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From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities . MjM Software Design, Gleneden Beach, Or PowerPoint Presentation
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From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities . MjM Software Design, Gleneden Beach, Or - PowerPoint PPT Presentation


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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. Table 9.1. Domain of input and range of output from transformations. Monotonic transformations Power transformation.

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slide1

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

slide3

Monotonic transformations

      • Power transformation
slide4

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.

slide6

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:

slide7

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

slide8

Arcsine transformation

bij = 2/p * arcsin(xij)

Arcsine squareroot transformation

bij = 2/p * arcsin

slide10

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.
slide11

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).
slide12

Box 9.1. (cont.) Example of Beals smoothing

Construct new matrix B containing values transformed with Beals smoothing function:

Data after transformation (B):

slide13

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
slide14

Relativizations

"To relativize or not to relativize, that focuses the question." (Shakespeare, ????)

slide15

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.

slide16

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.

slide17

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.

slide18

General relativization

By rows: By columns:

for a matrix of n rows and q columns.

slide19

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.
slide21

Binary with respect to mean

bij = 1 if xij > , bij = 0 if xij

slide22

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.

slide23

Binary with respect to median

bij = 1 if xij > median, bij = 0 if xij median

slide24

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.
slide25

Information function of ubiquity

  • where

and pj = Nj /N with Nj and N as defined above.

slide26

Double relativizations

  • Bray and Curtis (1957):
    • First relativized by species maximum, equalizing the rare and abundant species.
    • Then they relativized by SU total
slide27

"contingency deviate" relativization

Austin and Greig-Smith (1968)

slide28

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).

slide29

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.

slide30

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

slide31

First difference of time series

bij = aij,t+1 - aij,t

for a community sampled at times t and t+1.

slide32

First difference of time series

bij = aij,t+1 - aij,t

for a community sampled at times t and t+1.

  • Absolute differences, creating a matrix of species’ contributions to community change, without regard to the direction of the change:
        • bij = | aij,t+1 - aij1,t |
slide33

A general procedure for data adjustments

Species data

Table 9.3. Suggested procedure for data adjustments of species data matrices.

slide34

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,Whittaker`s 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 0.1000E+04

--------------------------------------------------------------------------

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-

--------------------------------------------------------------------------

slide35

A general procedure for data adjustments

Species data

Table 9.3. Suggested procedure for data adjustments of species data matrices.

< 50 = S

50-100 = M

100-300 = L

> 300 = XL

slide36

A general procedure for data adjustments

Species data

Table 9.3. Suggested procedure for data adjustments of species data matrices.

slide40

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,Whittaker`s 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 0.1000E+04

--------------------------------------------------------------------------

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-

--------------------------------------------------------------------------