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Are there other terms I should add to the list? Can you provide any additional or better definitions? If so, please let me know at: mike.palmer@okstate.edu

Note: Pielou (1984) has a good glossary. Although they do not contain glossaries, Legendre and Legendre (1998) and ter Braak and Šmilauer (1998) have good definitions of key terms.

*Abundance:* any measure of the amount of an organism. Can include
density, biomass, frequency, cover, presence/absence, etc. See species
abundances in ordination.

*Arch effect *- a distortion or
artifact in an ordination diagram, in which the second axis is an arched
function of the first axis. It is caused by the unimodal
distribution of species along gradients. The arch appears in Correspondence Analysis and other ordination
techniques. One of the main purposes of Detrended Correspondence
Analysis is to remove the arch effect. Principal Components
Analysis creates a more serious artifact called the horseshoe
effect.

*Axis* - I haven't the foggiest
idea how to define this. Any suggestions? Axes are the basic structure of the
Cartesian coordinate system, and are usually portrayed as being at right angles
(i.e. orthogonal) to each other, though non-Euclidean coordinate systems also
exist. With respect to the use of more exotic coordinate systems in ordination, see Camiz (1991)

*Beta
Diversity *- Also called species turnover or differentiation
diversity. Beta diversity is a measure of how different samples are from each
other, and/or how far apart they are on gradients of species composition.
Alternatively, it is a measure of the "length" of an ecological
gradient or ordination axis, in terms of species composition. Total beta
diversity can be compared among gradients, but not per unit (e.g. one cannot
compare whether the rate of change is higher along a pH gradient than along a
moisture gradient, but the total change along the gradients can be assessed).
An axis or gradient with high beta diversity will have completely different
species compositions (i.e. share no species) at opposite ends (indeed, the ends
might be completely different from the middle). An axis or gradient with low beta
diversity will be similar in species composition at both ends. Some ordination
techniques (e.g. PCA) behave best at low beta diversity, and
others (e.g. DCA, CCA) behave best at
high beta diversity. Beta diversity is one of the most misunderstood concepts
in community ecology, and it has been defined in numerous ways in the past. The
rescaling algorithm of DCA provides a
measure of beta diversity. For more discussion, see Explorations
in Coenospace.

*BACI Design *- "Before -
After - Controlled - Impact" design. Refers to a study in which quadrats
(or other samples) are studied through time, and some of the quadrats are
subjected to an experimental treatment or treatments. Ideally, the statistics
used for a BACI design will be able to distinguish the effects of treatment and
time.

*Biplot - *an ordination
diagram which simultaneously plots species scores and
sample scores. It is only relevant for ordination techniques such as CCA, DCA, CA, PCA,
and RDA. Distance-based
ordination techniques do not result in the simultaneous ordination of both
species and samples.

*Biplot arrow -* a representation
of a variable (usually an environmental variable) on a biplot.
The arrow points in the direction of maximum correlation, and the length of the
arrow is related to the strength of the correlation. In general, the longer the
arrow, the more highly related that variable is to species composition.

*Bootstrap *- A reasonably new
computer-intensive method to obtain confidence intervals, to estimate
parameters, or in some cases to test hypotheses. The bootstrap is considered a
"Resampling method", and is allied to the Jackknife
and to randomization tests. Introductions to
bootstrapping for ecologists are given in Manly (1993) and
Potvin and Roff (1993) . Knox and Peet
(1989) apply bootstrapping to DCA.

*Bray-Curtis Ordination *- A
synonym of polar ordination

*CA* - The acronym
for Correspondence Analysis

*CANOCO* - A
computer program, modified from DECORANA by C.J.F. ter
Braak, which performs a wide variety of ordinations
such as Canonical Correspondence Analysis, Correspondence Analysis, Detrended
Correspondence Analysis, etc.

*CANODRAW *-
A computer program written by Petr Šmilauer designed to graph the output of
CANOCO. CANODRAW's functions are now incorporated into CANOCO.

*CanoImp - *A
computer program written by Petr Šmilauer that converts spreadsheet blocks,
copied into the Windows clipboard, into proper format for input into CANOCO. Canoimp is the console version, and WCanoImp
is the Windows95 version. These functions are now seamlessly
integrated into CANOCO.

*Canonical Analysis *- A term
which appears in the literature a number of times, often with different
meanings. It has been used as a synonym for canonical correlation and Canonical Correspondence Analysis. It is probably best reserved
as a generic term referring to any method which links one set of variables to
at least one other set of variables, and would thus include canonical
correspondence analysis, canonical correlation, redundancy analysis, etc.

*Canonical Correlation -
*Often confused with *Canonical Correspondence Analysis*.
It is a technique which finds the linear combination of
one set of variables which is maximally correlated with a linear
combination of another set of variables. Canonical correlation is closely
related to PCA. See Gittens (1987)
for an ecological application.

*Canonical Correspondence
Analysis *- A widely used method for direct gradient analysis, best
developed by C.J.F. ter Braak (see Jongman et al. 1987, ter Braak and Prentice 1988, and many of the links
on the ordination web page). CCA assumes that species have unimodal
distributions along environmental gradients.

*CANOPOST *- A Windows program
that takes the results of CANODRAW and produces
publication-quality output.

*Categorical
Variable *- A variable that is represented by several different
types; for example: lake/river/stream, farm/pasture/unmanaged, pitfall
trap/fence trap/direct sighting. For most multivariate analyses, categorical
variables must be converted to *k*-1 dummy variables (where
*k* = the number of categories). See Environmental
variables in CCA

CCA - The Acronym for Canonical Correspondence Analysis

*Centroid* -
the (weighted) mean of a multivariate data set. Can be represented by a vector.
For many ordination techniques, the centroid is a vector of zeros (that is, the
scores are centered and standardized). In a direct gradient analysis, a categorical variable is often best represented
by a centroid in the ordination diagram. See Centroids and
Inertia.

*Classification
- *The act of putting things in groups. Most commonly in community
ecology, the "things" are samples or communities. Classification can
be completely subjective, or it can be objective and computer-assisted (even if
arbitrary). Hierarchical classification means that the groups are nested within
other groups. There are two general kinds of hierarchical classification:
divisive and agglomerative. A Divisive method starts with the entire set of
samples, and progressively divides it into smaller and smaller groups. An
agglomerative method starts with small groups of few samples, and progressively
groups them into larger and larger clusters, until the entire data set is
sampled. Pielou (1984) gives a good introduction to
various classification methods.

