It appears from the contours on my red-blue plot that the index values for the gaps is greater than those for the patches. Does this mean my spatial pattern really is more gappy than it is patchy?

Perhaps so, but this might be only what is to be expected, particularly if the data were skew and there were more sample units that were larger than the mean (potential patches) than were smaller than the mean (potential gaps). (See the FAQ on the 'Non-parametric SADIE method' for further details of this effect.)  In other words - it could be just a chance effect.




OK, but is there a way to assess if the degree of patchiness exceeds gappiness, or vice-versa?

Yes, there is a simple way. Like most evaluations in SADIE, it is based on comparing the observed pattern with randomizations, in this case formed from rearrangements of the observed counts amongst the sample units.

To understand the test let's first look at an example. Below is the red-blue plot for data of the aphid Metapolophium dirhodum, taken from Winder et al. (2001) Ecology Letters, in which there are clearly more gaps than patches. But, more importantly, the patch and gap indices differ in magnitude. The mean value of the patch indices is about 1.62, but the mean value of the gap indices negatively exceeds this at -1.8.






We can plot this pair of values, the mean patch and the mean gap index, on, respectively, the x- and y-axes, together with the corresponding values from all the randomizations that were done. The plot of mean gap indices versus mean patch indices is shown below. In the graph, the observed pair of values are shown by the red open circular symbol, and each of the randomizations pairs is shown by a black cross.

(You can find all these values, both observed and randomized, printed out about halfway down the output file rbno9.dat. The output reads typically like this:


"
Mean of clustering index, v_j, over inflows: -1.802
Mean of clustering index, v_i, over outflows: 1.621
Mean of v_i and abs. |v_j|, over all flows: 1.723



Mean clustering indices of permuted data are:
(-ve inflows, +ve outflows, abs(flows))
-1.0142 .9147 .9889
-.7563 .7618 .7577
-.8619 .8984 .8711
-.9754 1.0149 .9854
-1.1675 1.2277 1.1828
      .            .            .
      .            .            .
      .            .            .
"
and then this list continues for many rows)


The values plotted are those in the first two rows above (not the third), plus all of those from the first two columns of the second, long list of values. Notice the strong correlation in the graph between the randomized values; relatively small negative values of the gap index occur with relatively small positive values of the patch index, and similarly for the larger values. A moments thought will confirm that we should expect this. It is another way of saying that, for a given set of counts, if most of the larger counts are close to one another and far from the smaller counts, then it is most likely that most of the smaller counts will also be close to one another, and vice-versa. So values that lie along the faint green line towards the centre of the cloud of randomized values do not appear particularly unusual. Along that green line, the absolute values of the indices are roughly equal. However, if a point lies a long way from the green 'axis', i.e. if its projection onto the red 'axis', the faint line perpendicular to the green axis, is unusually large and positive (or unusually large and negative), then that is an indication that the mean patchiness is larger (or smaller) than the mean gappiness.

In the example, the observed value shown by the open red circular symbol does indeed appear to lie outside of the cloud of points formed by the randomized values.




But I want a formal test - is there one?

Yes, there is a simple test. The green axis and the red axis in the plot above are, respectively, the first and second major axes of a principal components analysis of the randomized values. For a formal test, just do a principal components analysis of the randomized values, and save the 'scores' on both axes; the scores are the projections of the rotated points onto the principal component axes. Here is a graph of the first two principal components for the above data. Once again, the observed value is also plotted for comparison, and is shown here by the open red circular symbol. Having saved these scores, just consider those on the second principal axis (the y-axis here, and the red line in the graph earlier up the page) and form a frequency distribution from these second axis scores. Then, compare the value of the observed score on the second axis with that frequency distribution, using the usual two-tailed test. For our example (see histogram below) the observed pair of mean indices have a value of -0.113 when projected onto the second, red axis. This value is ranked 19^th smallest out of all the 5968 randomizations, and so the probability of such an extreme value would be 38/5968 or 0.0064. Hence there is significantly more gappiness than we would expect at the P<0.01 level.





So how am I going to do this principal components analysis?

Most statistical software packages will allow you to do a principal components analysis. GenStat is one example




Who do I thank for this?

My post-doc Colin Alexander who developed the method and produced the example output from GenStat and graphs.