


New Graphs

 Hilltop Plot
A hilltop plot is a way of showing more than one nonlinear response surface at a time on a
single graph. The first use that we are aware of was in Nelson et al. (2015). In PCORD, response surfaces are superimposed on an
ordination as an overlay. This enables simultaneous measurement and display of one
and twodimensional, nonlinear community–trait–environment associations.
For each selected overlay variable, we trace a particular contour that is specified as a
percentage of that variable's range. Each contained area is a "hilltop",
and multiple partially transparent hilltops are superimposed on one ordination. The
resulting diagram shows the maxima of many nonlinear overlay variables (e.g. traits,
species abundances, or environmental variables) in a single figure.
Because the hilltops are based on a contoured response surface, you can better understand
the basis for hilltop plots by reading about contour overlays. The chief difference
in use between the hilltops and the contour plots is that contour plots can be shown for
only one variable at a time, while one can graph many hilltops on the same ordination.
This comes at a loss of information in that most of the contour plot is discarded
when converting to a hilltop, but has the advantage of representing multiple nonlinear
relationships at once..
 Contour Overlay
When you select a Q variable as an overlay, you are given a table of how the fit varies
with the flexibility (smoothing parameter) used in the nonparametric regression of the
overlay variable against the axis scores. If you choose "Optimize" flexibility, PCORD will select the flexibility with the highest
crossvalidated fit (xR²). If you wish to
choose a specific value for flexibility, select "Specify"
flexibility. Use the table presented under "Optimize" to help you choose a
specific flexibility, if desired. Finally, you get a contour overlay onto the
ordination space that you can customize to include color shading and/or contour interval
labeling.
 Convex Hulls
Filled Polygons
A convex hull is an overlay that uses a polygon to enclose all of the points in a group.
The purpose of this overlay is to show the outline of a group by using the outermost
points of that group in an ordination or a scatterplot. This can help the viewer to
discern if and how groups are separated or overlap in the scatterplot.
The basic rule for forming the convex hulls is that the outer points in a group are
connected in a closed polygon such that adjoining segments always make an interior angle
less than or equal to 180 degrees. At least three points are needed to draw a convex
hull. A convex hull for just three points must, however, have each of those points
as a vertex (corner) of the polygon.
 NMS Stress by
Iteration
A realtime display of how stress is changing with each iteration of NMS. The
display is updated with every single step that attempts to improve the configuration of
points in the ordination space.
This graph is meant to be both entertaining and informative. The entertainment is
that you get a dynamic, colorful, interesting window into the progress of NMS. For every
run and dimensionality you can see how stress is reduced as the iterations increase.
Each dimensionality is color coded, and separate panels are displayed for the real
and randomized runs.
Informativeness of these graphs comes from the insights into stability and consistency of
alternative NMS solutions. Unstable configurations show up as vertical zigzags.
Consistency shows up as minimum stress plateaus at a given level. The
importance of dimensionality shows up as a decreasing series of final stress as
dimensionality increases.


Traits
PCORD 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 PCORD menu items, the
Traits menu provides several operations specific to this kind of data.

 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 sunloving 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.
 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).
 Categorical to Binary
If, for a given variable, there are n unique categories (i.e., class levels with unique
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.
 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. nonnative
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 nonnative
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) nonnative annuals,
(4) nonnative perennials.
 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 mintomax, then calculating abundanceweighted
trait averages in each sample unit.
 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.


New Matrix File Format
Interface 
 
 Row and column indentifiers maximum length increased
from 8 characters to
12 characters.
 Categorical values can now be either numerical
(ShadeTolerQ in above example) or text (Dispersal, Leaf, Leaf, LearPersist, and
ShadeTolerC in above example) with a maximum of
20 characters.
 Matrix size increased from 32,000 rows x 32,000 columns
to
2,000,000 rows x 2,000,000 columns with a maximum of 536,848,900
elements with some limitations.
 Note: You can still import Excel old .wk1 files as well as .xls and .xlsx
files.



New Analyses

 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
 Distancebased Redundancy Analysis (dbRDA)
Distancebased redundancy analysis (dbRDA) is similar to redundancy analysis (RDA), except
that the main matrix is replaced by its principal coordinates, using a distance measure of
your choice. The purpose of this variant is to allow you to choose nonEuclidean
distance measures such as Sorensen (BrayCurtis), that have proven effective in community
ecology.
 Categorical Counts
Categorical Counts provides a way to track the number of cases (rows, usually sample
units), with a given categorical value. By default this is done for all categorical
variables in the selected matrix. This provides a quick assessment of the frequency
of categories, which is useful for such issues as balance in experimental designs,or
sampling effectiveness in different categories.
 Functional Diversity
Functional diversity analyzes the combination of the sample unit x species matrix with a
species x trait matrix. The rationale and use of functional diversity measures in
PCORD are described in the following topics.


