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of the table contains the probabilities of selecting an object having the various states of descriptor b, assuming that this object has the state of descriptor a corresponding to the row. The first row, for example, contains all the objects with dominant species a1, and it gives their probability distribution on descriptor b. The sum of each row is 1, since each frequency in the contingency table (Table 6.1, for the example) has been divided by the (marginal) total of the corresponding row.

To analyse a table of conditional probabilities, one identifies the conditional probabilities that are larger than the corresponding unconditional probabilities, in the column headings. Under the null hypothesis of independence of the two descriptors, each conditional probability distribution (i.e. each row of the table) should be approximately equal to the unconditional distribution (heading row). Thus, in Table 6.7, b has a probability of 0.50 among the objects with dominant species a1, while this same state b has a probability of 0.25 among all objects. A similar table of conditional probabilities could have been computed with descriptor a conditional on b; the cells of interest, and thus the ecological conclusions, would have been exactly the same.

Test of It is also possible to test the significance of the difference between Oj and Ej in

Oij = Eij each cell of the contingency table. Ecologists may be interested in any difference, whatever its sign, or only in cases where Oij is significantly higher than Ej (preference) or significantly lower (avoidance, exclusion).

Bishop et al. (1975: 136 et seq.) describe three statistics for measuring the difference between O and E. They may be used for two-way or multiway contingency

2 2 tables. The three statistics are the components of XP , components of XW , and

Freeman-Tukey deviates:

These statistics are available in various computer packages. A critical value has been proposed by Bishop et al. (1975) for testing the significance of statistics 6.26 and 6.28:

Eij is said to be significantly different from Oj when the absolute value of the statistic, for cell (i, j), is larger than the critical value. According to Sokal & Rohlf (1995), however, the above critical value often results in a type I error much greater than the nominal a level. These authors use instead the following approximate criterion to test Freeman-Tukey deviates:

In cells where the (absolute) value of the Freeman-Tukey deviate is larger than the criterion, it is concluded that E¡j ^ Oj at significance level a. Neu et al. (1974) recommend to test only the cells where 5 < E¿j < (n - 5). It is also recommended to apply a Bonferroni or Holm correction (Box 1.3) to significance level a in order to account for multiple testing. An example is provided in Table 6.8.

Alternatively, the following statistic (adapted from Neu et al., 1974) may be computed:

where n is the total number of observations in the contingency table. When statistic Z is larger than the critical value z[1 _ (a / 2 no. cells)] read from a table of standard normal deviates, it is concluded that Oij is significantly different from Eij at probability level a/2 (one-tailed test); the further division by the number of cells is the Bonferroni correction (Box 1.3). Statistics higher than the critical value z, in Table 6.9, are in boldface type. As is often the case, the conclusions drawn from Tables 6.8 and 6.9 are not the same.

Comparing Tables 6.4 and 6.7 to Tables 6.8 and 6.9 shows that considering only the cells where Oj > Ej may lead to conclusions which, without necessarily being

Table 6.8 Statistics (Freeman-Tukey deviates, eq. 6.28) for testing the significance of individual cells in a contingency table. The observed and expected values are in Table 6.4. Absolute values larger than the criterion (eq. 6.29) [9 x2i n 05] /16]1/2 = [9 x 3.84 / 16]1/2 = 1.47 are in boldface type. A

Bonferroni-corrected criterion [9 X^ n n5/16] / 16]1/2 = [9 x 9.5 / 16] = 2.31 would have led to the same conclusions with the present example. Values in boldface print identify the cells of the table in which the number of observations (Oj) significantly (p < 0.05) differs (higher or lower) from the corresponding expected frequencies (Ej). The overall null hypothesis (Ho: complete independence of descriptors a and b) was rejected first (Table 6.4) before testing the significance of the observed values in individual cells of the contingency table.

Table 6.8 Statistics (Freeman-Tukey deviates, eq. 6.28) for testing the significance of individual cells in a contingency table. The observed and expected values are in Table 6.4. Absolute values larger than the criterion (eq. 6.29) [9 x2i n 05] /16]1/2 = [9 x 3.84 / 16]1/2 = 1.47 are in boldface type. A

Bonferroni-corrected criterion [9 X^ n n5/16] / 16]1/2 = [9 x 9.5 / 16] = 2.31 would have led to the same conclusions with the present example. Values in boldface print identify the cells of the table in which the number of observations (Oj) significantly (p < 0.05) differs (higher or lower) from the corresponding expected frequencies (Ej). The overall null hypothesis (Ho: complete independence of descriptors a and b) was rejected first (Table 6.4) before testing the significance of the observed values in individual cells of the contingency table.

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