# Optimal Tests for Combining \(p\)-Values
## 4 Real Data Examples
### Example 2: A Survival Analysis from a Clinical Trial  
The second data set is from the randomized, double-blinded Digoxin Intervention Trial [34]. In this trial, patients with left ventricular ejection fractions of 0.45 or less were randomly assigned to digoxin (3397 patients) or placebo (3403 patients) groups. A primary outcome was the mortality due to worsening heart failure (see Figure S20 in the Supplementary Materials). In the original study, the authors used the log-rank (LR) test and obtained a \(p\)-value of 0.061, indicating that the evidence of the effectiveness of digoxin, in terms of reducing the mortality due to worsening heart failure, is at most marginal. However, it is well known that the LR test may fail to detect the difference between two survival functions if their hazard rate functions are crossing [35,36]. We apply the two-stage approach [35,36] to this data set and obtained two \(p\)-values of 0.06 and 0.04 for the two stages. Since, under the null hypothesis, the two \(p\)-values from the two stages are asymptotically independent [35,36], we can combine them using the proposed test \(T_{CLRT}\) and the gamma distribution-based tests. The \(p\)-values are 0.078, 0.017, 0.0099, and 0.011 from the Min \(p\), Fisher, z test, and \(T_{CLRT}\), respectively. The proposed test obtained the second smallest \(p\)-value, which is slightly larger than the smallest one obtained by the z test. In addition, the drug effect may differ between males and females. To investigate the possible interaction between sex and treatment, we divide the data into four groups based on the combinations of sex and treatment: Male--Placebo (MP), Male--Drug (MD), Female--Placebo (FP), and Female--Drug (FD). The sample sizes for the four groups are 2639, 2642, 764, and 755, respectively. We then compare the survival functions in the following three pairs of groups: MP vs. MD, (MP + MD) vs. FP, and (MP + MD + FP) vs. FD, where (MP + MD) is a new group with pooled data from groups MP and MD, and (MP + MD + FP) includes all the subjects from groups MP, MD, and FP (see Figure S21 in the Supplementary Materials). For each comparison, the two-stage approach is applied. We obtain the following six \(p\)-values: 0.019, 0.026, 0.504, 0.092, 0.975, and 0.050. It has been shown that under the null hypothesis, the six \(p\)-values obtained from the above approach are asymptotically independent [36]. Applying the gamma distribution-based tests, along with \(T_{CLRT}\), to the six asymptotically independent \(p\)-values, we obtain \(p\)-values \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \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 \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. and [PERSON] et al. of 0.11, 0.0067, 0.020, and 0.013 from the Min \(p\), Fisher, z test, and \(T_{CLRT}\), respectively. It noticeable that the proposed test obtained the second smallest \(p\)-value, while Fisher test obtained the smallest one among those methods. These results show that except for the Min \(p\) test, all other tests obtain \(p\)-values less than 0.05. In addition, the \(p\)-values from \(T_{CLRT}\) in general are close to the smallest ones, while the popular tests, Fisher and z test, may get quite different \(p\)-values under different situations (two groups vs. four groups). Compared with the original analysis, the results of the proposed optimal test \(T_{CLRT}\) using \(p\)-values from the two-stage approach applied to either two groups (placebo vs. drug) or four groups (combinations of sex and drug) provide stronger evidence against the null.