1. From: McCune, B. & J. B. Grace. 2002. Analysis of Ecological Communities. MjM Software Design, Gleneden Beach, Oregon http://www.pcord.comhttp://www.pcord.com Tables, Figures, and Equations

2. Table 9.1. Domain of input and range of output from transformations.

3. Monotonic transformations Power transformation

4. 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.

5. Logarithmic transformation

6. 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:

7. 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 b ij = log(x ij + d) - c

8. Arcsine transformation b ij = 2/  * arcsin(x ij ) Arcsine squareroot transformation b ij = 2/  * arcsin

9. Figure 9.2. Effect of several transformations on proportion data.

10. 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 S i is the number of species in sample unit i, M jk is the number of sample units with both species j and k, and N k is the number of sample units with species k.

11. Box 9.1. Example of Beals smoothing Data matrix X before transformation (3 sample units  5 species): S i = number of species in sample unit i. N j = number of sample units with species j. Construct matrix M, where M jk = number of sample units with both species j and k. (Note that where j = k, then M jk = N j ).

12. Box 9.1. (cont.) Example of Beals smoothing Construct new matrix B containing values transformed with Beals smoothing function: Data after transformation (B):

13. Example for sample unit 1 and species 2: b 1,2 = 1/4 (1/2 + 0/1 + 0/2 + 0/1) b 1,2 = 0.25 (0.5) b 1,2 = 0.125 (rounded to 0.13 in matrix above) Example for sample unit 3 and species 2: b 3,2 = 1/2 (1/2 + 1/1) b 3,2 = 0.5 (1.5) b 3,2 = 0.75 Box 9.1. (cont.) Example of Beals smoothing

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

15. 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. CV, %Variability among rows (or columns) < 50Small. Relativization usually has small effect on qualitative outcome of the analysis. 50-100Moderate (with a correspondingly moderate effect on the outcome of further analysis). 100-300Large. Large effect on results. > 300Very large.

16. 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.

17. 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.

18. General relativization By rows:By columns: for a matrix of n rows and q columns.

19. Relativization by maximum b ij = x ij /xmax j where rows (i) are samples and columns (j) are species, xmax j is the largest value in the matrix for species j.

20. Adjustment to standard deviate

21. Binary with respect to mean b ij = 1 if x ij >, b ij = 0 if x ij 

22. 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.

23. Binary with respect to median b ij = 1 if x ij > median, b ij = 0 if x ij  median

24. 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: N j is the number of samples in which species j occurs and N is the total number of samples.

25. Information function of ubiquity where and p j = N j /N with N j and N as defined above.

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

27. "contingency deviate" relativization Austin and Greig-Smith (1968)

28. 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 r 2, 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).

29. 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.

30. Difference between two dates If a ij1 and a ij2 are the abundances of species j in sample unit i at times 1 and 2, then the difference between dates is: b ij = a ij2 - a ij1

31. First difference of time series b ij = a ij,t+1 - a ij,t for a community sampled at times t and t+1.

32. First difference of time series b ij = a ij,t+1 - a ij,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: b ij = | a ij,t+1 - a ij1,t |

33. A general procedure for data adjustments Species data Table 9.3. Suggested procedure for data adjustments of species data matrices.

34. ****************************** 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- -------------------------------------------------------------------------- Example data set profile from PC-ORD 5.

35. 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

36. A general procedure for data adjustments Species data Table 9.3. Suggested procedure for data adjustments of species data matrices.

37. Species data, cont.

40. ****************************** 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- -------------------------------------------------------------------------- Example data set profile from PC-ORD 5.

41. Environmental data