*Clustering *- sometimes simply a
synonym of classification, but more usually referring
to agglomerative classification.

*Coenocline -*
a simultaneous portrayal of all species response curves along
an environmental gradient (presumably, an important one). This is probably the
most common category of graphs in all of community ecology. *Ecological
continuum* is a synonym. See Explorations
in Coenospace.

*Coenoplane -* a simultaneous
portrayal of all species response surfaces along two
(presumably important) environmental gradients.

*Coenospace - *a simultaneous
portrayal of all species response surfaces along an
unspecified number of gradients. It is difficult for mere mortals to visualize more
than three such gradients simultaneously. Fortunately, there are rarely more
than three important axes or dimensions in most ecological data sets. The
concept of coenospace is closely allied to Hutchinson's multidimensional niche.

*Correlation *-
A method which determines the strength of the relationship between variables,
and/or a means to test whether the relationship is stronger than expected due
to the null hypothesis. Usually, we are interested in the relationship between
two variables, *x* and *y*. The correlation
coefficient *r* is one measure of the strength of the relationship.

*Correlation
Coefficient* - usually abbreviated *r*. A number which reflects
the strength of the relationship between two variables. It varies between -1
(for a perfect negative relationship) to +1 (for a perfect positive
relationship). If variables are standardized to have zero mean and a unit
standard deviation, then *r* will also be the slope of the relationship.
The value *r*^{2} is known as the coefficient of determination; it
varies between 0 and 1. The coefficient of determination is loosely interpreted
as "the proportion of variance in *y* which can be explained by *x*".

*Correlation Matrix *-
a square, symmetric matrix consisting of nothing but correlation coefficients. The rows and the
columns represent the variables. The diagonal elements are all equal to 1, for
the simple reason that the correlation coefficient of a variable with itself
equals 1. The correlation matrices given in CANOCO
usually differ slightly from those calculated in basic statistical
packages. This is because CANOCO uses weighted correlations (i.e. samples
with a higher summed abundance of all species will have more influence in the
calculation).

*Correspondence
Analysis* - An eigenanalysis-based
ordination method, also known as reciprocal averaging. See Correspondence
Analysis.

- Correspondence Analysis has been discovered independently by different scientists.
- Reciprocal Averaging means that sample scores are calculated as a weighted average (or centroid) of species scores, and species scores are calculated as a weighted average (or centroid) of sample scores, and iterations continue until there is no change. However, other algorithms are possible.
- Correspondence Analysis simultaneously ordinates species and samples. There are as many axes as there are species or samples, whichever is less.
- The number of axes worth interpreting is a matter of taste, but the size of eigenvalues can be a guide.
- Correspondence Analysis maximizes the correlation between species scores and sample scores. The eigenvalue is equal to the correlation coefficient. The eigenvectors are either species scores or sample scores.
- An eigenvalue of 1.0 implies that one sample (or group of samples) shares no species with all other samples.
- One can put new points in a Correspondence Analysis without affecting the ordination.
- As with all the other eigenanalysis techniques, it is possible to define "passive samples" or "passive species".
- Correspondence Analysis has a problem: the arch effect. This effect is caused by nonlinearity of species response curves.
- The arch is not as serious as the horseshoe effect of PCA, because the ends of the gradient are not convoluted.
- Another related problem of Correspondence Analysis is that the ends of the gradient are compressed.
- Detrended Correspondence Analysis was designed to correct for the arch effect and gradient compression, as described above.

*Covariable*
- refers to a variable (in the context of DGA, an environmental variable) which
for some reason the investigator wishes to "factor out". This is usually
either a nuisance variable, or an important variable which is not of immediate
interest. A covariable can be used to specify a block effect or site effect (in
which case it is usually a dummy variable), if treatments
are of most interest. See Partial Analysis or Partial
Ordination

*Covariance Matrix*
- a square, symmetric matrix in which the rows and
columns are variables, and the entries are covariances. The diagonal elements
(i.e. the covariance between a variable and itself) will equal the variances.

*DCA *- The acronym
for Detrended Correspondence Analysis

*DECORANA -*
A computer program, now outdated by CANOCO, for
performing Detrended Correspondence Analysis and Correspondence Analysis.

*Detrended Canonical Correspondence
Analysis (DCCA) *- The detrended form of Canonical Correspondence
Analysis.

*Detrended Correspondence
Analysis (DCA) *- an eigenanalysis-based
ordination technique derived from correspondence analysis (Hill
and Gauch 1980) DCA performs detrending to
counteract the arch effect, a defect of correspondence
analysis. DCA also (optionally) performs rescaling of
ordination axes, so that the spacing of sample (and species) scores along the axes are scaled in units of beta diversity. See Detrended
Correspondence Analysis.

*Detrending -
*A method employed in DCA and DCCA to
remove the arch effect. Axes are divided into segments, and
the sample scores of higher axes are reassigned to be centered around the centroid. See Detrended
Correspondence Analysis for a brief description. More thorough descriptions
are given in Gauch 1982, Pielou 1984 and
Kent and Coker 1992.

*Dimension* - This is a difficult
term to define precisely in a comprehensible way. However, it is possible to
grasp at a more intuitive level. It is the number of axes in a Cartesian
coordinate system or the number of variables (unless some variables are linear combinations of other variables). Even though there
are often a large number of dimensions, there are usually only a small number
of important dimensions. A related concept to dimension is the *rank* of a
matrix. The rank is "the number of dimensions of a
space in which the data points lie" (Pielou 1984)

*Direct Gradient
Analysis *- Any gradient analysis in which the important gradients
are known and measured. Direct gradient analysis is commonly performed using
nonlinear regression, or using a technique such as Canonical
Correspondence Analysis. In contrast, see indirect
gradient analysis.

*Discriminant Analysis* - A
technique related to ordination, which is used in
many fields other than ecology. Digby and Kempton (1987)
provide a good discussion. Discriminant Analysis tells us whether a particular
set of variables is useful in *discriminating* previously delineated
groups. Canonical Variates Analysis (CVA) is a form of discriminant analysis
which is actually a special case of Canonical Correspondence
Analysis in which the classes are coded as dummy variables.

*Dissimilarity Matrix - *see distance matrix.

*Distance Decay *- the property
by which two nearby points have more similar characteristics than two distant points.
Distance decay violates the basic statistical assumption that samples are
independent, and is therefore a special case of pseudoreplication.
Distance decay can be quantified using *geostatistics*.

