# Optimal Concordant Tests
## 3 Results
### Real Data Application  
In this section, we demonstrate the usefulness of the proposed tests by applying them to a real-world problem. A data set from a meta-analysis is used. In that meta-analysis, results from 12 randomized trials--each examined the effect of patient rehabilitation designed for geriatric patients versus usual care on improving functional outcome at 3-12-month follow-ups--were collected [15; 16]. The 12 estimated odds ratios (ORs) from the individual studies are listed in Table 1.  
The commonly used fixed-effect model for meta-analyses was inadequate for this data set, as the \(p\)-value from the [PERSON]'s test for homogeneity was 0.021. Hence, the random-effect model of the meta-analysis was applied, which estimated the overall OR as 1.36 with a 95% CI (1.08, 1.71) [16]. However, when we applied the goodness-of-fit test for the random-effect model, we obtained a \(p\)-value of 0.025 [3], which indicates a lack of fit of the random-effect model for this data set. We then applied the \(p\)-value combination methods to test whether there is an overall effect among these 12 independent trials.  
To use the \(p\)-value combination methods, we need the individual \(p\)-values. For each trial, the corresponding \(p\)-value is calculated based on the reported 95% CI of the OR. If we use U and L to denote the upper and lower limits of the 95% CI, then the test statistic can be approximated as \(z=\ln(U\times L)/\sqrt{4\ln(U/L)/3.92}\), which is asymptotically distributed as \(N(0,1)\) under the null hypothesis that there is no difference between the new treatment and the control. Since the sample sizes of these 12 trials were relatively large, ranging from 108 to 1388, their \(p\)-values can be reasonably estimated using the asymptotic null distribution by comparing the test statistic with \(N(0,1)\). For example, the right-sided \(p\)-value can be calculated as \(P[Z>z]\), where \(Z\) has a standard normal distribution, \(N(0,1)\), and \(z\) is the test statistic calculated as described above.  
We first calculate the one-sided \(p\)-value for each trial and then use the values for the concordant tests. The \(p\)-values from the concordant tests \(T_{G(0)}^{C}\), \(T_{G(1)}^{C}\), \(T_{G(\infty)}^{C}\), and \(T_{CLRT}^{\mathbb{C}}\) are 0.013, 0.00017, 0.00031, and 0.00030, respectively. For comparison, we also applied the gamma distribution-based tests to this data set, using two-sided \(p\)-values calculated from each individual trial. The resulting \(p\)-values are 0.013, 0.0068, 0.075, and 0.0047 from \(T_{G(0)}\), \(T_{G(1)}\), \(T_{G(\infty)}\), and \(T_{CLRT}\), respectively. As expected, under the situation that a majority of the trials have the same direction (10 out of the 12 estimated ORs were greater than 1), concordant tests are usually more powerful than the corresponding gamma distribution-based tests using two-sided \(p\)-values. Noticeably, the z-test-based concordant test, \(T_{G(\infty)}^{\mathbb{C}}\), obtained a much smaller \(p\)-value (0.00031) compared with the z-test using two-sided \(p\)-values (0.075). Among the concordant tests, the proposed test has the second smallest \(p\)-value (0.00030).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Study** & **OR** & **95\% CI** & **Study** & **OR** & **95\% CI** & **Study** & **OR** & **95\% CI** \\ \hline
1 & 1.11 & 0.51, 2.39 & 5 & 0.88 & 0.39, 1.95 & 9 & 1.06 & 0.63, 1.79 \\
2 & 0.97 & 0.78, 1.21 & 6 & 1.28 & 0.71, 2.30 & 10 & 2.95 & 1.54, 5.63 \\
3 & 1.13 & 0.73, 1.72 & 7 & 1.19 & 0.69, 2.08 & 11 & 2.36 & 1.18, 4.72 \\
4 & 1.08 & 0.42, 2.75 & 8 & 3.82 & 1.37, 10.60 & 12 & 1.68 & 1.05, 2.70 \\ \hline \end{tabular}
\end{table}
Table 1: Estimated odds ratio and its 95% CI from each study in a meta-analysis with 12 trials. Data were taken from [PERSON] et al. [15] and [PERSON] et al. [16].