# _meta.shrinkage_: An R Package for Meta-Analyses for Simultaneously Estimating Individual Means
## 5 Data Example  
This section analyzes a dataset to illustrate the methods in the proposed package, demonstrating their possible advantages over the standard meta-analysis. Appendix B provides the R code, which can reproduce the following results.  
We used the blood pressure dataset containing \(G=10\) studies, where each study examined the effect of a treatment to reduce blood pressure. The dataset is available in the R package _mvmeta_. Available online: [[https://CRAN.R-project.org/package=mvmeta](https://CRAN.R-project.org/package=mvmeta)]([https://CRAN.R-project.org/package=mvmeta](https://CRAN.R-project.org/package=mvmeta)) (accessed on 14 November 2021). Each study provided the treatment's effect estimate on the systolic blood pressure (SBP) and the treatment's effect on the diastolic blood pressure (DBP), as shown in Table 1. In the following analysis, we focus on the treatment's effect estimates for the SBP and regard those for the DBP as covariates.  
We aimed to improve the individual treatment effects on the SBP by the methods in the R package _meta.shrinkage_. For this purpose, we utilized the covariate information to implement meta-analyses under ordered means (Sections 3.2 and 3.3).  
To apply the proposed methods, we changed the order of the 10 studies by the increasing order of the covariates (see the first column of Table 2). Under this order, we defined \(Y_{i}\), where \(i=1,\ldots,10\) as the treatment effect estimates on the SBP (see \(\mathbf{Y}\) in the second column of Table 2). Table 2 shows a good concordance between \(\mathbf{Y}\) and the covariates; the smallest covariate (\(-7.87\)) yielded the smallest outcome (\(Y_{1}=-17.93\)), and the largest covariate (\(-2.08\)) yielded the largest outcome (\(Y_{11}=-6.55\)). However, the values of \(Y_{i}\) were not perfectly ordered.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{1}{c}{**Treatment**} & \multicolumn{1}{c}{**SE**} & \multicolumn{1}{c}{**Treatment**} & \multicolumn{1}{c}{**SE**} \\  & **Effect on SBP** & **SE** & **Effect on DBP** & **SE** \\ \hline Study 1 & \(-6.66\) & \(0.72\) & \(-2.99\) & \(0.27\) \\ Study 2 & \(-14.17\) & \(4.73\) & \(-7.87\) & \(1.44\) \\ Study 3 & \(-12.88\) & \(10.31\) & \(-6.01\) & \(1.77\) \\ Study 4 & \(-8.71\) & \(0.30\) & \(-5.11\) & \(0.10\) \\ Study 5 & \(-8.70\) & \(0.14\) & \(-4.64\) & \(0.05\) \\ Study 6 & \(-10.60\) & \(0.58\) & \(-5.56\) & \(0.18\) \\ Study 7 & \(-11.36\) & \(0.30\) & \(-3.98\) & \(0.27\) \\ Study 8 & \(-17.93\) & \(5.82\) & \(-6.54\) & \(1.31\) \\ Study 9 & \(-6.55\) & \(0.41\) & \(-2.08\) & \(0.11\) \\ Study 10 & \(-10.26\) & \(0.20\) & \(-3.49\) & \(0.04\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The 10 studies from the blood pressure data. Each study provided the treatment’s effect on the systolic blood pressure (SBP) and the treatment’s effect on the diastolic blood pressure (DBP).  
We therefore considered the order-restricted estimators (\(\delta^{\text{RML}}\), \(\delta^{\text{RJS}}\), and \(\delta^{\text{RJS+}}\)) by imposing the assumption that the true treatment effects were ordered. Table 2 shows that these restricted estimators satisfied \(\delta_{1}\leq\ldots\leq\delta_{10}\) (see the fifth through the seventh columns of Table 2). Since it was reasonable to impose a concordance between the treatment effects on the SBP and DBP, these estimators may have been advantageous over the standard estimates (\(\mathbf{\gamma}\)). In this data example, the JS estimators (\(\delta^{\text{JS}}\) and \(\delta^{\text{JS+}}\)) were almost identical to the standard estimates \(\mathbf{Y}\). Hence, there would be little advantage to shrinkage in the JS estimators.  
By the \"rml(.)\" function, we tested if the \(\mu_{i}\) values were ordered by a sequence of covariates. Figure 4 shows a correlation-based test and the LOWESS plot. The test confirmed that the means were ordered by an _increasing_ sequence (\(x_{1}<\ldots<x_{10}\)). The P-value was highly significant (\(p=0.002\)). Therefore, we validated the assumption of ordered means.