*Distance Matrix*
- A square and (usually) symmetric matrix in which the
rows and the columns represent (usually) samples. The entries represent some
index of the difference between samples; the measure could be Euclidean
distance, Manhattan (City Block) Distance, Bray-Curtis dissimilarity, the
Jaccard coefficient, or any of a huge number of possibilities. The diagonal
elements (the difference between a sample and itself) is usually zero. Distance
matrices are necessary prerequisites for distance-based
ordination methods such as Polar Ordination
and Nonmetric Multidimensional Scaling. Distances matrices
are closely related to (and easily converted to) similarity
matrices.

*Downweighting* - An option in
many ordination programs to dampen the effects of rare species. Downweighting
gives weights to species which are related to their abundances. Correspondence
Analysis and its derivatives are sensitive to rare species which occur in
species-poor areas (see, e.g. ter Braak 1987);
downweighting reduces but does not eliminate this problem.

*Dummy Variable *-
a binary variable of 1's and 0's, which is one if the observation belongs to a
category and zero if it does not. See also categorical
variable and environmental
variables in CCA.

*Eigenanalysis*
- the process of finding eigenvectors and eigenvalues. See eigenvector and eigenvalue.

*Eigenvalue *-
a central concept in linear algebra (i.e. matrix algebra). A semitechnical
definition is as follows:

- "Let
**A**be a*p*by*p*matrix and**w**a*p-element*vector. If it is true that**Aw**= l**w**for some scalar l , then**w**is an*eigenvector*of**A**and l is the corresponding*eigenvalue*. That is, an*eigenvector*of a matrix is a vector such that when we multiply the matrix by the vector we get the vector back again except that it has been multiplied by a particular constant, called the*eigenvalue*." - Cliff (1987). - The process of finding the
eigenvectors and eigenvalues of a matrix is known as
*eigenanalysis.*For a square matrix, there are as many eigenvectors and eigenvalues as there are rows and columns in the matrix. The eigenvalues are usually ranked from highest to lowest, and termed the first, second, third, etc. eigenvalues or*latent roots*. - Why do we care about the eigenvectors and eigenvalues? It turns out that many ordination techniques are based on eigenanalysis. For example, PCA is based upon an eigenanalysis of the correlation or covariance matrix. For eigenanalysis-based methods, the sample scores (or species scores) are typically the eigenvectors of some matrix, and eigenvalues measure the strength of an ordination axis.

*Eigenvector*
- a central concept in linear algebra. Sample scores are often eigenvectors.
See eigenanalysis.

*Environmental Gradient
*- a spatially varying aspect of the environment which is expected to be
related to species composition. Environmental gradients are the *x-axes*
for coenoclines. I do not know whether gradients
which vary only through time could properly be called environmental gradients,
though to some degree they can be treated as such in ordination methods.
Differences in resource use within a site cannot be considered gradients.
Human-imposed effects can be considered environmental gradients.

*Environmental Variable* - a
measure of the environment which is presumably related to an environmental
gradient. Environmental variables can be continuous, or they can be represented
by dummy variables.

*Euclidean Distance*
- the straight line distance between two points in a Cartesian coordinate
system. The Euclidean distance can be determined using the Pythagorean Theorem.
In two dimensions, the Euclidean distance is [(*x*_{1}-*x*_{2})^{2}
+ (*y*_{1}-*y*_{2})^{2}]^{0.5}.
Usually, the points represent samples and the axes of the Cartesian coordinate
system represent the abundances of species. Gauch (1982)
has a good description of the various kinds of data space.

*Exploratory Analysis *- a
general term for an analysis in which the chief objectives is to find pattern
in the data. Often, exploratory analysis conflicts with hypothesis testing. For
example, stepwise regression is permissible in exploratory analysis, but can
cause serious problems if you are interested in testing hypotheses. See Hypothesis-Driven
and Exploratory Data Analysis.

*Factor
Analysis *- This is a term which has been variously defined. In some
treatments it seems to be a synonym of ordination.
Sometimes (as in some statistical software) it includes principal
components analysis. The following discussion is from Morrison
(1967); my comments are in brackets.

- "It would seem clear
that a new class of techniques [Morrison had just finished discussing
partial correlation, multiple correlation, and canonical correlation] will
be required for picking apart the dependence structure when the responses
are symmetric in nature or no
*a priori*patterns of causality are available [amongst all variables of interest]. Those methods fall under the general heading of*factor analysis*, for by them one attempts to descry those hidden factors which have generated the dependence or variation in the responses. That is, the observable, or*manifest*variates are represented as functions of a smaller number of*latent*factor variates.

According to Gauch (1982), there is a subtle distinction between ordination and factor analysis, which appears consistent with Morrison:

*"Factor analysis*is similar to principal components analysis, except that instead of trying to account for as much of the total variance as possible, only correlations between variables are of interest as reflecting putative underlying causes or factors"

*FORTRAN - *
one of the earliest computer languages in widespread use in
ecology. Most ordination programs were originally written in FORTRAN,
including the Cornell Ecology Programs such as DECORANA.

*Fuzzy
Sets and Fuzzy Set Ordination *- Fuzzy sets are sets which allow
grades of membership. For example, the set of all high-elevation plots may
include no plots at sea level, and all plots on mountain tops, but what about
plots at intermediate elevations? Classical set theory would have us define an
arbitrary elevation or threshold, above which all plots must belong, and below which
no plots belong. Fuzzy set theory would allow a plot to belong with 25%
membership (for a relatively low elevation) or 75% (for a relatively high
elevation). Fuzzy set theory is currently being used in robotics, computer
vision, and artificial intelligence. The application of fuzzy set theory to
ecology was developed by Roberts (1986). Fuzzy set
ordination is probably best classified as a direct gradient
analysis technique, and it bears strong similarities to polar ordination.

*Gaussian
Curve* - The simplest model for a unimodal species response curve (see explorations
in coenospace). It has only three parameters, and the equation is:

*y* = A*e*^{-(x-B)^2/C}

where A is the maximum height of the curve, B is the modal location of the curve,
and C is a measure of the breadth of the curve (often called niche breadth,
tolerance, or standard deviation). The curve is bell-shaped. The difference
between a Gaussian Curve and a Normal Distribution is that the latter is a
statistical distribution, and hence the area under the curve is constrained to
be one, and the *y*-axis represents frequency.

*Gaussian Ordination* - A
little-used ordination technique which arranges
samples along ordination axes such that the fit of the species
response curves to the Gaussian curve is maximized.
The fit can be measured by *r*^{2}. Ter Braak
(1987) shows that CCA can be considered (asymptotically)
a special case of Gaussian Ordination.

