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.And also the data in nominal i.e. soil and vegetation type names in these respective polygons.

Since you don't have the primary data, you don't have actual observations which rules out nominal data tests like Phi, Cramer's V or other 'correlation' measures, which are all derived from Chi-square tests of difference/association. Do you know how the maps were actually produced and whether the polygons drawn do indeed represent the spatial patterns of the data? Has there been simplification/generalization of the data at any level or stage? Why do you need to quantify your observation when a qualitative statement might suffice? I am not sure what you mean by scientifically present the relationship. I am sure you have resisted the temptation of calculating co-located areas and expressing that as the correlation, so perhaps more information as to the purpose would be useful.