# What are the reasonable properties of p-value combination methods when all p-values are the same?

Suppose we are testing a null hypothesis $H_0$, by combining evidence from $n$ independent tests, using a p-value combination method $M$. The combined p-value is $M(p_1,...,p_n)$. Assume the extreme case when we get the same p-value from all individual tests, which of the following properties of $M$ are more reasonable:

1. $M_n(p,...,p)\leq p$;
2. $M_n(p,...,p)\geq p$;
3. $M_n(p,...,p)$ decreases with $n$;
4. $M_n(p,...,p)$ increases with $n$;
5. all of the above depends on the value of $p$.

Take the Fisher's combination method as an example: $$M_n(p,...,p)=P\{\chi^2_{2n}>-2n \ln(p)\}\approx P\{Z>-\sqrt{2n}\ln(p)-\sqrt{2n}\}.$$The combined p-value decreases when $p<e^{-1}=0.367$.

At the same time, with the minimum $p$-value method, $$M_n(p,...,p)=P\{U_{(1)} < p\}=1-(1-p)^n,$$ which always increases with $n$.

In this special case of $p_i=p$, which method is more "reasonable"?

• Is this a verbatim copy of a question from some test or assignment? – amoeba Aug 11 '15 at 11:01
• Of course it's not – yliueagle Aug 11 '15 at 12:36
• Okay, sorry for suspicion. +1. – amoeba Aug 11 '15 at 12:48
• @Amoeba It would nevertheless make a great question to ask of anyone who is learning about p-values. One could add some other interesting possibilities, such as (6) it depends on the particular test; (7) it depends on the null and alternative hypotheses; and (8) it depends on whether the test is two-sided or one-sided. – whuber Aug 11 '15 at 14:57