*Geostatistics*
- A body of analytical techniques for the study of spatial pattern.
Geostatistics were largely developed for the mining industry, but they are now
widely used in ecology. For an introduction to geostatistics in ecology, see Burrough (1987). There are two interrelated components to
geostatistics: variography and spatial interpolation
(kriging). See Rossi et al. (1992)
for a good introduction to geostatistical applications in ecology.

*Gradient* - see *Environmental Gradient*

*Gradient Analysis *- the study
of species distributions along gradients. See *direct
gradient analysis* and *indirect gradient analysis.*

*Guttman Effect *- A synonym for
the horseshoe effect.

*Horseshoe
Effect *- a distortion in ordination
diagrams. It is more extreme than the arch effect because
the ends of the first gradient are involuted. The horseshoe effect can be
observed for very long gradients in PCA.

*Indirect
Gradient Analysis *- : gradients are unknown *a priori*, and are
inferred from species composition data. The species tell us what the gradients
are. Usually performed using an ordination technique
such as Detrended Correspondence Analysis.

*Inertia* - a
measure of the total amount of variance in a data set. It is directly related
to the physical concept of inertia, which is the tendency for an object in
motion to stay in motion, and the tendency for an object at rest to stay at
rest. For weighted averaging methods such as DCA and CCA, the inertia is related to the spread of species modes (or
optima) in ordination space, rather than the variance
in species abundance.

*Iteration*
- Often, a mathematical operation must be repeated again and again (using the
output of the operation as the input into the operation the next time around).
Each incidence of this operation is termed an iteration.

*Jackknife*
- A (usually) computer-intensive method to estimate parameters, and/or to gauge
uncertainty in these estimates. The name is derived from the method that each
observation is removed (i.e. cut with the knife) one at a time (or two at a
time for the second-order Jackknife, and so on) in order to get a feeling for
the spread of data. See Manly (1993) and Dixon
(1993) for reviews of the use of the Jackknife and similar methods in
ecology.

*Kriging*
- a method of spatial interpolation based upon geostatistics.
By "spatial interpolation", we mean estimating the value of a
variable at an unsampled location based upon measured values of the same value
at known locations. The most common application of kriging is mapping. For
example, one can use kriging produce maps of DCA Axis 1 across a landscape even
if the landscape is incompletely sampled. CANODRAW is
capable of performing some kinds of kriging, mostly for the purpose of drawing
isoclines in ordination space. See Burrough (1987) for
a brief introduction to kriging.

*Latent Root *-
another name for eigenvalue.

*Latent Value *- another name for
eigenvalue.

*Linear Combination *-
a linear combination of a set of variables is a new variable (*y _{i}*)
which can be expressed as follows:

*Linear Least
Squares *- the principle or method by which the fit of a function to
data is such that the sum of the squared residuals is minimized. In linear
regression, the function is a line.

*Mantel Test* - a
method for comparing matrices to each other, also called
"matrix correlation". See Legendre and Fortin
(1989) for an introduction to Mantel tests for spatial pattern.
Significance can be evaluated using randomization
methods.

*Matrix* - a
set of numbers arranged in rows and columns. "An n by m matrix is a
rectangular array of elements with n rows and m columns in which not only is
the value of an element important, but also its position in the array"
(Burden et al. 1981). It is very common to encounter a matrix with the same
number of rows as columns; this is called a square matrix. A square symmetric
matrix is one which is identical if you "transpose" the matrix (i.e.
switch the rows and the columns). The correlation matrix is
an example of a square symmetric matrix.

*MCPT* - an acronym for Monte Carlo Permutation Test

*MDS* - an acronym for Multidimensional
Scaling, but perhaps it is better to avoid this acronym since it has been
variously used in the past. See Terminology in
Ordination.

*Monotonic
Distributions* - describes species response curves in
which species only increase along environmental gradients, or only decrease
along environmental gradients. A monotonic distribution can be linear or more
complex. Also, species with unimodal distributions may
appear to have monotonic distributions may appear to have short gradients if
only a small portion of the gradient is sampled. See Explorations
in Coenospace. If most species have a monotonic distribution, then it is
best to use PCA and RDA, but if most
species have unimodal distributions, then it is best to use DCA
and CCA.

*Monte Carlo Tests -
*a synonym of *randomization tests* (at
least as commonly used by ecologists). A Monte Carlo *permutation* test is
when the actual data values are maintained, but they are randomly permuted in
order to obtain the distribution of the test statistic. Exactly how they
are permuted depends on the null hypothesis to be tested. In the simplest
use of Monte Carlo permutation tests in CCA, the values for the environmental
variables are randomly reassigned to the values for the species data.

*MRPP *- an acronym for Multiresponse Permutation Procedure.

*Multicolinearity -* describes
the situation in which a number of variables (or perhaps all of them) are
highly correlated with each other. This is often considered a problem, and
indeed it makes inferential statistics difficult. But it can also be considered
a blessing, because redundant data are useful in identifying patterns.

*Multidimensional Scaling
*- nowadays, this is often a synonym for nonmetric
multidimensional scaling, but it previously referred to Principal
Coordinates Analysis.

*Multiple Regression
*- See multiple
regression. A method (usually based on the least
squares principle) which attempts to describe or "fit" a measured
dependent variable as a function of multiple measured independent variables.

*Multiresponse
Permutation Procedure*- usually abbreviated MRPP. A
randomization test that evaluates differences in species composition, based on
some distance measure. See, for example Biondini et al.
(1988).

*Multiscale Ordination *- an
ordination method which analyses species composition at multiple spatial scales
simultaneously. See, for example, Ver Hoef and Glenn-Lewin
(1989).

*Multivariate Analysis* - any
analysis which attempts to simultaneously examine the behavior of more than one
dependent variable. A multiple regression is not
considered a multivariate analysis, since only one dependent (response)
variable is studied at a time. Ordination, classification, canonical
correlation, and factor analysis are considered
multivariate methods. Why use a multivariate analysis instead of multiple
univariate analyses? For several reasons:

- If there are numerous variables (for example, hundreds of species) multiple univariate analyses are tedious, and the problem of multiple comparisons emerges
- Multivariate methods take advantage of joint structure (e.g. intercorrelations) between variables.
- Some multivariate methods provide statistical tests of all response variables simultaneously.
- If one is interested in
*community*ecology, one must be interested in all species simultaneously, rather than one at a time. - Of course, if one only has a few variables (e.g. 1-3) then it is somewhat artificial to force a multivariate analysis to process the data. As with all of statistics, one should use the simplest analysis possible to answer the question posed.

