# Optimal Tests for Combining \(p\)-Values
## 2 Methods
### Constrained Likelihood Ratio Tests
#### 2.3.1 CLRT-Based Tests with Known \(\alpha\) Values  
Under (8), with known \(\alpha=\alpha_{0}\), the CLRT-based tests can be constructed based on the constrained MLE of \(c\) obtained through maximizing \(l(\alpha_{0},c)=l(c)=n\alpha_{0}ln(1-c)+c\sum_{i=1}^{n}F_{G(\alpha_{0})}^{-1} (1-P_{i})\). More specifically, we define the following test statistic:  
\[T_{CLRT,\alpha_{0}}(P_{1},\cdots,P_{n})=2\alpha_{0}ln\big{(}1-\mathcal{E}_{ CLRT,\alpha_{0}}\big{)}+2\big{(}\mathcal{E}_{CLRT,\alpha_{0}}\big{)} \sum_{i=1}^{n}F_{G(\alpha_{0})}^{-1}(1-P_{i}), \tag{9}\]  
where \(\mathcal{E}_{CLRT,\alpha_{0}}\) is the constrained MLE of \(c\) through maximizing the log-likelihood \(l(\alpha_{0},c)\) with the constrain \(0<c<1\). We will reject the overall null hypothesis if the test statistic is large. In other words, a one-sided \(p\)-value will be calculated from the test.  
For the above CLRT-based test \(T_{CLRT,A_{0}}\) in (9), we have the following result [30]:  
**Proposition 4**.: _The asymptotic distribution of the test \(T_{CLRT,A_{0}}\) is a mixture of chi-square distributions \(\sum_{i=0}^{1}w_{i}\chi_{i}^{2}\), where \(\chi_{i}^{2}\) is the chi-square distribution with \(df=i\), \(\chi_{0}^{2}\) is the random variable with probability 1 of being 0, and the weights \(w_{0}\), \(w_{1}\) are determined by the null and the alternative hypothesis._  
In practice, the \(p\)-values of the test \(T_{CLRT,A_{0}}\) can be estimated through resampling methods (e.g., see Section 2.3.2 for an example). However, the following result shows that this test is tightly connected with the gamma distribution-based test \(T_{G(a_{0})}\), whose \(p\)-value can be calculated directly.  
**Theorem 3**.: _Let \(t_{CLRT,A_{0}}\) be the observed statistic of test \(T_{CLRT,A_{0}}\); the \(p\)-value of \(T_{CLRT,A_{0}}\) is determined by the gamma distribution-based test \(T_{G(a_{0})}=\sum_{i=1}^{n}F_{G(a_{0})}^{-1}(1-P_{i})\) as follows:_  
\[Pr\big{[}T_{CLRT,A_{0}}>t_{CLRT,A_{0}}\big{]}=\left\{\begin{array}{c}P_{T_{G (a_{0})}}\text{ if }t_{CLRT,A_{0}}>0\\ Pr\Big{[}T_{G(a_{0})}<n\alpha_{0}\Big{]}\text{ if }t_{CLRT,A_{0}}=0\end{array}\right. \tag{10}\]  
The proof is given in the Appendix A.  
**Proposition 5**.: _Under the conditions specified in (8), if the parameter \(\alpha=\alpha_{0}\) is known, then asymptotically \(T_{G(\alpha_{0})}\equiv T_{CLRT,A_{0}}\)._  
Proof.: From the proof of Theorem 3, we know that under (8), when \(0\leq c<1\), \(Pr[T_{G(\alpha_{0})}<n\alpha_{0}]=Pr[\hat{c}_{\alpha_{0}}\leq 0]\to 0\ (as\ n\to\infty)\), and hence, the two \(p\)-values from tests \(T_{G(a_{0})}\) and \(T_{CLRT,A_{0}}\) are asymptotically equal with probability 1. \(\square\)  
**Theorem 4**.: _Under the conditions specified in (8), if the parameter \(\alpha=\alpha_{0}\) is known, then the gamma distribution-based test \(T_{G(a_{0})}\) and the CLRT-based test \(T_{CLRT,A_{0}}\) are both asymptotically UMP._  
Proof.: When \(\alpha=\alpha_{0}\) is known, it can be shown that the constrained MLEs for \(c\) is consistent (see, for instance, Theorem 1 of Self and Liang [30]). Hence, from Theorem 2, \(T_{G(a_{0})}\) is asymptotically UMP. From Proposition 5, \(T_{CLRT,A_{0}}\) is asymptotically UMP. \(\square\)