# How the combined test under Fisher's method is generally stronger than individual tests, mathematically?

I have gone through the definition of Fisher's method in several materials but it is unclear how Fisher's method is stronger than individual tests. I guess it is not always the case, i.e., there are scenarios that the combined test may perform worse, but not sure in which scenarios. To verify it, I tried setting up the following simple setting and want to compare type-II errors in the individual and combined tests.

Assume there are 2 individual tests having the same hypothesis $$H_0: \mu=\mu_0$$, $$H_a:\mu > \mu_0$$ with p-values $$p_1$$ and $$p_2$$ respectively. Fisher's method combines the p-values by calculating the following test statistic: $$\chi^2=-2(ln(p_1)+ln(p_2))$$ Let $$\mu_a$$ be the value that satisfies $$H_a$$. Let $$\beta_f$$ and $$\beta_1$$ respectively be the type-2 error of the combined test and the first test. I want to know in which conditions, $$\beta_f<\beta_1$$ and vice versa, but am not sure how to continue. Is the above formulation complete or are any further assumptions needed?

• 1. Are you seeking to simulate? Or something else? 2. One thing missing is that you don't seem to have the d.f. on your chi-squared variate; it should be 4 in this case. 3. Ponder this little thought: say you have a sample of size 2m (independent observations) and consider two cases: a: treat it as one large sample. b: you treat it as two samples and combine log-likelihoods by addition (because they're independent, so the combined likelihood is a product). What is the overall likelihood under each scenario? This should give you some clues about what to expect for many tests you might consider Commented Jul 11 at 1:56
• I should have mentioned that in the simplest split case you take it as two samples of size m, but perhaps that's a natural enough thing to do. Fisherian testing would typically use the likelihood as the test statistic (or some equivalent), making the analogy to information content from a larger sample or by combining likelihood from two samples particularly pertinent but the conclusions for combining tests work somewhat more generally than when the test is Fisherian in style. Commented Jul 11 at 2:05