Dear colleagues,

Here, with a bit of delay, a big thank for all the replies re.
statistical methods to test for and quantify spatial aggregation. I paste below
a summary of the replies we received.

Best regards and thanks very much again for your kind assistance.

Pauline Manhes.

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How to measure aggregation

- The most popular seems to be the Ripley’s K function. Information about it can be found on http://
zappa.nku.edu/~longa/geomed/stats/ripley/ripley.html.It is available in R software in the spatstat 
package called Kest.

-Spatial aggregation can also be tested with the package SADIE, free available (http://
www.rothamsted.bbsrc.ac.uk/pie/sadie/SADIE_home_page_1.htm) or GenAlex 2.1. developped by 
Peakall et al (2003) that use autocorrelogramms.

-One could use a randomisation technique : randomly assign x,y coordinates to n points (n= the 
number of objects in your real data), calculate the mean nearest-neighbour distance (i.e. for each point 
find the closest point to it and calculate the distance between those, do that for all points and take the 
average). Repeat all these steps m times (1000 times for example) and compare to the mean nearest-
neighbour distance for your real data.

-The programme PASSAGE can be downloaded at http://lsweb.la.asu.edu/rosenberg/Passage

-A Mantel test or a two tailed chi-square test can also be used.
Chi-square : uniform distribution = chi-square smaller than expected by chance, clumped distribution 
= chi-square larger than expected by chance. 

-A Procedure is available in SAS : proc modeclus. It is a nonparametric density clustering procedure that 
can tell you how many significant clusters are present in multivariate data. Information on the website : 
http://www.id.unizh.ch/software/unix/statmath/sas/sasdoc/stat/chap42/sect1.htm

-If you are interested in aggregation in genetics data, you can read the manuscript of Mark Miller where 
he describes ซ Allelic Aggregation Index Analyses ป. A new software package should be available soon.

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Bibliography

-  Allain, C. & Cloitre, M. (1991). Characterizing the lacunarity of random and deterministic fractal sets. 
Physical Review A. 44 (6), 3352-3358 that gives a method used in Plotnick, R E. Gardner, R H. & O'neill, 
R V. (1993). Lacunarity indices as measures of landscape texture.Landscape Ecology. 8, 201-211.
and With, K.A. & King, A.W. (1999b). Dispersal success on fractal landscapes: a consequence of 
lacunarity thresholds. Landscape Ecology.14, 73-82.

-  Grabarnik, P. & Chiu, S.N. (2002), Goodness-of-fit test for complete spatial randomness against 
mixtures of regular and clustered spatial point processes. Biometrika 89 (2): 411-421.

- J M Elliot. 1977. Some methods for the statistical analysis of samples of benthic macroinvertebrates. 
2nd ed. Freshwater Biological Association.

-  Peter Diggle's book, ซ Statistical Analysis of Spatial Point Patterns ป

- The text by Charles Krebs, Ecological Methodology (1999), has a chapter on "Spatial pattern and 
indices of dispersion."

- The statistical analysis of spatial pattern by M.S. Bartlett

- Cluster analysis and the identification of aggregations, Richard E. Strauss, Animal behavior 2001, 61, 
481-488. 

- Frequency and spatial patterning of clonal reproduction in louisiana iris hybrid populations, Burke et 
al, 2000, Evolution 54(1), 137-144.

- On the social structure of offspring rearing in the burrower bug, Sehirus cinctus (Hemiptera: 
Cydnidae), Agrawal, Brown & Brodie, 2004,  Behav Ecol Sociobiol 57:139–148
(Monte Carlo simulations).

- Monte Lloyd, J. Anim. Ecol. 36:1-30, 1967.

- Analysis of aggregation, a worked example: numbers of ticks on red grouse chicks, Elston et al., 
2001, parasitology 122, 563-569.

pmanhes@ens-lyon.fr