Ordinations
Bray-Curtis (Polar)
We offer numerous options and improvements beyond Bray and Curtis' original method, such
as perpendicularized axes and variance-regression endpoint selection.
Canonical Correspondence Analysis (CCA)
CCA is unique among the ordination methods in PC-ORD in that the ordination of the
main matrix (by reciprocal averaging) is constrained by a multiple regression on variables
included in the second matrix. In community ecology, this means that the ordination
of samples and species is constrained by their relationships to environmental
variables. CCA is most likely to be useful when: (1) species responses are unimodal
(hump-shaped), and (2) the important underlying environmental variables have been
measured.
Detrended Correspondence Analysis (DCA,
DECORANA)
DCA is an eigenanalysis ordination technique based on reciprocal averaging (RA;
Hill 1973). DCA is geared to ecological data sets and the terminology is based on
samples and species. DCA ordinates both species and samples simultaneously.
Non-metric Multidimensional Scaling
(NMS)
Non-metric Multidimensional Scaling (NMS, MDS, NMDS, or NMMDS) is an ordination method
that is well suited to data that are nonnormal or are on arbitrary, discontinuous, or
otherwise questionable scales. NMS is generally the best ordination method for
community data. Our auto-pilot feature makes it easy to use. A Monte Carlo
test of significance is included.
NMS Scree Plot graph example
NMS Scores
NMS Scores provides a prediction algorithm for non-metric multidimensional scaling
(NMS). This is not prediction in the sense of forecasting, but rather statistical
prediction in the same way as using multiple regression to estimate a dependent variable.
NMS Scores calculates scores for new items based on prior ordinations.
Principal Components Analysis (PCA)
Principal Components Analysis is the basic eigenanalysis technique. It maximizes the
variance explained by each successive axis. Although it has severe faults with many
community data sets, it is probably the best technique to use when a data set approximates
multivariate normality. PCA is usually a poor method for community data, but it is
the best method for many other kinds of multivariate data. Broken-stick eigenvalues
are provided to help you evaluate statistical significance.
Principal Coordinates Analysis (PCoA)
Principal Coordinates Analysis is an eigenanalysis technique similar to PCA, except that
one extracts eigenvectors from a distance matrix among sample units (rows), rather than
from a correlation or covariance matrix. In PCoA one can use any square symmetrical
distance matrix, including semi-metrics such as Sorensen distance, as well as metric
distance measures such as Euclidean distance.
Reciprocal Averaging (RA) =
Correspondence Analysis (CA)
Reciprocal averaging is also known as correspondence analysis (CA). It is
performed in PC-ORD by selecting options in program DCA adapted from the Cornell Ecology
Program series. Reciprocal averaging (RA) yields both normal and transpose
ordinations automatically. Like DCA, RA ordinates both species and samples
simultaneously.
Redundancy Analysis (RDA)
Redundancy Analysis models a set of response variables as a function of a set of predictor
variables, based on a linear model. RDA thus applies to the same conceptual problem
as canonical correspondence analysis (CCA). RDA is, however, based on a linear model
among response variables and between response variables and predictors. CCA, on the
other hand, implies a unimodal response to the predictors.
Weighted Averaging
The simplest yet often effective method of ordination is weighted averaging.
The essential operation is the same: a set of pre-assigned species weights (or
weights for species groups) are used to calculate scores for sites (sample units).
The calculation is a weighted averaging for species or species groups actually present in
a sample unit. Weighted averaging used in Federal Manual and numerous ecological
indices.
Fuzzy Set (FSO)
Fuzzy set ordination applies fuzzy set theory to direct gradient analysis in
ecological ordination. This ordination method requires the user to hypothesize the
relationship between species communities and environmental variables or other predictors.