*MRPP - *an
acronym for Multiresponse Permutation Procedures

*NMDS* - an acronym
for Nonmetric MultiDimensional Scaling

*Noise* - This term is very
difficult to define, but in general it refers to chance variation in nature
which interferes with our ability to see pattern and infer processes. In its
simplest form, noise is the same thing as statistical error (e.g. the error
term in a regression). See Gauch (1982) for a more
thorough discussion. Ideally, an ordination method will represent real,
important gradients as its first, second, third, etc. axes. Axes which
predominantly summarize noise should be among the last axes.

*Nominal Variable *- A variable
which can be represented as a binary: yes/no, on/off, present/absent. A Nominal
variable is usually summarized by a dummy variable.

*Nonmetric
Multidimensional Scaling (NMDS)* - The most widely used distance-based
ordination method. The user needs to prespecify the number of dimensions,
and then the method will minimize the stress (a measure
of poorness of fit between the ordination and measured ecological distances).
See also distance matrix.

*Normal Equations *- The
equations by which the solution to regression problems are found. The
"normal" comes from the concept of "normal lines" in
physics, i.e. vectors which are at right angles, and therefore uncorrelated.

*Ordination*
- The simplest definition is "Putting Things in Order", which
explains the titles of a series of papers (Wartenberg et
al. 1987, Peet et al. 1988, Jackson
and Somers 1991, Palmer 1993). For some opinions on
what makes a good ordination method, see The ideal
ordination method. The origin of the term "ordination" in ecology
is attributed to Goodall (1954).

- "Ordination is the collective term for multivariate techniques that arrange sites along axes on the basis of data on species composition" (ter Braak 1987)
- "The term 'ordination' derives from early attempts to order a group of objects, for example in time or along an environmental gradient. Nowadays the team is used more generally and refers to an 'ordering' in any number of dimensions (preferably few) that approximates some pattern of response of the set of objects. The usual objective of ordination is to help generate hypotheses about the relationship between the species composition at a site and the underlying environmental gradients" (Digby and Kempton 1987)
- "
**Ordination**- The ordering of a set of*data points*with respect to one or more axes. Alternatively, the displaying of a swarm of data points in a two or three-dimensional coordinate frame so as to make the relationships among the points in many-dimensional space visible on inspection" (Pielou 1984).

*Orthogonal
- *At right angles to, or completely uncorrelated with. Usually in
ordination, axes are orthogonal to each other. Two orthogonal variables will
have a correlation coefficient, (and, for that
matter, covariance), equal exactly zero. If two orthogonal variables are standardized, the sum of the products of the variables
will equal zero. In many ordinations, it may appear that two axes are
correlated with each other (this often creeps up in DCA). However, note it is
the WEIGHTED correlations will equal zero - so a single sample with a high
weight (i.e. high abundance of all species combined) can counteract the effects
of a number of samples with low weight.

*Partial Analysis *-
an analysis (e.g. regression, correlation, ANOVA, ordination) in which the
effects of covariables are "factored out" or nullified. Examples of
partial analysis include partial correlation, partial DCA,
partial CCA, ANCOVA, etc. See Partial
Ordination

*PC-ORD *- A computer program developed
by Bruce McCune which provides a wide variety of statistical tests and
analyses.

*PCA *- The acronym for "Principal Components Analysis"

*PCoA *- The acronym for "Principal Coordinates Analysis"

*Permutation Test *- a special
case of *randomization test*

*Phytosociology*** **- following
Kent and Coker (1992), "[the process of] recognizing
and defining plant communities". According to some (such as Kent
and Coker), the discipline requires a Clementsian world view. However, some
would argue that phytosociology is possible within a Gleasonian framework, and
that it is necessary for mapping vegetation. Worldwide, the Braun-Blanquet
method is the most widely practiced kind of phytosociology. In it, communities
are given Latin names just like species are in the Botanical Code and the
Zoological Code.

*Polar
Ordination* - Also known as
Bray-Curtis ordination. See distance-based
ordination methods. One of the first ordination methods to be widely used
in ecology. Two sites are chosen as endpoints for each axis (or artificial
endpoints can be established), and all the other sites are ordinated relative
to these endpoints, based upon their similarity to these endpoints.

*Principal Components* - The axes of a Principal Components
Analysis. The first Principal Component will, ideally, represent the
dominant gradient. The second Component will be orthogonal
to the first, and will explain some of the residual variation. The third will
be orthogonal to the first and second components, and so on.

*Principal Components
Analysis* - see Eigenanalysis-based
ordination methods and Principal
Components Analysis. Principal Components Analysis (PCA) is an ordination technique which involves an eigenanalysis of the correlation
matrix or the covariance matrix. PCA suffers from a
serious problem for gradient analysis: the horseshoe effect. This problem is caused by unimodality in the species response curve.

*Principal Coordinates
Analysis (PCoA)* - A distance-based
ordination method in which the distances between sites in the ordination
diagram is maximally correlated with the ecological distances. Almost any distance matrix can be used, (see *Similarity,
Difference and Distance*) but if the distance measure is Euclidean, PCoA = PCA.

*Procrustes Analysis *- See *Procrustes
rotation*

*Procrustes Rotation *- Suppose
you had the same objects arranged in two different coordinate systems (e.g. based
on two different ordination procedures, or based on
different years of data). How can you figure out how well the different
ordinations correspond to each other?

- The most obvious solution is to test whether the first axis of one ordination is correlated with the first axis of the second, and so on with the second axis, third axis, and higher axes. However, this can be problematic. For example, if the first axis of one technique is highly correlated with the second axis of another, and vice versa, you might still have two ordination which are very similar, but just rotated.
- Procrustes rotation solves this problem. Procrustes rotates both data sets, and expands and contracts axes, such that the distance between data points in the two data sets is minimized.
- "Procrustes was the leader of a band of robbers in Greek mythology. He was in the habit of putting his victims in a bed - whether they fit in this bed or not. If they were too long for it, he performed radical surgery on their legs to improve the fit. If they were too short, he stretched their limbs so that they became the right length. One of the most important fundamentals of multivariate analysis is respect for the data. The investigator should therefore try to ensure that his rotation to a target is not literally Procrustean. When tempted by the very human desire to confirm our expectations, it may help to remember Procrustes's fate: The hero Theseus "fitted" Procrustes to his own bed as Procrustes had fitted others" - Cliff (1987).
- A good discussion of Procrustes rotation of ordination diagrams is given in Digby and Kempton (1987).

