The first order tests are performed by considering the permutation distribution of the estimated distances TSi and TLSi.
While the q p-values λi are obtained from the marginal permutation distribution of the first order tests, the p-value of the global test is derived from the permutation distribution of the second order test, which is related to the q-variate distribution in the first step.
Note that, no further permutations are needed at this stage, because we use, for the global test, the results on the p-values obtained from each permuted sample that define the permutation distributions of the first order tests [ 4].
The q first order tests are then calculated on each permuted sample and, at each time point, the partial p-values λi's for the q observed TSi or TLSi (i=1,…,q) are obtained from the empirical permutation distribution.
This second order test compares the number of vents within distance d of another vent, with the number expected for a random distribution with the same density (Fig. 3c) widehat{K}(d) = frac{A}{N^2}{displaystyle {sum}_{j=1}^N{displaystyle {sum}_{ine j}^Nfrac{Hleft d-{d}^Nfrac{Hleft d-{di,j}}}} (6).
The cross-correlation is a second order test.
Y is at first the empty set (zero-order test), then each variable X d in turn following order(V) (first-order test), then all possible pairs of potentials variables (X d, X e ) following order(V) (second-order test) and so on, until a Y∗ is identified or possible conditions have been exhausted.
If these movements are so important, why are they mentioned only as a second-order test of the model and not in any of the studies of the behaviors?
In order to test the global null hypothesis H0, the partial tests Ti are subsequently combined in a unidimensional second-order test statistic T. This combination has to be done nonparametrically, since the q univariate tests are dependent, and it is applied to the p-values λi, which are permutationally equivalent to the partial tests [ 4].
