# Optimal Concordant Tests
## 3 Results  
### Numeric Results from Simulation Study  
In this section, we assess the performance of the proposed tests through a simulation study. In the simulation, we compare the proposed optimal concordant test, \(T_{CLRT}^{C}\), with gamma distribution-based concordant tests, \(T_{G(0)}^{C}\), \(T_{G(1)}^{C}\) (i.e., the Owen's Pearson concordant test), and \(T_{G(\infty)}^{C}\), using one-sided \(p\)-values. In the simulation study, we want to combine fifty (\(n=50\)) randomly simulated independent \(p\)-values using different methods. We assume that a number, \(m\)\((m=10,\ 20,\ 40,\ 50)\), out of these \(n\)\(p\)-values are from the true individual alternative hypotheses, and the rest (\(n-m\)) are from the true individual null hypotheses. The n-m \(p\)-values from the true null hypotheses are randomly sampled from the uniform distribution with range \(0\) and \(1\). For a true individual alternative hypothesis, \(H_{i1}\)\((i=1,\cdots,m)\), the \(p\)-value \(p_{i}\) is obtained via a normal variable, \(z_{i}\sim N(\mu_{i},1)\). To consider the situations where the true effects from different studies may have opposite directions (positive and negative), we randomly set \(k\) of the \(m\)\(\mu_{i}\)\(s\) to have the same direction (positive or negative), and the rest of \(m-k\) has the other direction. A one-sided (right-sided) \(p\)-value for each true individual alternative hypothesis is obtained via the standard z-test, i.e., \(p=P[Z>z]\), where \(Z\) is the standard normal distribution, \(N(0,1)\), and \(z\) is the simulated number as described above. For the hypotheses under true alternative, we consider three different scenarios for the effects of \(\mu_{i}\)'\(s\). Scenario 1: We use \(\left|\mu_{i}\right|=\mu v_{i}/\sum\limits_{i=1}^{m}v_{i}\), where \(v_{i}=10^{\prime i}\), \(r_{i}\sim N(0.3,1)\), and \(\mu=0.8,\ 0.6,\ 0.4,\ 0.3\) when there are \(10\), \(20\), \(40\), and \(50\) true individual alternatives, respectively. Scenario 2: We set \(\left|\mu_{i}\right|=\mu v_{i}/\sum\limits_{i=1}^{m}v_{i}\), where \(v_{i}\sim U(1,100)\) and \(\mu=1.2,\ 1.0,\ 0.6,\ 0.5\) when the number of true individual alternatives is \(m=10,\ 20,\ 40,\ \text{and}\ 50\), respectively. Scenario 3: We consider \(|\mu_{i}|=\mu/m\), \(\mu=1.5,\ 1.2,\ 0.8,\ \text{and}\ 0.6\) when \(m=10,\ 20,\ 40,\ \text{and}\ 50\), respectively. Note that, in the simulation, we consider the following: (i) The constants (e.g., \(\mu\), the parameters in the normal distribution for \(r_{i}\) and the uniform distribution for \(v_{i}\)) are chosen in a way that the empirical powers are appreciable for comparison. (ii) The sum of the absolute effect sizes is equal to \(\mu\) for all the three scenarios. (iii) The degree of heterogeneity of the effect sizes among the true individual alternatives decreases from Scenario 1 to Scenario 3 for each given \(m\). The empirical power values of the tests are estimated as the rejection proportions based on 1000 replicates, with 0.05 as the significance level. For the CLRT-based concordant test, \(T_{CLRT}^{C}\), we use \(B=10^{5}\) samples to estimate the \(p\)-values from the resampling method described in Section 2.3. When all of the 50 \(p\)-values are from null hypotheses, the empirical power is the type 1 error rate. All methods considered here were able to control the type I error rate: their empirical powers under this situation were close to 0.05 when the significance level 0.05 was used. Figures 1-3 plot the empirical power values for the concordant tests based on minimal \(\text{p}\), \(T_{G(0)}^{C}\) (denoted as Min_CC), Fisher \(T_{G(1)}^{C}\) (denoted as Fisher_CC, i.e., [PERSON]'s concordant test), z-test \(T_{G(\infty)}^{C}\) (denoted as Z_CC), and the proposed concordant test, \(T_{CLRT}^{C}\) (denoted as LRT_CC), when one-sided \(p\)-values under Scenarios 1 to 3, respectively, are used. From the simulation study, we have the following observations.