# Significance test for a pattern in union of sets given that the pattern is significant in the individual sets

We have a pattern like $$P = \{f_{A},f_{B}\}$$ which is statistically significant in multiple sets $$S = \{S_{1}, S_{2}, S_{3}, ...\}$$.

We can think of the features $$f_{i}$$ to be retail establishments & the sets $$S_{i}$$ to be geographic regions like counties.

The features $$f_{A}, f_{B}$$ are related by some association measure like geographic distance. Now the statistical significance of the pattern is computed against a null hypothesis which models the intensity of the features in the sets ($$S_{i}$$) but are randomly populated in the regions ($$S_{i}$$).

From experiments I have ensured that the pattern is statistically significant in some of the sets $$S_{j}$$ which is a subset of $$S_{i}$$.

Now if I have to check for the significance of the pattern in the union of the sets $$S_{j}$$, i.e. the region obtained by combining the individual regions within which the pattern is statistically significant, I am again calculating the p-value of the pattern in the new region obtained by the union of $$S_{j}$$.

Is there any property of p-value/significance tests via which I can check if the pattern is statistically significant in the union of the $$S_{j}$$ sets given that I know that the pattern is statistically significant in the individual regions $$S_{j}$$?

Thanks for your suggestions!

It is not clear from your question whether you appreciate that by selecting $$S_j$$ in that way you have to also allow for the remainder of $$S_i$$ but just in case, yes you do.
I think you may find one of the methods for dealing with thresholding helpful. Specifically Zhang and colleagues in a paper at arXiv:1801.04309 entitled "TFisher tests: optimal and adaptive thresholding for combining $$p$$-values" refer to a number of methods and present them in the framework of thresholding $$p$$-values. In your case presumably the threshold would be 0.05 or whatever significance level you chose. Their paper is far too long and complex to summarise here.