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I am doing a buffer analysis of incinerators and looking at the output areas within those buffers, is there a way of doing a statistical analysis of several buffers to see if there is a significant pattern between the data within the buffers?

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Do you want to compare the areas around different incinerators, or do you want to compare areas around incinerators vs areas not around incinerators? – Andy W Jul 27 '11 at 15:50
To make this question less vague, please indicate what you mean by a "significant pattern." What is being compared to what, on what basis? For example, are you looking for trends in concentration over distance? Differences in population densities between incinerator neighborhoods? Association between residential locations and areas downwind of incinerators? – whuber Jul 27 '11 at 16:09
Do you by "several buffers" mean stepped distance (i.e. 1/4 mile, 1/2 mile) or do you mean buffers of several sites? – ako Jul 27 '11 at 16:56
I apologise for not being specific, I have created shapefiles that identify the location of operational incinerators and areas that have refused incinerators planning permission and wish to compare the buffers of each. – dpc1989 Jul 28 '11 at 8:49
by several buffers i maen serveral sites and i wish to see if there is a pattern between the incinerator sites and non-incinerator sites – dpc1989 Jul 28 '11 at 8:51

Because this question is buried in comments, let's assemble them in one place to see what's the issue:

(1) I have created shapefiles that identify the location of operational incinerators and areas that have refused incinerators planning permission and wish to compare the buffers of each.

(2) I also wish to ... see if there is any correlation between the distance from an incinerator and the geodemographics... .

In short, you want to relate various geodemographic (response) variables to two factors: (1) whether an incinerator was refused or accepted and (2) distance to some point location relevant to (1).

This calls for a regression model. Normally, you don't perform regression modeling in a GIS if you can help it: although it can be done, no GIS supports all the diagnostic, cross-checking, and reporting needed for rigorous analysis, nor have any of them undergone the extensive testing needed to assure correct results. Instead, use the GIS to output the data (as a list of tuples containing the refuse/accept indicator, the distance, and the geodemographic variables). Pass these to a statistical program or code library and perform the regression.

(Note that actually creating buffers in the GIS is unnecessary. Note, too, that you don't have to leave the GIS environment to do this work: Python supports regression analysis (if you're willing to do a bit of programming) and there are R interfaces to ArcGIS.)

This leaves us with a subtle issue, but perhaps it's the most important: because you include spatial-analyst as a tag, it suggests your data are in a gridded form. "Geodemographic variables" are rarely collected in grids. Instead, they are reported by administrative units such as blocks, tracts, or counties. Although these units can be represented by collections of cells on a grid, it would be a mistake to treat each cell as if it had independent data. (Doing to would make it appear you have far more data than you really have, which would greatly inflate the apparent significance of any regression coefficients.) You get only one collection of geodemographic values per unit, period. That, therefore, will be the basic unit of statistical analysis. Accordingly, you need to summarize the distances within each administrative unit. One way is to compute zonal statistics of a Euclidean distance grid. This will give you a mean distance, extreme distances, and some measures of variation of distance within each unit. Any of these--or even combinations of these--can be used as independent (explanatory) variables in the regression. Thus, the data to be output by the GIS should correspond, one-to-one, to the administrative units and not to the grid cells.

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ecellent exploration of the issue. – BWill Oct 3 '11 at 10:31

Turns out this is not the question the poster is asking, but I will leave the information here as it might be useful to others who come across this question:

I -think- you might be looking for an analysis of spatial auto-correlation at different distance bands?

Check out this series of esri instructional videos:

The 2nd one might be particularly useful for you, as it discusses Moran's I, which I think might be the statistic you will be interested in.

Really though, we need a lot more detail on the exact question you are trying to answer. There are a wide range of statistical tools available that are appropriate to different problems.

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