Du is to Dio what D2 is to D7. Because, in Dw, the difference for each descriptor is first expressed as a fraction, before squaring the values and summing them, this coefficient may be used with species abundance data. As in coefficient D8,however, double-zeros must be excluded from the computation and their number subtracted from p. Unless one intends to use this coefficient as a basis for ordination (Chapter 9), it is better, for species abundance data, to use the semimetric DM described below.
Another coefficient, which is related to Dw, was developed by Pearson (1926) for anthropological studies under the name coefficient of racial likeness. Using this coefficient, it is possible to measure a distance between groups of sites, like with the Mahalanobis generalized distance D5, but without eliminating the effect of correlations among descriptors:
for two groups of sites wj and W2 containing respectively «1 and «2 sites; yij is the mean of descriptor j in group i and s- is the corresponding variance.
Other measures, related to %2, are available to calculate the distance among sites using species abundances or other frequency data; no negative value is allowed in the data. The first of these coefficients is called the %2 metric. The sum of squares of differences is calculated between profiles of conditional probabilities in two rows (or columns) of a frequency table, weighting each term of the sum of squares by the inverse of the frequency of the column (or row) in the overall table. This measure has been used by Roux & Reyssac (1975) to calculate distances among sites described by species abundances.
In order to compute the %2 metric, the data matrix must be transformed into a matrix of conditional probabilities; see Table 6.7. If the metric is computed between the rows of the matrix, the conditional probabilities are computed by rows. The elements of the matrix become new terms yij/yi+ where yi+ is the sum of frequencies in row i. In the numerical example below, rows of the data matrix, on the left-hand side, are sites and columns are species. The matrix of conditional probabilities (by rows) on the right is used below to compute the association between rows (sites):
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