1. CHAPTER 22 Reliability of Ordination Results 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. Bootstrapped ordination Calculate variance in rank of species scores across bootstrap replicates. These variances were averaged across species. The average variance was then rescaled to range from 0 to 1:

3. Pillar’s (1999b) method: 1.Save the usual ordination scores for k axes from the complete data set (n  p). Call the n  k scores the original ordination. 2.Draw a bootstrapped sample of size n. 3.Ordinate the sample. 4.Perform Procrustes rotation of the k axes from the bootstrapped ordination, maximizing its alignment with the original ordination. 5.Calculate the correlation coefficient between the original and bootstrapped ordination scores, saving a separate coefficient for each axis. The higher the correlation, the better the agreement between the scores for the full data set and the bootstrap. 6.Repeat steps 1-5 for a randomization of the original data set. The elements of the complete data set are randomly permuted within columns. 7.For each axis, if the correlation coefficient from step 5 for the randomized data set is greater than or equal to the correlation coefficient from the nonrandomized data set, then increment a frequency counter, F = F + 1. 8.Repeat the steps above many times (B = 40 or more). 9.For the null hypothesis that the ordination structure of the data set is no stronger than expected by chance, calculate a probability of type I error: p = F/B

4. Wilson's method Definitions w 0 = the true underlying species ranking = an estimate of the true ranking, based on species scores on an ordination axis X(w 0,w) = the number of discordant pairs between two rankings, w 0 and w.  = Kendall's tau, a rank correlation coefficient, which is a linear function of X. q = the number of rankings (subsets) k = the number of objects (species) = the value of w to minimize

5. The measure of overall disagreement between the observed rankings based on subsets of the data and the maximum likelihood estimated ranking is

6. The expected value of Kendall's rank correlation (  ) between the true underlying species ranking and the ordination species ranking is estimated by Kendall's  ranges from -1 (complete disagreement) to 1 (complete agreement), and it can be used as a measure of accuracy of the ordination.

7. The consistency of the ordination is measured as the ratio of the observed variation to the expected variation:

8. Procedure 1.Randomly partition the sample into q subsets. 2.By ordination, produce q rankings of the p species. 3.Test for overall independence of the rankings. If the hypothesis of independence is not rejected, stop. 4.Calculate the maximum likelihood estimate of the true species ranking. 5.Measure the accuracy (  ) of the ordination rankings. 6.Measure the consistency (C) of the rankings. 7.Wilson (1981) also recommended testing the fit of the observations to the model, by comparing observed and expected frequencies of X with a Kolmogorov-Smirnov or chi-square test. If the model is inappropriate, reject the analysis and stop.