1. Tables, Figures, and Equations
From: McCune, B. & J. B. Grace Analysis of Ecological Communities. MjM Software Design, Gleneden Beach, Oregon

2. Figure 21. 1. Decision tree for using CCA for community data
Figure Decision tree for using CCA for community data. Assume that we have a site  species matrix and a site  environment matrix and that chi-square distances are acceptable. RDA is a constrained ordination method based on a linear model (see “Variations” below).

3. Table Questions about the community (A) and environmental or experimental design (E) matrices that are appropriate for using CCA.

4. The basic method 1. Start with arbitrary but unequal site scores, x.
The species data matrix Y contains nonnegative abundances,
yij, for i = 1 to n sample units and j = 1 to p species.
y+j indicates species totals
yi+ indicates and sample unit (site) totals
The environmental matrix Z contains values
n sites by q environmental variables.
1. Start with arbitrary but unequal site scores, x.

5. 2. Calculate species scores, u, by weighted averaging of the site scores:
a = user-selected scaling constant as described later.

6. 2. Calculate species scores, u, by weighted averaging of the site scores:
Score for species j
a = user-selected scaling constant as described later.

7. Score (weight) for site i Score for species j
2. Calculate species scores, u, by weighted averaging of the site scores:
Score (weight) for site i
Score for species j
a = user-selected scaling constant as described later.

8. 3. Calculate new site scores, x
3. Calculate new site scores, x*, by weighted averaging of the species scores:
a = user-selected scaling constant as described later.

9. Score (weight) for species j Score for site i
3. Calculate new site scores, x*, by weighted averaging of the species scores:
Score (weight) for species j
Score for site i
a = user-selected scaling constant as described later.

10. 4. Obtain regression coefficients, b, by weighted least- squares multiple regression of the sites scores on the environmental variables. The weights are the site totals stored in the diagonal of the otherwise empty, n  n square matrix R.

11. 4. Obtain regression coefficients, b, by weighted least- squares multiple regression of the sites scores on the environmental variables. The weights are the site totals stored in the diagonal of the otherwise empty, n  n square matrix R.
Environmental matrix
WA scores

12. 5. Calculate new site scores that are the fitted values from the preceding regression:
These are the "LC scores" of Palmer (1993), which are linear combinations of the environmental variables.

13. 6. Adjust the site scores by making them uncorrelated with previous axes by weighted least squares multiple regression of the current site scores on the site scores of the preceding axes (if any). The adjusted scores are the residuals from this regression.

14. 7. Center and standardize the site scores to a mean = 0 and variance = 1.

15. 8. Check for convergence on a stable solution by summing the squared differences in site scores from those in the previous iteration. If the convergence criterion (detailed below) has not been reached, return to step 2.

16. 9. Save site scores and species scores, then construct additional axes as desired by going to step 1.

17. Axis scaling
Centered with Unit Variance. The site scores are rescaled such that the mean is zero and the variance is one. Three steps:
where
xi* is the new site score
wi* is the weight for site i
(wi* = yi+ / y++)

18. Hill's scaling standardizes the scores such that:
In CCA, Hill's scaling is accomplished by multiplying the scores by a constant based on la / 1-l (see below). Thus it is a linear rescaling of the axis scores.

19. Table Constants used for rescaling site and species scores in CCA. Combining the choices for axis scaling and optimizing species or sites results in the following constants used to rescale particular axes. Lambda (l) is the eigenvalue for the given axis. Alpha (a) is selected as described in the text.

20. Interpreting output
1. Correlations among explanatory variables
Table Correlations among the environmental variables.

21. 2. Iteration report. ITERATION REPORT
Calculating axis 1
Residual = E+04 at iteration 1
Residual = E-01 at iteration 2
Residual = E-01 at iteration 3
Residual = E-01 at iteration 4
Residual = E-02 at iteration 5
Residual = E-02 at iteration 6
Residual = E-02 at iteration 7
Residual = E-02 at iteration 8
Residual = E-03 at iteration 9
Residual = E-03 at iteration 10
Residual = E-05 at iteration 20
Residual = E-07 at iteration 30
Residual = E-09 at iteration 40
Residual = E-11 at iteration 50
Residual = E-13 at iteration 58
Solution reached tolerance of E-12 after 58 iterations.
Calculating axis 2
Residual = E+01 at iteration 1
Residual = E-03 at iteration 2
etc....