The predictors are most commonly environmental variables, but they can also be a
secondary set of species communities, or any other quantitative data set with the same
number of rows as the community matrix. The community data are placed in the main
matrix, and the secondary set is in the second matrix. The resulting ordination is
an ordination of sample units in species space. Species can be superimposed on the
ordination by a single weighted averaging step
Compare Scores (Compare Ordinations)
Evaluate the similarity of two ordinations, independent of any rotation,
reflection, units for axis, and number of dimensions. This is accomplished by
evaluating the correlation between the interpoint distances of two ordinations. Squaring
this correlation expresses the redundancy between two ordinations. A formal test of
the hypothesis of no relationship between the two ordinations is provided by a Mantel
test.
Groups
Cluster Analysis
We offer eight fusion strategies and eight distance measures, for hierarchical,
polythetic, agglomerative cluster analysis. Results are given for each step in the
analysis, along with a publication-quality final dendrogram. Cluster Analysis graph example
Two-way Cluster Analysis
The purpose of our two-way clustering (also known as biclustering) is to graphically
expose the relationship between cluster analyses and your individual data points.
The resulting graph makes it easy to see similarities and differences between rows in the
same group, rows in different groups, columns in the same group, and columns in different
groups. You can see graphically how groups of rows and columns relate to each
other. Two-way clustering refers to doing a cluster analysis on both the rows and
columns of your matrix, followed by graphing the two dendrograms simultaneously, adjacent
to a representation of your main matrix. Rows and columns of your main matrix are
re-ordered to match the order of items in your dendrogram.
Two-way Cluster Analysis graph example
| Group Linkage Methods |
- Nearest Neighbor
- Farthest Neighbor
- Median
- Group Average
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- Centroid
- Ward's Method
- Flexible Beta
- McQuitty's Method
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| Ward's is also know as Orloci's and Minimum Variance Method |
Multi-Response Permutation Procedures
(MRPP)
MRPP is a non-parametric procedure for testing the hypothesis of no difference between two
or more groups of entities. The groups must be a priori. For example, one
could compare species composition between burned and unburned plots to test the hypothesis
of no treatment effect. Discriminant analysis is a parametric procedure that can be
used on the same general class of questions. However, MRPP has the advantage of not
requiring assumptions (such as multivariate normality and homogeneity of variances) that
are seldom met with ecological community data. Eight distance measures options are
available.
Blocked Multi-Response Permutation
Procedures (MRBP)
Randomized block experiments or paired-sample data can be analyzed with a variant of MRPP
called MRBP or blocked MRPP. PC-ORD allows up to 1000 blocks and 100 groups.
Indicator Species Analysis
Dufrêne and Legendre’s (1997) method provides a simple, intuitive solution to the
problem of evaluating species associated with groups of sample units. It combines
information on the concentration of species abundance in a particular group and the
faithfulness of occurrence of a species in a particular group. It produces indicator
values for each species in each group. These are tested for statistical significance
using a Monte Carlo technique.
Blocked Indicator Species Analysis
Dufrêne and Legendre’s (1997) method for Indicator Species Analysis can be adapted
to a randomized block experiment or a paired-sample design. The data are
pre-relativized by species within blocks (or pairs), such that the sum across groups
equals one for each block. If a species is absent from a block, the abundances are
maintained at zero. The relativization alters the relative abundance portion of the
Indicator Value (IV) index to focus on within block differences. Then the ISA is run
as usual. The randomization test differs from regular ISA in that instead of an
unconstrained permutation of group identifiers, groups are randomly permuted within
blocks.
Phi Coefficient for Indicator Species
Tichý and Chytrý's (2006) phi coefficient is a method for evaluating the indicator value
(or diagnostic value) of a species with respect to a one-way grouping of sample units.
It applies only to presence-absence data. If have quantitative data you
choose this option in the Indicator Species Analysis Setup, then the data are
automatically converted to presence-absence. Any value greater than zero is
transformed to 1, while values less than or equal to zero are transformed to zero.
Tichý and Chytrý's method corrects for unequal sample sizes among groups. The
adjusted phi coefficient also allows comparisons across studies with different sample
sizes.
