## Spatial dependence Schlaengglidata revisited

This is an example of real-world data illustrating the application of the Mantel test, its directed version and also Moran's I. The investigation area (see for example Figure 8.5 and Appendix B) has the interesting property that it is almost quadratic in shape and trends can therefore be evaluated in different spatial directions. Furthermore, there are many site factors available; some of these correlate with vegetation while others do not (see Section 7.2.3).

From Figure 8.3, upper-left graph, it can be seen that there is a strong floristic gradient in the vertical direction (a = 90 °). This suggests that the species pattern is likely to be space-dependent. Using the correlation matrix of the releves and the matrix of Euclidean distances computed from the x-and y-axes in space, the Mantel test yields r = -0.5204 and p = 0.0000. Hence, spatial dependence is highly significant.

Since this dependence has its origin in a gradient, direction is a major issue. To evaluate different directions, the spatial distances have to be projected on one line. This yields new distances from which Mantel's r or Moran's I is computed. The way distances are projected at an angle of a = 45 ° is shown in Figure 7.4. In this direction, the Mantel test yields r = -0.4980 and p = 0.000, a highly significant trend. At an angle of a = 0 °, where the vertical component of the space is suppressed and only the horizontal expansion is considered, the test yields r = -0.0884 and p = 0.014. This means that there is still a trend, but much weaker than that in the vertical direction (a = 90 °).

A full evaluation of all directions in the range of 0 ° < a < 180 ° allows identification of the direction in which the gradient is strongest. This is shown in Figure 7.5. The maximum is achieved at a & 75 °, indicating that the main gradient points from the upper-left to the lower-right corner; that Figure 7.4 Projecting distances in one direction. Without considering direction, the neighbouring data points Located on the same arrow are separated by 1.414 grid units. When projected at an angle of 45 they are at the same location. Figure 7.5 Evaluating the direction of the floristic gradient. The gradient is strongest at a ~ 75

is, in an almost vertical direction. Perpendicular to this, at a & 165 , the floristic gradient vanishes (r = -0.0072).

It can be seen in Figure 7.4 that the sample space, when projected, is one-dimensional only. This is also the case when taking one single site factor instead of spatial axes. In Section 7.2.3 all site factors have been evaluated based on their potential in predicting a specific classification of the releves. In the present context, this potential is measured independent of classification, based on the full similarity matrices of the releves. In Table 7.6 the same site factors used in Figure 7.2 are subjected to the Mantel test. For the purpose of comparison, the F-values from Section 7.2.3 are also shown (from the classification based on the transformation x' = x025). Mantel's r is always negative because the site factors are compared by distance, whereas for the releves the correlation coefficient (a similarity measure) is used. The results are significant in all cases except the last, where a random variable is used instead of a site measurement.

Moran's I can be computed as well and it will yield a correlogram instead. In the example shown in Figure 7.6 the distance (dissimilarity) matrices are classified. The classes are formed by dividing the longest distance encountered into 10 segments, from which Moran's I is calculated. Figure 7.6 shows correlograms of four different site factors in one graph. Although this helps in the comparison of the curves, interpretation has to be carried out with care: the distance matrices differ among site factors and Moran's I values are based on a slightly different number of data pairs. Generally, the results at distance classes 9 and 10 become unreliable due to insufficient sample size.

The strongest dependence occurs with pH. From distance classes 2-6 the change of Moran's I is almost linear. Only then does the trend level off, indicating nonlinearity occurs at larger differences in pH. The water level yields a similar overall shape of the correlogram, but much less pronounced. Random fluctuation plays a more visible role than in pH. At distance class

 No. Site factor F-value Mantel's r Permutation p 1 pH peat 33.87 -0.649 0.000 3 Ca (mg/100g peat) 19.09 -0.634 0.000 9 Base saturation (%) 40.63 -0.737 0.000 12 Waterlevel, av. (cm) 9.58 -0.346 0.000 14 Peat depth (log(cm)) 9.27 -0.400 0.000 15 Slope (log(deg.)) 4.15 -0.132 0.029 17 Conductivity (log(Ohm/cm)) 24.38 -0.597 0.000 18 Ca in water (log(0.1ppm)) 55.80 -0.755 0.000 21 random variable 0.57 0.050 0.228

distance class distance class Figure 7.6 CorreLograms of site factors. The random variable is an example illustrating the shape when there is no relationship.

1, where small differences are taken into account, Moran's I is still rather reliable. Slope, on the other hand, is an example for weak dependence. The correlogram ends at distance class 8 because n is too small to calculate I at distance classes 9 and 10. The deviations from the zero line hardly exceed what can be expected from a random number, which is also included to illustrate the lack of relationship.