# _meta.shrinkage_: An R Package for Meta-Analyses for Simultaneously Estimating Individual Means
## 6 Conclusions and Future Extensions  
This article introduced an R package _meta.slirinkage_ ([[https://CRAN.R-project.org/package=meta.shrinkage](https://CRAN.R-project.org/package=meta.shrinkage)]([https://CRAN.R-project.org/package=meta.shrinkage](https://CRAN.R-project.org/package=meta.shrinkage)) (accessed on 15 November 2021)), which we made freely available on CRAN. It was first released on 19 November 2021 (version 0.1.0), following our original methodological article published on 20 October 2021 (2021). We hope that the timely release of our package facilitates the appropriate use of the proposed methods for interested readers. As the precision and reliability of the developed statistical methods are important, we conducted extensive simulation studies to validate the proposed R package (Section 4). We also analyzed a data example to show the ability of the R package (Section 5). To implement isotonic regression in our R package, we proposed a matrix-based algorithm for the PAVA (Figure 1). This algorithm is easy to program in the R environment, \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Covariate** & \(\mathbf{Y}\) & \(\delta^{\text{JS}}\) & \(\delta^{\text{JS+}}\) & \(\delta^{\text{RML}}\) & \(\delta^{\text{RJS}}\) & \(\delta^{\text{RJS+}}\) \\ \hline Study 2 & \(-7.87\) & \(-17.93\) & \(-17.91\) & \(-17.91\) & \(-16.05\) & \(-16.05\) & \(-16.05\) \\ Study 8 & \(-6.54\) & \(-10.26\) & \(-10.25\) & \(-10.25\) & \(-16.05\) & \(-16.05\) & \(-16.05\) \\ Study 3 & \(-6.01\) & \(-14.17\) & \(-14.16\) & \(-14.16\) & \(-12.88\) & \(-12.88\) & \(-12.88\) \\ Study 6 & \(-5.56\) & \(-11.36\) & \(-11.35\) & \(-11.35\) & \(-10.6\) & \(-10.6\) & \(-10.6\) \\ Study 4 & \(-5.11\) & \(-8.71\) & \(-8.70\) & \(-8.70\) & \(-9.76\) & \(-9.76\) & \(-9.76\) \\ Study 5 & \(-4.64\) & \(-10.6\) & \(-10.59\) & \(-10.59\) & \(-9.76\) & \(-9.76\) & \(-9.76\) \\ Study 7 & \(-3.98\) & \(-8.7\) & \(-8.69\) & \(-8.69\) & \(-9.76\) & \(-9.76\) & \(-9.76\) \\ Study 10 & \(-3.49\) & \(-12.88\) & \(-12.87\) & \(-12.87\) & \(-9.76\) & \(-9.76\) & \(-9.76\) \\ Study 1 & \(-2.99\) & \(-6.66\) & \(-6.65\) & \(-6.65\) & \(-6.66\) & \(-6.66\) & \(-6.66\) \\ Study 9 & \(-2.08\) & \(-6.55\) & \(-6.54\) & \(-6.54\) & \(-6.55\) & \(-6.55\) & \(-6.55\) \\ \hline \end{tabular}
\end{table} 
Table 2: The treatment effect estimates on the SBP based on the 10 studies from the blood pressure data. The 10 studies are ordered by the covariates (treatment effect estimates on the DBP). Figure 4: The LOWLES plot based on the treatment effect estimates on the SBP (\(y\)) and those on the DBP (\(x\)) from the blood pressure dataset with 10 studies. Shown are Kendall’s tau and the test of no association between \(x\) and \(y\). where matrices are convenient to manipulate. However, if one tries to implement the PAVA in other programing environments, the matrix-based algorithm may be inefficient, especially for a meta-analysis with a very large number of studies. The number \(G>500\) makes the proposed algorithm slow to compute the output. However, real meta-analyses rarely have more than 100 studies, so this issue may not arise at the practical level. An extension of the present R packages to multiple responses is an important research topic. An example includes a meta-analysis of verbal and math test scores [45], consisting of two responses. A similar instance involves math and stat tests [44; 46]. There is much room for meta-analyzing bivariate and multivariate responses [47; 48; 49; 50; 51; 52; 53; 54; 55; 56]. Multivariate shrinkage estimators of multivariate restricted and unrestricted normal means, such as those in [57; 58; 59; 60], can be considered for this extension. The proposed R package can only handle normally distributed data; it cannot handle data that are non-normally distributed, asymmetrically distributed, or discrete-valued. To analyze such data, one should consider extensions of the meta-analysis methods in [20] toward asymmetric distributions for skewed or discrete response variables.