Mantel Test
The Mantel test evaluates the null hypothesis of no relationship between two dissimilarity
(distance) or similarity matrices. The Mantel test is an alternative to regressing
distance matrices that circumvents the problem of partial dependence in these
matrices. Example applications are: evaluating the correspondence between two groups
of organisms from the same set of sample units or comparing community structure before and
after a disturbance. Two methods are available in PC-ORD: Mantel’s asymptotic
approximation and a randomization (Monte Carlo) method.
Partial Mantel Test
The partial Mantel test requires three matrices, the main matrix, a second matrix,
and a control matrix. The null hypothesis is of no relationship between the main and
second matrices, after controlling for the relationship with the third (control) matrix.
If we call the main matrix Y, the second matrix X, and the control matrix C, then
we seek the partial correlation between X and Y while controlling for C.
PerMANOVA
PerMANOVA performs distance-based multivariate analysis of variance, also known as
nonparametric MANOVA or npMANOVA. Hypothesis are evaluated with permutation tests,
rather than by reference to an assumed distribution. Options include one-way,
factorial, nested, and blocked designs.
SumF
A simple but surprisingly effective method of comparing two or more groups of sample units
is to calculate a univariate F statistic for each variable, sum those F statistics, then
compare the resulting sum to the distribution of F statistics based on randomizing the
data under the null hypothesis. This is the core of the SumF method, as suggested by
Edginton (1995). Good performance of this method, as compared to distance-based
methods, was found by Warton and Hudson (2004). An advantage to this method is that
by aggregating a simple, well-known test statistic, the F ratio, into a summary statistic
across multiple variables, we simultaneously obtain information about differences between
groups both across all variables and for individual variables. Thus for the generic
question, "Do communities differ between groups?", the SumF method allows us to
report an answer for communities as a whole as well as for individual species.
TWINSPAN
TWINSPAN simultaneously classifies species and samples. At its core, TWINSPAN is
based on dividing a reciprocal averaging ordination space. One of the most useful
features of TWINSPAN is the final ordered two-way table. Species names are arrayed
along the left side of the table, while sample numbers are along the top. The
pattern of zeros and ones on the right and bottom sides define the dendrogram of the
classifications of species and samples, respectively. The interior of the table
contains the abundance class of each species in each sample. Abundance classes are
defined by pseudospecies cut levels.
Import/Export File Formats
- Excel (*.xls) and Excel 2007 (*.xlsx)
- spreadsheet (*.wk1)
- compact
- database
- Cornell condensed
- comma-separated values (*.csv)
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Data Modifications
| Transformations |
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Relativizations |
- power transformation
- logarithmic
- arcsine
- arcsine squareroot
- Beals smoothing
- presence - absence
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- by totals
- by proportion of maximum
- rank
- deviation from mean
- binary with respect to median
- ubiquity
- deviation from mean
- binary with respect to mean
- info function of ubiquity
- Hellinger
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Traits
PC-ORD 7 provides ways to
relate data on species traits (trait matrix) to community samples (main matrix) and
environmental data (second matrix). While many of these operations can be done in the
other PC-ORD menu items, the Traits menu provides several operations specific to this kind
of data.
- Traits | Categorical to Binary
If, for a given variable, there are n unique categories (value labels), then n new binary
(0/1) variables will be generated. Each new variable will be designated as a Q
variable with value 0 or 1.
- Traits | Create Trait Combinations
Create one new categorical variable by combining categories from two existing
variables. Each combination of categories from the two selected variables is taken
as a new category in the new variable. The resulting new variable is always
categorical. The existing variables are left intact, but you can easily remove them
with Modify | Delete Columns.
For example, say you had two categorical variables, one coding for native vs. non-native
species, and one coding for annuals vs. perennials. That might work well in the
analyses, but what if species having a combinations of those, for example the non-native
annuals, is particularly different ecologically from all remaining species? You
might, therefore, wish to create a new categorical variable with all four combinations of
those trait categories: (1) native annuals, (2) native perennials, (3) non-native annuals,
(4) non-native perennials.
