Most parametric and nonparametric tests assume sample independence...so if your sample variable is spatially autocorrelated then ANOVA could be problematic. If you provide a bit more detail, I might be able to point you in the right direction.
If your data are normally distributed so that if you can do an analysis of variance for comparison of means and further analysis with a geostatistical kriging interpolation. If your data does not have a normal distribution, you can do an analysis of variance to compare medians, also you can make a indicator kriging interpolation if you have a threshold (cut off).
A compariosn of means, either parametric as ANOVA or non-paramatric as for example Kruskal-Wallis on data points grouped by some spatial container like administrative boundaries should be sensible. The choice must be informed by your sample sizes for each class, the distribution etc. as has been mentioned. But if you first interpolate by Kriging and then use that as input to the ANOVA etc. that would be a problem as far as I understand.
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