Reliability of Ordination Results

1
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
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

• Pillars (1999b) method
• Save the usual ordination scores for k axes from
  the complete data set (n ? p). Call the n ? k
  scores the original ordination.
• Draw a bootstrapped sample of size n.
• Ordinate the sample.
• Perform Procrustes rotation of the k axes from
  the bootstrapped ordination, maximizing its
  alignment with the original ordination.
• 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.
• Repeat steps 1-5 for a randomization of the
  original data set. The elements of the complete
  data set are randomly permuted within columns.
• 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.
• Repeat the steps above many times (B 40 or
  more).
• 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 w0 the true underlying species
ranking an estimate of the true ranking,
based on species scores 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)
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
(t) between the true underlying species ranking
and the ordination species ranking is estimated by
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.
7
The consistency of the ordination is measured as
the ratio of the observed variation to the
expected variation
8

• Procedure
• Randomly partition the sample into q subsets.
• By ordination, produce q rankings of the p
  species.
• Test for overall independence of the rankings.
  If the hypothesis of independence is not
  rejected, stop.
• Calculate the maximum likelihood estimate of the
  true species ranking.
• Measure the accuracy (t) of the ordination
  rankings.
• Measure the consistency (C) of the rankings.
• 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.