From: McCune, B'

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Tables, Figures, and Equations
From McCune, B. J. B. Grace. 2002. Analysis
of Ecological Communities. MjM Software Design,
Gleneden Beach, Oregon http//www.pcord.com
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Table 9.1. Domain of input and range of output
from transformations.
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Monotonic transformations Power transformation
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Figure 9.1. Effect of square root and higher
root transformations, b f(x). Note that roots
higher than three are essentially
presence-absence transformations, yielding values
close to 1 for all nonzero values.
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Logarithmic transformation
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If the lowest nonzero value in the data is one
(as in count data), then it is best to add one
before applying the transformations
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If the lowest nonzero value of x differs from one
by more than an order of magnitude, then The
following transformation is a generalized
procedure that (a) tends to preserve the original
order of magnitudes in the data and (b) results
in values of zero when the initial value was
zero. Given Min(x) is the smallest nonzero
value in the data Int(x) is a function that
truncates x to an integer by dropping digits
after the decimal point c order of magnitude
constant Int(log(Min(x)) d decimal constant
log-1 (c) then the transformation is
bij log(xij d) - c
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Arcsine transformation
bij 2/p arcsin(xij)
Arcsine squareroot transformation bij 2/p
arcsin
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Figure 9.2. Effect of several transformations on
proportion data.
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Beals smoothing
The index evaluates the favorability of a given
sample for species i, based on the whole data
set, using the proportions of joint occurrences
between the species that do occur in the sample
and species i.
where Si is the number of species in sample unit
i, Mjk is the number of sample units with both
species j and k, and Nk is the number of sample
units with species k.
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Box 9.1. Example of Beals smoothing Data matrix
X before transformation (3 sample units ? 5
species)
Si number of species in sample unit i. Nj
number of sample units with species j. Construct
matrix M, where Mjk number of sample units with
both species j and k. (Note that where j k,
then Mjk Nj).
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Box 9.1. (cont.) Example of Beals smoothing
Construct new matrix B containing values
transformed with Beals smoothing function
Data after transformation (B)
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Box 9.1. (cont.) Example of Beals smoothing
Example for sample unit 1 and species 2 b1,2
1/4 (1/2 0/1 0/2 0/1) b1,2 0.25
(0.5) b1,2 0.125 (rounded to 0.13 in matrix
above) Example for sample unit 3 and species
2 b3,2 1/2 (1/2 1/1) b3,2 0.5
(1.5) b3,2 0.75
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Relativizations "To relativize or not to
relativize, that focuses the question."
(Shakespeare, ????)
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Table 9.2. Evaluation of degree of variability
in row or column totals as measured with the
coefficient of variation of row or column totals.
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Figure 9.3. Effect of various transformations on
relative weighting of species. Species abundance
was measured on a continuous, quantitative scale.
Rank is the order of species ranked by their
abundance.
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Figure 9.3. (cont.) Effect of various
transformations on relative weighting of species.
Species abundance was measured on a continuous,
quantitative scale. Rank is the order of
species ranked by their abundance.
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General relativization By rows By columns
for a matrix of n rows and q columns.
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Relativization by maximum bij xij /xmaxj where
rows (i) are samples and columns (j) are
species, xmaxj is the largest value in the
matrix for species j.
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Adjustment to standard deviate
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Binary with respect to mean
bij 1 if xij gt , bij 0 if xij ?
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Rank adjustment
Matrix elements are assigned ranks within rows or
columns such that the row or column totals are
constant. Ties are assigned the average rank of
the tied elements. For example, the values 1, 3,
3, 9, 10 would receive ranks 1, 2.5, 2.5, 4, 5.
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Binary with respect to median
bij 1 if xij gt median, bij 0 if xij ?
median
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Weighting by ubiquity
If rows are samples, columns are species, and
relativization is by columns, more ubiquitous
species are given more weight. Under these
conditions Nj is the number of samples in which
species j occurs and N is the total number of
samples.
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Information function of ubiquity
where
and pj Nj /N with Nj and N as defined above.
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Double relativizations
Bray and Curtis (1957) First relativized by
species maximum, equalizing the rare and abundant
species. Then they relativized by SU total
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"contingency deviate" relativization Austin and
Greig-Smith (1968)
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Deleting rare species
Figure 9.4. Correlation between ordination axis
scores and environmental variables can often be
improved by removal of rare species. In this
case, the strength of relationship between
hydrologic variables and vegetation, as measured
by r2, is maximized with removal of species
occurring in fewer than 5-15 of the sample
units, depending on the hydrologic variable. The
original data set contained 88 species 59, 35,
16, and 9 species remained after removal of
species occurring in fewer than 5, 15, 40, and
45 of the sample units, respectively. Data are
courtesy of Nick Otting (1996, unpublished).
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Figure 9.5. Response of A statistic (blocked
MRPP) to removal of rare species from small
mammal trapping data. A measures the effect size
of the treatments, in this case different stand
structures.
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Difference between two dates
If aij1 and aij2 are the abundances of species j
in sample unit i at times 1 and 2, then the
difference between dates is bij aij2 - aij1
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First difference of time series
bij aij,t1 - aij,t for a community sampled
at times t and t1.
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First difference of time series
bij aij,t1 - aij,t for a community sampled
at times t and t1.
Absolute differences, creating a matrix of
species contributions to community change,
without regard to the direction of the
change bij aij,t1 - aij1,t
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A general procedure for data adjustments
Species data
Table 9.3. Suggested procedure for data
adjustments of species data matrices.
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Example data set profile from PC-ORD 5.
Data Set Profile
Main matrix
StreamRestoration.wk1



