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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!

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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.

Zhang and colleagues provided an R package for this https://cran.r-project.org/package=TFisher. In my CRAN package metap I provide a wrapper for a subset of what their package provides

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