Originally posted by

**Maaneli**View PostConsider what would happen if you conducted the binomial test without meeting the minimal assumptions of the test. Then a rejection of the null hypothesis leads to the useless conclusion that the data are unlikely under the null. Okay, but what else can be concluded? Nothing. It does not follow that there is a p ≠ p₀, for all or even some of the data. All we can conclude is that the data weren't derived from a binomial(p₀) distribution.

And of course, when one assumes the null is true in an experiment such as the ganzfeld, it automatically implies independence and constant hit probability for each ganzfeld trial, because of how the ganzfeld trials are designed.

*is*that the data are independent with equal probability of success. It does not "imply" it "because of how the ganzfeld trials are designed."

That's why one can use the exact binomial test on a single ganzfeld experiment, and that's why one can apply the binomial test to the pooled hits and trials of a ganzfeld meta-analysis (assuming all studies have 4-choice design).

*can't*(or, rather, shouldn't) apply the test to a collection of ganzfeld experiments if there is evidence of heterogeneity—because heterogeneity means that the trials are

*not*independent and identically distributed.

Here is what you are claiming: that if a collection of k independent experiments are each binomial(p_i) for i = 1 to k, with common probability of success p₀ under the null hypothesis, then a binomial test of the combined data that rejects the null implies that p_i ≠ p₀ for some i. That is an interesting conjecture that sounds superficially plausible, but it does not follow from your argument. It needs to be rigorously proved, and maybe it has been, but I have not seen a proof of it.

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