NNCTNiihama National College of Technology (Japan)
NNCTNagaoka National College of Technology (Japan)
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The NNCT for this data set is presented in Table 13.
Notice that the test statistics and the corresponding p values imply that the allocations of the tree species are asymmetric in mixed and shared NN structure (as was suggested in the NNCT and Q-symmetry contingency table), since the corresponding p values for Dixon's symmetry test and Pielou's second type of symmetry test and Fisher's exact test are significant (p valuesbasedonMonte Carlorandomization are significant for all tests).
For example, for the unrestricted pairwise tests for Dixon's symmetry test, we use the off-diagonal entries [N.sub.ij] and [N.sub.ji] in the NNCT in Table 13 and the test statistic [Z.sup.ij.sub.D] in (16).
Pielou's first type of symmetry test and Dixon's symmetry tests are for symmetry in mixed NN structure and are based on the nearest neighbor contingency table (NNCT), while Pielou's second type of symmetry test is for symmetry in shared NN structure and is based on the Q-symmetry contingency table.
Among the symmetry tests, we demonstrate that versions of Pielou's first type of symmetry test are extremely conservative when used with the asymptotic critical value for the McNemar's test, due to dependence between base-NN pairs and the underlying framework for the NNCT. Hence, these tests should be avoided in practice with the a symptotic critical values but can be used with Monte Carlo randomization.
For example, Pielou [2] constructed nearest neighbor contingency tables (NNCTs) which yield tests of segregation (positive or negative), symmetry, and niche specificity, and a coefficient of segregation in a two-class setting.
Pielou's first type of symmetry and Dixon's symmetry test are based on the NNCTs that are constructed using the NN frequencies.
which may have various forms based on the assumed underlying frameworks for the contingency tables in general and for the NNCTs.
Dixon [10] also suggested a symmetry test for testing the equality of frequency of mixed NNs (or between class NNs), that is, the equality of the expected values of the off-diagonal entries in the 2x2 NNCTs. So the null hypothesis is given by
The use of exact tests on NNCTs for testing segregation/association is discussed in Ceyhan [18].
Notice that, under Case I patterns, the off-diagonal entries, [N.sub.12], [N.sub.21], in the NNCTs tend to be much smaller than expected under [H.sub.o] : E[[N.sub.12]] = E[[N.sub.21]] = [n.sub.1][n.sub.2]/n = 20 and [N.sub.12] values tend to be larger than [N.sub.21] values which suggests asymmetry in the mixed NN structure.
Notice that [N.sub.12] and [N.sub.21] in the NNCTs tend to be similar and larger than expected and [N.sub.12] values tend to be slightly larger than [N.sub.21] values.