspondence
analysis.
The
program has been
extended
to
cover
also
principal
components
analysis
(PCA)
and
the
canonical
form
of
PCA,
called
redundancy
analysis
(RDA).
Redundancy
analysis
(Van
den Wollenberg, 1977;
Isra~ls,
1984)
is
also
known
under
the
names
of
reduced-rank
regression.
(Davies and Tso,
1982),
PCA
of
y
with
respect
to
x (Robert and
Escoufier,
1976) and
mode
C
partial
least
squares
(Wold, 1982), For
these
linear
methods
there
are
options
for
centring/standardization
by
species
and
by
sites
and
for
the
method
of
scaling
the
species
and
site
scores
for
use
in
the
biplot.
The
eigenvalues
reported
in
PCA/RDA
are
fractions
of
the
total
variance
in
the
species
data
(percentage
variance
accounted
for).
Principal
coordinates
analysis
and
canonical
variates
analysis
are
also
available.