- Traits | Calculate SU x Traits Matrix
Calculating a sample unit x trait matrix provides a flexible first step in
analyzing the relationships between species traits and explanatory variables. This
matrix is obtained by multiplying a sample unit x species matrix by a species x trait
matrix, but the content of the resulting matrix depends on whether and how traits are
standardized and whether or not the multiplication is followed by a weighted averaging
step (McCune 2015). To maximize versatility of the SU x trait matrix, including
comparability among traits, and usability with a wide range of distance measures, we
recommend first standardizing traits by min-to-max, then calculating abundance-weighted
trait averages in each sample unit.
- Traits | Species Distances in Trait Space
Species can be compared in their traits by calculating a distance matrix among
species, starting with a species x trait matrix. Mathematically this is the same as
calculating a distance matrix among sample units in species space, except that in this
case the objects are species and their attributes are traits, rather than objects being
sample units and the attributes species. The same distance measures are offered from
the traits menu as for distances between sample units in species space.
- Traits | Functional Diversity
Functional diversity analyzes the combination of the sample unit x species matrix
with a species x trait matrix. Functional diversity measures attempt to describe the
diversity of species functional traits represented in a sample unit, rather that simply
species diversity. For example, a plot containing three species, all having the same
traits, are considered no more diverse than a single species. Similarly, two species
with very different functional traits contribute more functional diversity than two
species that are similar in their functional traits. For example, if we have two
species in a plot and one is a weedy sun-loving pioneer plant species, and the other is a
shade tolerant species with poor colonizing ability, that plot would have more functional
diversity than two different species that were both shade tolerant poor colonizers.
- Traits | Fourth Corner Analysis
The methodological question of linking species traits to environmental variables, via the
sample unit x species matrix, is called the fourth corner problem because of the
arrangement of four basic matrices (see the traits x environment positions in Dray and
Legendre (2008, Fig. 1a) and McCune and Grace (2002, Fig. 2.1). Fourth Corner
Analysis provides statistical tests of the strength of the links between these matrices.
For a detailed explanation of the theory and mathematics of fourth corner analysis
see Legendre et al. (1997), Dray and Legendre (2008), Ter Braak et al, (2012), and Dray et
al. (2014).
- Fuzzy Set (FSO)
Fuzzy set ordination applies fuzzy set theory to direct gradient analysis in
ecological ordination. This ordination method requires the user to hypothesize the
relationship between species communities and environmental variables or other predictors.
The predictors are most commonly environmental variables, but they can also be a
secondary set of species communities, or any other quantitative data set with the same
number of rows as the community matrix. The community data are placed in the main
matrix, and the secondary set is in the second matrix. The resulting ordination is
an ordination of sample units in species space. Species can be superimposed on the
ordination by a single weighted averaging step
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Summaries
Descriptive Statistics and Diversity
Indices
Summarize attributes of your rows or columns (mean, standard deviation, sum, minimum,
maximum, skewness, kurtosis), and measures of diversity: richness, equitability, Simpson
index, and Shannon index).
Outlier Analysis
Detect multivariate outliers. These are frequent in ecological data and they often
exert undue influence over the results of multivariate analyses.
Species-area Curves
Species-area curves are constructed by randomly subsampling a data set.
Species-area curves are frequently used during study design to help determine sample
sizes. Species-area Curves graph example
Species Lists
Easily produce species lists from your spreadsheets, based on a species file.
This file associates your species acronyms with full species names, as they will
appear in your lists. Request a species list for each sample unit, or for your
combined sample units. You can include key summary statistics, such as frequency and
abundance of each species.
Write Distance Matrix
Although many analyses in PC-ORD calculate a distance matrix and offer to write
the distance matrix to the result file, these have limited formats and options.
Consider Write Distance Matrix if you wish to use your distance matrix in other software,
save it for further analysis, or simply calculate a distance matrix with no other
analysis.
Shuffle
Randomly reassign values in columns to new positions in the same column. The
resulting data set has the same column totals, matrix total, and number of elements
containing zeros, but it effectively randomizes the data set. Why shuffle your data?
Explore how multivariate methods can appear to detect pattern from nonsense.
Generate null models for comparison with your unshuffled data.
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