Transcript Chapter 22

CHAPTER 22
Reliability of Ordination Results
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
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:
scaled rank variance 
observed variance in rank
expected variance in rank
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
Wilson's method
Definitions
w0 = the true underlying species ranking
~ = an estimate of the true ranking, based on species scores
w
i
on an ordination axis
X(w0,w) = the number of discordant pairs between two
rankings, w0 and w.
t = 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)
q
w 0 = the value of w to minimize
~
 X ( w, w )
i
i=1
The measure of overall disagreement between the
observed rankings based on subsets of the data and
the maximum likelihood estimated ranking is
q
X = (1 / q )
~
X
(

,
w
w
 0 i)
i=1
The expected value of Kendall's rank correlation (t)
between the true underlying species ranking and the
ordination species ranking is estimated by
-1
k

E [t ] = 1- 2   X
 2
Kendall's t ranges from -1 (complete disagreement)
to 1 (complete agreement), and it can be used as a
measure of accuracy of the ordination.
The consistency of the ordination is measured as the ratio
of the observed variation to the expected variation:
C = sx /
var(X)
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 (t) 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.