Second
matrix StreamRestoration2.wk1




-------------------------------------------------
-------------------------
Main matrix Second
Matrix -------------------------------------------
------------------------------- zeros
78.2
11.1 Average distance - Sorensen 55.97955
Rela.Eucl. 0.41136 -------------------------
-------------------------------------------------
Beta diversity,Whittakers 3.6
--- Beta diversity,ave.1/2 changes
1.2 --- Range(orders
magnitude base10) 1.3
7.0 Lowest nonzero value 0.0436
0.1000E-03 Highest value
0.8165 0.1000E04 ------------
--------------------------------------------------
------------ Rows
Columns Rows Columns Contents
54 Sites 67 Attribut 54 sites
14 attribut Skewness Average 3.2
3.8 2.0 1.6
Maximum 5.1 7.3
3.6 6.9 Minimum 1.5
-0.5 1.3 -0.1 CV of
totals, 26.10 179.03 42.24
160.31 -----------------------------------
--------------------------------------- Potential
Outliers Distance measure Sorensen
Rela.Eucl.
SD-Item SD-Item SD-Item
SD-Item 4.4-Pott Crk 0.0-
3.9-Brushy F 2.4-Vol LWD/
2.5-Little W 0.0- 3.2-Lindley
0.0- 2.0-yates
mi 0.0- 2.4-Pott Crk 0.0-
-----------------------------------------------
---------------------------
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A general procedure for data adjustments
Species data
Table 9.3. Suggested procedure for data
adjustments of species data matrices.
lt 50 S 50-100 M 100-300 L gt 300 XL
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A general procedure for data adjustments
Species data
Table 9.3. Suggested procedure for data
adjustments of species data matrices.
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Species data, cont.
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Species data, cont.
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Species data, cont.
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Example data set profile from PC-ORD 5.
Data Set Profile
Main matrix
StreamRestoration.wk1



Second
matrix StreamRestoration2.wk1




-------------------------------------------------
-------------------------
Main matrix Second
Matrix -------------------------------------------
------------------------------- zeros
78.2
11.1 Average distance - Sorensen 55.97955
Rela.Eucl. 0.41136 -------------------------
-------------------------------------------------
Beta diversity,Whittakers 3.6
--- Beta diversity,ave.1/2 changes
1.2 --- Range(orders
magnitude base10) 1.3
7.0 Lowest nonzero value 0.0436
0.1000E-03 Highest value
0.8165 0.1000E04 ------------
--------------------------------------------------
------------ Rows
Columns Rows Columns Contents
54 Sites 67 Attribut 54 sites
14 attribut Skewness Average 3.2
3.8 2.0 1.6
Maximum 5.1 7.3
3.6 6.9 Minimum 1.5
-0.5 1.3 -0.1 CV of
totals, 26.10 179.03 42.24
160.31 -----------------------------------
--------------------------------------- Potential
Outliers Distance measure Sorensen
Rela.Eucl.
SD-Item SD-Item SD-Item
SD-Item 4.4-Pott Crk 0.0-
3.9-Brushy F 2.4-Vol LWD/
2.5-Little W 0.0- 3.2-Lindley
0.0- 2.0-yates
mi 0.0- 2.4-Pott Crk 0.0-
-----------------------------------------------
---------------------------
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Environmental data
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Environmental data
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Environmental data