## Example Ordering Ellenbergs data

For the purpose of illustration the data set of Ellenberg (1956) (also published in Mueller-Dombois & Ellenberg 1974) is used in Figure 6.6 to demonstrate the effect of different methods. The number of releve groups chosen is three, the same as chosen by Ellenberg. In many cases, taking the square root of the number of releves turned out to yield pleasing results (i.e. about 15 for a table of size 200, 30 for a table of size 1000, etc.). The number of species groups in this example is eight. Because the correlation among species is generally low, the groups should be small in size (i.e. three to six species). Under many circumstances, dividing the number of species by about four may be a good choice for the number of species groups. The number of species displayed is 30 out of 94.

In Figure 6.6 (a) the result of clustering is shown. As can be expected, the blocks of high-frequency non-zero scores are dispersed. Figure 6.6

Figure 6.6 Structuring the meadow data set of Ellenberg (Mueller-Dombois & Ellenberg 1974). (a) Ordering based on cluster analysis only. (b) Blocks rearranged by analysis of concentration. (c) Within-group order changed according to correspondence analysis (complete ordering). (d) Ordering with four instead of three releves groups.

(b) exhibits the same classification, but the blocks are rearranged by analysis of concentration. In (c), within-group order is changed according to correspondence analysis. In some cases, small within-group gradients can be identified. This slightly improves the appearance of the diagonal structure of the table, but does not improve on the result of classification. In (d), the number of releve groups is increased to four. This example demonstrates that the releve groups determine the list of differentiating species. The selection of differentiating species has changed completely. The last group consists of one releve only, and species occurring only there achieve high F-values. It has to be noted that such a solution with a single releve in a group is not practical as the variance within the group is not defined. Clearly, releve number 19 is an outlier (see Section 11.3) and should be removed from the set prior to analysis (Wildi 1989).