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I am working on a research- trying to model vegetation types based on factors affecting their distribution. In that case I have observed a strong relation between vegetation and soil types, from just overlaying the vegetation type polygon shapefile with the soil type polygon shapefile, now the problem is i want to quantify this relationship between the vegetation and soil types i.e.the vegetation-soil type correlation analysis so as to scientifically present this relationship. Please i want to know how to do that using ArcGIS 10.1, I have tried to search for solutions to this problem but with no success. I do not have the primary data which was used to create those polygon shapefiles so I am hoping there is way of just working with just the polygons.

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A Pearson, Kendall or Spearman correlation (R) between two nominal variables is nonsensical. You want to explore techniques such as contingency analysis (Phi coefficient, Cramér's V, Persons C, Kappa, ...), factor analysis, canonical correspondence analysis or perhaps something in the family of Jaccard's similarity statistics. If you put one variable (y) in the context of a Binominal [0,1] process you could specify a logistic OLS regression and look at the odds ratio.

Your best bet, and simplest approach, would be a chi square testing the hypothesis of independence and homogeneity in a contingency table. In this case, a contingency table represents the frequencies of agreement between the two nominal variables.

Sorry, but without stepping out to Python, there is no tractable way to do this type of analysis in ArcGIS, which is not a statistical software. Although, you can set up the analysis in ArcGIS by performing an intersect of the polygon datasets to associate the attributes. I should note that there are additional concerns that you should explore regarding spatial uncertainty, aggregation errors, and change of support (MAUP).

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If you have just one type of vegetation, and one type of soil (i.e., if you have one polygon layer representing the spatial extent of one type of vegetation, and a polygon layer representing one type of soil), I think one could use a Jaccard similarity measure. This would entail converting the polygon layers into two different rasters, where cells corresponding to vegetation (and, by the same token, to soil type) are given 1, the other 0. Then, with a bit of Map Algebra (not too difficult), one can calculate Jaccard similarity (i.e., assessing the degree of overlap of vegetation and soil). I found this solution while googling a little. Source and full description here https://www.researchgate.net/post/How_to_statistically_compare_two_maps#view=589c358148954c3b2370faa3

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