*Pseudoreplication: *a term popularized by Hurlbert
(1984). It refers to (usually) field data in which samples are not
independent. A hypothetical extreme example is a study of the diatoms in a
polluted lake and an unpolluted lake If 1000 samples are taken from each lake,
we cannot consider these to be true replicates to test for a pollution effect.
This is because we do not know whether the lakes are different for reasons
other than pollution (actually, we do know: no two lakes can be identical!). A
less extreme example of pseudoreplication is ordinary spatial dependence. Pseudoreplicated data are
rampant in ecology, and the problem is to some degree unavoidable.

*Q-Mode* - Q-mode
and R-mode refer to ordinations of sites and species, respectively. These terms
are most used in the context of correspondence
analysis and related methods. It turns out that R-mode and Q-mode analyses
give identical results in correspondence analysis; see Digby
and Kempton (1987) and ter Braak (1987).

*R-Mode* - see Q-Mode

*RA* - an acronym for "Reciprocal Averaging"

*Randomization
Test - *See also Randomization Tests. The
purpose of inferential statistics is to evaluate whether a number which
summarizes something of interest, is greater than (or less than) one would
expect just due to chance (i.e. if *H0* is true). This number can be one
of the well-known parametric statistics (*t, F*, chi-squared, *r*,
etc.), or nonparametric statistics (Mann-Whitney *U*, Spearman* r*,
etc.), BUT

- Sometimes there is no theory relating a statistic to a distribution, or
- The problem is too difficult for nonparametric statistics to be developed, or
- Distributional assumptions cannot be met.

Often it is possible to get around this problem by the use of randomization tests (also called Monte Carlo tests or permutation tests). These are related, but not the same thing, as the Bootstrap and the Jackknife methods. Randomization tests are good for statistical inference, but not so good for developing confidence intervals or for model building.

The procedure for a randomization test is:

- Devise a test statistic which is large if your hypothesized process is strong, and small if it is weak (you could do it the other way around, but let us ignore this for now).
- Define your null hypothesis.
- Create a new data set consisting of your data, randomly rearranged. Exactly how it is rearranged depends on your null hypothesis.
- Calculate your test statistic for this data set, and compare it to your true value.
- Repeat steps 3 and 4 many times (preferably several hundred).
- If your true test statistic
is greater than 95% of the random values, then you can reject the null hypothesis
at
*p*<0.05. (be careful about whether you are performing a one tailed vs. two tailed test - if the latter, you will need to use a 97.5% cutoff).

This method may seem somewhat magical, or even circular - how can you get any information out of randomness? It is because you are answering the question directly: "How likely is it that if the null hypothesis were true, I would observe a value this extreme just due to chance." It is worth knowing that Fisher used randomization tests to test the value of the t-test, F-tests, etc. Many people are now promoting the use of randomization tests even when parametric and nonparametric tests exist. Statistical Educators are beginning to use randomization tests as the introduction to statistics, because in many ways it is easier to grasp. See Manly (1992) for more information about randomization tests in ecology.

*RDA -* an acronym for Redundancy Analysis

*Reciprocal Averaging
*- another name for Correspondence
Analysis, OR one particular algorithm for obtaining the Correspondence Analysis solution.

*Redundancy* - The property of
data with much repeated information. Ecological data are typically quite
redundant. For example, a stream which has a certain fish species which likes
fast currents is likely to have other fish species which like fast currents.
Also, it is somewhat less likely to have species which like slow currents. A
statistical tendency for certain groups of species to be either negatively or
positively associated causes this redundancy, and the most common cause of
redundancy is that species have particular environmental requirements. If
redundancy did not exist, multivariate methods would fail.

*Redundancy Analysis (RDA)*
- a multivariate direct gradient analysis method in which
species are presumed to have linear relationships to environmental
gradients (i.e. linear species response curves). Like CCA, the results of RDA can be expressed in a triplot, i.e. a
plot of sample scores, species scores, and environmental arrows. Unlike CCA,
the species scores in RDA are most accurately represented by arrows (that is,
the direction in which that species is increasing in abundance).

*Regression*
- A function (alternatively, a method by which this function is found) relating
one or more dependent variables to one or more independent variables. CCA and multiple regression are only
two of many kinds of regression. The original use of the term
"regression" refers to "regression back to the mean". For
example, it was observed that the sons of tall fathers tended to be shorter, on
average, than their fathers. This regression effect poses some serious problems
for ecological monitoring (See Palmer 1993b).

*Regression Coefficient* - A
parameter which is estimated in most kinds of regression. In multiple
regression, CCA, and RDA, there is a
regression coefficient associated with each independent variable.

*Rescaling* -
The stretching and compression of coenoclines to a
standardized beta diversity. The ability to rescale allows
us to convert skewed species response curves to symmetrical
ones. It can also allow us to change leptokurtic (tightly peaked) or
platykurtic (flat-topped) curves to normal ones. This is why we sometimes don't
need to care too much if the Gaussian model doesn't work perfectly. A form of
rescaling is performed in Detrended Correspondence Analysis (optionally).

*Residual *-
the observed value minus the expected, predicted, or modeled value. In least-squares
regression methods, a line is fit to data such that the sum of the squares of
the residuals is minimized.

*Sample
Score *- same as site score and stand score. A coordinate along an
ordination axis specifying the location of a sample. It is the goal of an
ecologist to determine whether sample scores are related to environmental
gradients. Ideally, sample scores represent the position of communities along
the coenocline.

*Segments* -
In DCA, axes are divided into segments prior to detrending.
It is thought that the choice of number of segments will have a large impact on
the results of DCA. Also, there has been a minor bug reported in the detrending
algorithm of DECORANA and CANOCO.
See the ordination
web page for links related to this bug.

*Semivariogram -* see variogram

*Similarity Index*
- A measure of the similarity of species composition between two samples.
Examples include the Sørensen coefficient and the Jaccard coefficient. Most
similarity indices have values of zero for samples that share absolutely no
species, and 1 or 100% for samples which have identical species composition.

*Similarity Matrix*
- a square and (usually) symmetric matrix in which the
entries are similarities between samples. Similarity
matrices are easily produced from, or converted into, distance
matrices. The diagonal entries are usually 1 or 100%, meaning a sample is
usually 100% similar to itself.

