I have a volume of brain which contains thousands of connections between hundreds of cells. Suppose I observe that Cell type A connects three times more to Cell type B than Cell type C, and calculate the probability (p-value) of observing this pattern if all of the connections were in fact randomly distributed amongst the cells (the null hypothesis), for instance, by running 10,000 simulations using the same cell positions with randomised connections.
My confusion comes when I think about how I would test for the replicability of the finding across different brain samples. I could obtain two or three more biologically independent samples, repeat the exact same analysis, and see what the p-value is in each case - and if the p-value is very low in each case, then is this adequate to demonstrate replicability? If the p-value in each case were 0.001, would an 'overall p-value' for three independent samples be 0.001*0.001*0.001 = 0.1e-9? It seems too simple, and I don't think I've ever seen P-values combined in this way.
Alternatively, if the test statistic is the ratio (number of cell type A to cell type B connections):(number of cell type A to cell type C connections), then perhaps I would actually need to repeat the same experiment across many independent biological replicates to establish the distribution of this statistic, which would enable me to calculate a 95% confidence interval, and then see whether it overlaps with 1 (which would be the null hypothesis where cell type A connects just as frequently to cell type B as to cell type C).