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Given a number of K-functions from data of two populations (e.g. 10 sets with points of population 1 and 10 sets with points of population 2; non-overlapping image areas; only one population per set, so data is not marked), how can I perform testing for group difference based on the K-curves alone? Is there an appropriate test for this?

If I have a single parameter (e.g. Clark Evans or minimum nearest neighbor distance) I can simply perform a T-Test or Mann-Whitney-U-Test (depending on data normality) for the two groups.

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  • $\begingroup$ (1) Ripley's K is far better suited for analysis and insight rather than tests. No single test will capture everything that can be learned from a careful look at the graph of $K$ (or better, a suitably standardized version of $L$). (2) I have never seen software for Ripley's K that did not also include a simulation-based confidence band. That can be very helpful to support visual assessments of differences. (3) OK, let's accept that you need a test. You must specify what behaviors you are comparing. What is the alternative hypothesis.? $\endgroup$ – whuber Aug 20 '15 at 14:23
  • $\begingroup$ Hi whuber, thanks for your comment. (1) You are right, the graph gives good insight and contains a lot of information. (2) I have already computed confidence bands an the basis of Monte Carlo simulations. But I am missing some comparability between data sets of different populations. (3) Well, the alternative hypothesis should be that the graph behaviour of population 1 differs from that of population 2. Maybe pop1 has small clusters that produce an elevation of the graph for small r, while the graphs of pop 2 are somewhat flatter indicating no distinct clustering? $\endgroup$ – Kardashev3 Aug 24 '15 at 10:02
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I only have time for a very short answer. In the spatstat package for R there is a vignette which explains a bit of the theory for replicated point patterns, so you might want to have a look at that: https://cran.rstudio.com/web/packages/spatstat/vignettes/replicated.pdf

Furthermore, a relevant test could be the studentised permutation test developed by Ute Hahn:

Hahn, U. (2012) A studentised permutation test for the comparison of spatial point patterns. Journal of the American Statistical Association 107 (498), 754-764.

This test is implemented in the spatstat function studpermu.test, so you might also try to read the documentation for that.

Finally, a relevant thing may be to pool all the estimates of the K-function for each group to get two overall K-functions to compare. Have a look at the help file for pool (specifically the method pool.fv) in spatstat.

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  • $\begingroup$ I will try the studentised permutation test for my data. Pooling of estimates seems to be another good idea! I will report back when I am done. $\endgroup$ – Kardashev3 Aug 24 '15 at 10:04
  • $\begingroup$ Now I have tried your suggestions and I am very pleased with the results :)! The vignette about replicated point patterns provides an excellent introduction to that topic. The studentised permutation test is exactly what I was looking for. Group differences can be calculated based on the actual point patterns without the need of deriving some numbers from the curves while omitting a lot of spatial information. Pooling of functions is another great thing as overall group information can be plotted and analyzed. Thank you very much for these great ressources :)! $\endgroup$ – Kardashev3 Sep 2 '15 at 15:19

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