# Hypothesis testing with p-values from similar tests as prior

I hope to work on a hypothesis testing problem to test $$\beta$$ against the null hypothesis ($$H_{null}$$) $$\beta=0$$, and we can consider Wald test with the test statistics $$W = \dfrac{\beta^2}{var(\beta)}$$ Then, with $$W$$, I can get p-values, which is essentially "getting a value of the test statistic as extreme as or even more extreme than what is observed by chance alone", so $$p(t > W | H_{null})$$.

However, the sample size I have to calculate $$\beta$$ and $$var(\beta)$$ is fairly small. The results, although p-values for some covariates are small, are not very convincing.

On the other hand, I have some p-values calculated for the same set of covariates, from a larger body of samples. Basically, I have a prior $$p_0(t>W_0|H_{null})$$. In addition, sometimes I also have $$\beta_0$$.

I wonder whether/how I can calibrate $$p(t > W | H_{null})$$ with $$p_0(t>W_0|H_{null})$$.

Some of my current efforts

• I learned a way to do this is to simulate a large amount of data that leads to $$p_0$$ if we conduct the test, and then combine the simulated data and my real ones to do the hypothesis testing, but I hope there are more efficient ways to do it.
• Bayesian hypothesis testing sounds relevant, but it assumes $$p_0(H_{null})$$ instead of $$p_0(t>W_0|H_{null})$$, and it leads to a MAP result. I hope we can still have a p-value at the end.