*Singular Matrix* - a square
matrix which cannot be inverted. In multivariate methods, a singular matrix can
occur if one variable is precisely a linear combination of
the other variables. This may occur if data are expressed in a percentage
basis, or there are is a categorical variable expressed as a series of dummy variables. See environmental
variables in CCA. Most multivariate methods are not able to cope with
singular matrices; this is the matrix equivalent of dividing by zero. CANOCO is able to recognize and make corrections for
singular matrices, but many other software packages are not.

*Singular Value Decomposition* -
a way of manipulating matrices which is similar to, and
ultimately equivalent to, eigenanalysis. This is
the approach illustrated in Digby and Kempton (1987).

*Site Score *- same as sample score.

*Spatial Autocorrelation *- a
synonym for spatial dependence.

*Spatial
Dependence *- the value of a variable at a given point depends on the
value of that variable at other points. Spatial dependence violates the basic
assumption of most statistics that observations are independent. It thus can
lead to pseudoreplication. The most common
form of spatial dependence is distance decay. Although problematic in some
senses, spatial dependence is indispensable for geostatistics
and spatial interpolation techniques such as kriging.

*Species Response Curve
*- a graphical portrayal of the abundance or
performance of a species as a function of an environmental gradient. Some
ordination methods assume that species response curves are linear, others
assume they are unimodal. See, for example, ter Braak
and Prentice (1988). A species response surface related the species abundance
to two or more gradients simultaneously.

*Species Score*
- A coordinate along an ordination axis specifying
the location of a species. In weighted-averaging ordination methods such as CA, CCA, and DCA, the species score represents the centroid
of the species, or the mode of the unimodal species response curve. Species scores help one interpret
ordination axes in indirect gradient analysis.

*Stand Score -* a synonym of sample score.

*Standardization
*- a way of scaling variables so that different variables, measured in
different units, can be compared. See also the end of Basic
Statistical Concepts.

- The most common forms of
standardization include ranking, logarithmic transformations, placing on a
0-1 scale (according to the formula [
*x-min]/[max-min*]; this is used in Fuzzy Set Ordination), and subtracting the mean and dividing by the standard deviation. The last two kinds of standardization produce variables which are perfectly correlated (*r*=1) with the raw data. The last kind is by far the most common, and unless otherwise stated, is what should be assumed when you hear "standardized variables". - Standardization is a form of
transformation, but not all transformations are standardizations. For
example, a square-root transformation retains units (e.g. if the raw data
are in km, the square root transform will result in units of the square
root of km). However, true standardization results in dimensionless
numbers. See
*transformation*. Also see Schneider (1994) for a more detailed (though difficult) discussion about units.

*Stepwise Analysis*
- A multiple regression method (including RDA and CCA, which are special cases of multiple
regression) in which explanatory (independent) variables are selected on
the basis of whether they explain a "significant" amount of variation
in your dependent variable(s). There are several flavors of stepwise analysis:

- Forward selection (implemented in CANOCO), in which variables are entered one at a time, until no more variables explain significant variation
- Backwards selection, in which nonsignificant variables are dropped one at a time.
- Combined Analysis, which contains elements of both forward and backward selection.

There are serious problems to the use of stepwise analysis coupled with inferential statistics (see Hypothesis Driven and Exploratory Data Analysis).

*STRESS* - 1)
a measure of the optimality of an ordination solution
(i.e. the relationship between the similarity in species composition and the
closeness in ordination space), used as part of the algorithm of NMDS.
2) What one often feels while performing multivariate analyses

*t-Value Biplot* -
An infrequently used biplot in RDA and
CCA of species scores and environmental variable scores in
low-dimensional (conventionally, 2-dimensional) space, in which it is possible
to infer the strength of the relationship between species and the environment.

*Tongue Effect *- a possible
statistical artifact in DCA, in which one end of the first axis
is artificially compressed along the second axis. The importance of this effect
is disputed, since many believe that most real data sets should exhibit
"true" compression along a secondary axis. That is, the most
important secondary gradient is different at opposite ends of the first
gradient.

*Trace* - the sum of the diagonal
elements of a square, symmetric matrix. In a correlation matrix, since the diagonals must equal one, the
trace equals the number of variables. The sum of the eigenvalues
of a matrix will usually equal the trace of the matrix. In the context of
ordination, inertia is a synonym for trace. In CCA, the trace will be related to the amount of variation
explained by all ordination axes. CANOCO performs a randomization test on the trace statistic, to test
whether the measured variables significantly explain species composition.

*Transformation*
- A mathematical operation performed on a variable (e.g. species abundances or
environmental variables), usually with the goal of making that variable more
useful in a subsequent analysis. Transformations are performed for a number of
purposes, including:

- To make the variable conform to a statistical distribution (usually the normal distribution) for hypothesis tests that require such a distribution
- To dampen the effects of outliers
- To put different variables on a more common footing (see standardization)
- To make a measured variable more biologically meaningful
- For more discussion about transformations, see Species Abundances in Ordination, On the transformation of species abundances, and Environmental variables in CCA.

*Triplot - *In CCA and RDA, we have three sets of scores (species scores, sample scores,
and environmental variable scores). If we decide to plot all three
simultaneously, we have a *triplot*. Sample scores and species scores are
usually indicated by symbols or labelled points. Continuous environmental
variables are indicated by arrows, while categorical
variables are indicated by points on their centroids.

*TWINSPAN - *The Acronym,
Algorithm, AND Computer Program for Two Way Indicator Species Analysis - A classification method derived from Correspondence Analysis.

*Unimodal
Distribution* - A distribution with one mode. In the case of species response curves, a unimodal distribution means the
species has one optimal environmental condition. If any aspect of the
environment is greater or lesser than this optimum, the species will perform
more poorly (i.e. it will have a lesser abundance). Some ordination techniques
(such as DCA and CCA) perform best when
species have unimodal distributions, others (such as PCA and
RDA) perform better when species have monotonic
distributions along gradients (i.e. the species either increase or decrease,
but not both, as a function of environmental factors).

*Variogram
*- Also known as semivariogram. A plot of variance as a function of distance
of separation. The variogram will tell you expected variance at large scales
(i.e. the SILL), the amount of variance at infinitesimally small spatial scales
(the NUGGET) and the spatial scale at which samples can be considered
independent (the RANGE). The variogram is used in the field of geostatistics.