22. 3. Total variance in the species data.
It is the sum of squared deviations from expected values, which are based on the row and column totals. Let
eij = the expected value of species j at site i
y+j = total for species j,
yi+ = total for site i, and
y++ = community matrix grand total.
The variance of species j, var(yj), is

23. and the total variance is

24. 4. Axis summary statistics
Table Axis summary statistics

25. 5. Multiple regression results
Table Multiple regression results (regression of sites in species space on environmental variables).

26. 6. Final scores for sites and species
6. Final scores for sites and species. Ordination scores (coordinates on ordination axes) are given for each site, x, and each species, u (Tables 21.6, 21.7, 21.8).
Table Sample unit scores that are derived from the scores of species. These are the WA scores. Raw data totals (weights) are also given

27. Table Sample unit scores that are linear combinations of environmental variables for 100 sites. These are the LC Scores that are plotted in Fig

28. Table 21.8. Species scores and raw data totals (weights).

29. From: McCune, B Influence of noisy environmental data on canonical correspondence analysis. Ecology 78:

30. No noise LC Scores WA Scores
Figure Influence of the type and amount of noise in environmental data on LC site scores (left column) and WA site scores (right column) from CCA, based on analysis of simulated responses of 40 species to two independent environmental gradients of approximately equal strength.

31. Moderate noise added to two otherwise perfect environmental variables
LC Scores
WA Scores
Figure (cont.) A small amount of noise added to the two environmental variables.

32. 10 random environmental variables
LC Scores
WA Scores
Figure (cont.) The two underlying environmental variables replaced with ten random variables.

33. 7. Weights for sites and species
7. Weights for sites and species. Sites and species are weighted by their totals.
Table Species scores and raw data totals (weights).

34. 8. Correlations of environmental variables with ordination axes.
"interset correlations" are correlations of environmental variables with x*, the WA scores.
"intraset correlations" are correlations of environmental variables with x the LC scores.

35. Table Biplot scores and correlations for the environmental variables with the ordination axes. Biplot scores are used to plot the vectors in the ordination diagram. Two kinds of correlations are shown, interset and intraset.

36. 9. Biplot scores for environmental variables
The environmental variables are often represented as lines radiating from the centroid of the ordination. The biplot scores give the coordinates of the tips of the radiating lines (Fig. 21.3).

37. If Hill's scaling is used, then
The coordinates for the environmental points are based on the intraset correlations. These correlations are weighted by a function of the eigenvalue of an axis and the scaling constant ():
where
vjk = the biplot score on axis k of environmental variable j,
rjk = intraset correlation of variable j with axis k, and
α = scaling constant
If Hill's scaling is used, then

38. 10. Monte Carlo tests of significance
Ho: No linear relationship between matrices.
For this hypothesis, the rows in the second matrix are randomly reassigned within the second matrix.
Ho: No structure in main matrix and therefore no linear relationship between matrices.
For this hypothesis, elements in the main matrix are randomly reassigned within columns.

39. To evaluate the significance of the first CCA axis:
n = the number of randomizations (permutations) with an eigenvalue greater than or equal to the corresponding observed eigenvalue
N = the total number of randomizations (permutations)
then
p = (1 + n)/(1 + N)
p = probability of type I error for the null hypothesis that you selected.

40. Table Monte Carlo test results for eigenvalues and species- environment correlations based on 999 runs with randomized data.

41. Table 21.11. Comparison of CCA and NMS of the example data set.

42. Redundancy analysis Given matrix of response variables (A)
matrix of explanatory variables (E).
The basic steps of RDA as applied in community ecology are:
Center and standardize columns of A and E.
Regress each response variable on E.
Calculated fitted values for the response variables from the multiple regressions.
Perform PCA on the matrix of fitted values
Use eigenvectors from that PCA to calculate scores of sample units in the space defined by E.

43. Regression with multiple dependent variables
In the usual case of regressing a single dependent variable (Y) on multiple independent variables (X), the regression coefficients (B) are found by:
B = (XX)-1 X’Y
With multiple dependent variables, Y and B are matrices rather than vectors.