Additions to
Existing Analyses

 Gower Distance
Gower's (1971a) coefficient is highly unusual among measures of similarity or
dissimilarity, because it can be calculated based on qualitative (categorical) data,
quantitative data, or mixtures of the two. Categorical data are handled as a
matching problem: items that share a qualitative attribute receive a unit of similarity
from that attribute. See Legendre and Legendre (1998) for a detailed description of
the method.
 Gower Distance, ignore 0,0
Gower (1971) and Legendre and Legendre (1998) suggested an intriguing but largely
untested variant of Gower's coefficient of similarity that ignore 0,0 data pairs. If
these double zeros are considered ambiguous information, you can exclude them from the
computation of the coefficient. Sensitivity to double zeros is notorious in
community ecology in producing unwanted effects on an analysis (Legendre & Legendre
1989, p. 253; McCune & Grace 2002, p. 38, 51).
Legendre and Legendre (1998) presented a modified version of the Gower coefficient of
similarity (S19), calling it " asymmetrical" because matching zeros are handled
differently than nonzeros. It is the same as the Gower coefficient, except that for
quantitative variables (0,0) pairs are excluded and the sum of partial similarities is
thus divided not by p variables, but by p* the number of non (0,0) pairs.
Note that this sense of "asymmetric" is different than in matrix symmetry.
If Legendre's asymmetric version of Gower's similarity is converted to a distance
(or dissimilarity) and used to build a distance matrix, this is still a symmetric matrix.
In other words the distance between items A and B is the same as the distance
between B and A, even if using the asymmetric version of Gower's similarity. To
avoid that confusion PCORD uses the
terminology "Gower, Ignore 0,0" in the menu system and output files.
 MorisitaHorn Distance
Horn (1966) modified Morisita's (1959) similarity measure, resulting in what is now
known as MoristaHorn similarity or distance. The chief appeal of this distance
measure is that it is relatively insensitive to sampling effort (Wolda 1981). It is
thus most useful for cases where it is impossible to control the time, area, or volume of
sampling.
The MorisitaHorn index is only weakly influenced by minor species, which is largely the
reason that the method resistant to undersampling. At the same time, it makes the
method insensitive to pattern that is carried in minor species.
 Increased maximum number of blocks or groups in perMANOVA and Indicator Species Analysis
to 1000
 New Summary  Write distance matrix options:
 Bootstrapped confidence intervals for species area curve.
 PCA option to write prediction equations into text file.
 PCA option to write eigenvalues from randomizations into spreadsheet.
 Added R²perm adjusted variance represented in CCA using method of PeresNeto et al.
(2006).
 Added Ezekiel and R²perm adjusted variance represented in RDA, using methods of
PeresNeto et al. (2006).
 Added measures of fit, including randomizationbased methods, to NMS.
 Dust bunny index as measure of departure from multivariate normality toward the dust
bunny distribution.
 New Summary  Write distance matrix options:
 write subdiagonal distances only (for sequential samples)
 write every other subdiagonal distances (for paired samples)


Graph Enhancements

 Apply button to instantly preview changes without exiting Preferences dialog
 Save scores for group centroids to spreadsheet or text file
 Save graphs as TIFF
 Species Frequency Label Cutoff
 Overlay from third axis
 Control over number of tick marks
 Reorder legend
 Customization of toolbar


Improved Data
Format and Management

 New traits window for simultaneous display of species traits matrix.
 Text allowed in the categorical variable type. For example, instead of coding a C
variable for experimental treatments as 1, 2, and 3, you can code them as
"burned", "mowed", and control". The consequence of this is that
categorical variables can no longer be used in arithmetic operations, only to define
groups of items.
 Conversion between variable types:
 Categorical to Quantitative
 Quantitative to Categorical
 Categorical to Binary

 Row names and column names (acronyms) increased to 12 character maximum. See Summary of Limitations.
 Spreadsheet design allows up to 2 million rows or columns. See Summary of Limitations and
memory use.
 Streamlined import/export to Excel
 New Project redesigned for drag and drop setup of matrices and easier file management
 Save All to save groups of related new files with common base name.
 Export Project is a convenient way to move all the files and settings for a given
project from one machine to another.
 Modifed File  Import simple spreadsheet options to include defining a specific range of
cells for the matrix
 Compact format: Code numbers increased from 4digit max to 8digit max. Code names
(species acronyms) increased from 8 character max to 12character max.
 Click and drag to switch matrices.
 Convert main or second matrix to graph file. This provides an easy way to use ordination
coordinates produced outside PCORD. In version 6 this was available only as the addin,
"wk1togph".
 Recode categorical variables (numbers to text or text to text)


Software Management
 Builtin free update to latest version
 Distribution by download from web with license key



Multivariate Analysis for
Ecologists: StepbyStep

 

Second edition, custommade for the beginning data analyst, this book also includes techniques and tips
useful to advanced users. A 10step analysis process using tools available in
versions 57 guides both users who can, and cannot, participate in PCORD training. 
 

 
 
 