*Variography*
- the art of interpreting variograms. Variography, a
branch of geostatistics, tells you the spatial
scales at which your variable(s) varies, whether you have nested scales of
variation, and whether you have unresolved "nugget" variation at fine
spatial scales. Variograms were developed as part of geostatistics. See Rossi et al. (1992) for an introduction to the use of
geostatistics in ecology.

*Vector *- a matrix
which consists of either one row (i.e. a row vector), or one column (a column
vector). "A vector is a set of values, commonly denoting a point in a
multidimensional space.." - ter Braak (1987)

*WCanoImp* - The
Windows95 version of CanoImp, for inputting data from a
spreadsheet.

*Weighted average * - an
average, except that different observations are given differing importances or
"weights". In ordination, the weights are typically the abundances (perhaps transformed) of
species. The weighted average environment of a species (e.g. the weighted
average soil pH) can be used as an index of that species' environmental
preference. Weighted averages are intrinsic to Correspondence Analysis and related
methods.

*X *-

*Y *-

*Z *- The symbol used
to represent a "regionalized variable", which is a variable that
varies spatially. Regionalized variables are used in geostatistics.

See also Suggested References for Self Education

Biondini, M.E., P.W. Mielke, and K.J. Berry. 1988. Data
dependent permutation for the analysis of ecological

data. Vegetatio 75:161-168

Burrough, P.A. 1987. Spatial aspects of ecological data. Pp. 213-251 in Jongman, R.H., C.J.F. ter Braak and O.F.R. van Tongeren, editors. Data Analysis in Community Ecology. Pudoc, Wageningen, The Netherlands.

Camiz, S. 1991. Reflections on spaces and relationships in ecological data analysis: effects, problems, possible solution. Coenoses 6: 3-14.

Cliff, N. 1987. Analyzing Multivariate Data. Harcourt Brace Jovanovich, San Diego

Digby, P. G. N., and R. A. Kempton. 1987. Population and Community Biology Series: Multivariate Analysis of Ecological Communities. Chapman and Hall, London.

Dixon, P. M. 1993. The bootstrap and the jackknife: describing the precision of ecological indices. Pages 290-318 in S. M. Scheiner and J. Gurevitch, editors. Design and Analysis of Ecological Experiments. Chapman and Hall, New York.

Draper, N.R. & Smith, H. 1981. Applied Regression Analysis. 2^{nd}
ed. Wiley, NY.

Gauch, H. G., Jr. 1982. Multivariate Analysis in Community Structure. Cambridge University Press, Cambridge.

Gittens, R. 1981. Towards the analysis of vegetation
succession. *Vegetatio* 46:37-59

Goodall, D.W. 1954. Objective methods for the classification of vegetation. III. An essay on the use of factor analysis. Australian Journal of Botany 1:39-63.

Hill, M.O. and H.G. Gauch, Jr. 1980. Detrended
Correspondence Analysis: an improved ordination technique. *Vegetatio*
42:47-58.

Hurlbert, R.H. 1984. Pseudoreplication and the design
of ecological field experiments. *Ecological Monographs* 54:187-211.

Jackson, D. A., and K. M. Somers. 1991. Putting things
in order: the ups and downs of detrended correspondence analysis. *Am. Nat.*
137:704-12.

Jongman, R. H. G., C. J. F. ter Braak, and O. F. R. van Tongeren, editors. 1987. Data Analysis in Community and Landscape Ecology. Pudoc, Wageningen, The Netherlands.

Kent, M., and P. Coker. 1992. Vegetation description and analysis: a practical approach. Belhaven Press, London.

Knox, R. G. and R. K. Peet. 1989. Boostrapped ordination: a
method for estimating sampling effects in indirect gradient analysis. *Vegetatio*
80: 153-165.

Legendre, P. and M.-J. Fortin. 1989. Spatial pattern and ecological analysis. Vegetatio 80:107-138.

Legendre, P. and L. Legendre. 1998. Numerical Ecology. 2^{nd}
English edition. Elsevier, Amsterdam. 853 pages.

Manly, B.F.J. 1992. Randomization and Monte Carlo methods in biology. Chapman and Hall, New York. 281 pp.

Manly, B. F. J. 1993. A review of computer intensive multivariate methods in ecology. Pages 307 in G. P. Patil and C. R. Rao, editors. Multivariate Environmental Statistics. Elsevier, .

Morrison, D.F. 1967. Multivariate Statistical Methods. McGraw-Hill, New York. 415pp.

Økland, R. H. 1990. Vegetation ecology: theory, methods and applications
with reference to Fennoscandia. *Sommerfeltia* Supplement 1:1-233.

Palmer, M. W. 1993a. Putting things in even better order: the advantages of canonical correspondence analysis. Ecology 74:2215-30.

Palmer, M. W. 1993b. Potential biases in site and species selection for ecological monitoring. Environmental Monitoring and Assessment 26:277-282.

Peet, R. K., R. G. Knox, J. S. Case, and R. B. Allen. 1988. Putting things in order: the advantages of detrended correspondence analysis. Am. Nat. 131:924-34.

Pielou, E. C. 1984. The Interpretation of Ecological Data: A Primer on Classification and Ordination. Wiley, New York.

Potvin, C. and D. A. Roff. 1993. Distribution-free and
robust statistical methdos: viable alternatives to parametric statistics? *Ecology*
74: 1617-1628

Roberts, D. 1986. Ordination on the basis of fuzzy set theory. Vegetatio 66:123-131

Rossi, RE, DJ Mulla, AG Journel, and EH Franz. 1992.
Geostatistical tools for modeling and interpreting ecological spatial
dependence. *Ecol. Monogr*. 62(2):277-314.

Schneider, D. C. 1994. Quantitative Ecology: Spatial and Temporal Scaling. Academic Press, New York.

ter Braak, C. J. F. 1987. Ordination. P. 91-173 in Jongman, R.H., C.J.F. ter Braak and O.F.R. van Tongeren, editors. Data Analysis in Community Ecology. Pudoc, Wageningen, The Netherlands.

ter Braak, C. J. F., and I. C. Prentice. 1988.
A theory of gradient analysis. *Adv. Ecol. Res*. 18:271-313.

ter Braak, C. J. F., and P. Šmilauer. 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.

Ver Hoef, J.M. and D.C. Glenn-Lewin. 1989. Multiscale ordination: a method for detecting pattern at several scales. Vegetatio 82: 59-67.

Wartenberg, D., S. Ferson, and F. J. Rohlf. 1987. Putting things in order: a critique of detrended correspondence analysis. Am. Nat. 129:434-48.

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