# meta.shrinkage: An R Package for Meta-Analyses for Simultaneously Estimating Individual Means

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## Abstract

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## 1. Introduction

## 2. Background

#### 2.1. Meta-Analysis

#### 2.2. Improved Estimation of Individual Means

## 3. R Package meta.shrinkage

**,**and ${\mathit{\delta}}^{\mathrm{GPT}}$. We will divide our explanations into four sections: Section 3.1 for$\text{}{\mathit{\delta}}^{\mathrm{JS}}$ and ${\mathit{\delta}}^{\mathrm{JS}+}$, Section 3.2 for ${\mathit{\delta}}^{\mathrm{RML}}$, Section 3.3 for ${\mathit{\delta}}^{\mathrm{RJS}}$ and ${\mathit{\delta}}^{\mathrm{RJS}+}$, Section 3.4 for ${\mathit{\delta}}^{\mathrm{PT}}$, and ${\mathit{\delta}}^{\mathrm{GPT}}$.

- y: a vector for ${y}_{i}$s;
- s: a vector for ${\sigma}_{i}$s.

#### 3.1. James–Stein Estimator

#### 3.2. Restricted Maximum Likelihood Estimators under Ordered Means

#### 3.3. Shrinkage Estimators under Ordered Means

#### 3.4. Estimators under Sparse Means

- alpha1: significance level for ${\alpha}_{1}$ (0 < alpha1 < 1);
- alpha2: significance level for ${\alpha}_{2}$ (0 < alpha2 < 1);
- q: degrees of shrinkage for $q$ (0 < q < 1).

## 4. Simulation: Validating the R Package

#### 4.1. Simulation Design

**Scenario (a)**:- Ordered and non-sparse: $\mathit{\mu}=\left(-2,-2,-1,-1,0,0,1,1,2,2\right)$;
**Scenario (b)**:- Ordered and sparse: $\mathit{\mu}=\left(0,0,0,0,0,0,0,0,2,4\right)$;
**Scenario (c)**:- Unordered and non-sparse: $\mathit{\mu}=\left(1,-1,1,-1,1,-1,1,-1,1,-1\right)$;
**Scenario (d)**:- Unordered and sparse: $\mathit{\mu}=\left(0,0,0,1,0,0,0,1,0,0\right)$;
**Scenario (e)**:- Ordered and non-sparse: $\mathit{\mu}=\left(2,2,1,1,0,0,-1,-1,-2,-2\right)$;
**Scenario (f)**:- Ordered and sparse: $\mathit{\mu}=\left(0,0,0,0,0,0,0,0,-2,-4\right)$.

#### 4.2. Simulation Results

**,**which appropriately accounted for the ordered means. Thus, these ordered mean estimators provided some advantages over the standard estimator $\mathit{Y}$. Here, users needed to specify the option “decreasing = FALSE” (Scenario (a)) or “decreasing = TRUE” (Scenario (e)) to capture the true ordering of the means. On the other hand, ${\mathit{\delta}}^{\mathrm{PT}}$ and ${\mathit{\delta}}^{\mathrm{GPT}}$ produced unreasonably large TMSE values, since they wrongly imposed the sparse mean assumptions.

## 5. Data Example

## 6. Conclusions and Future Extensions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. R Code for Simulations

## Appendix B. R Code for the Data Example

## References

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**Figure 1.**The schematic diagram for implementing the pool-adjacent violators algorithm (PAVA) in the R function “rml(.)” in our R package.

**Figure 2.**The LOWLESS plot based on the estimates (y) and the proportions of males (x) from the COVID-19 data with 11 studies. Shown are Kendall’s tau and the test of no association between x and y.

**Figure 3.**Simulation results for the estimators $\mathit{Y}$, ${\mathit{\delta}}^{\mathrm{JS}}$, ${\mathit{\delta}}^{\mathrm{JS}+}$,${\mathit{\delta}}^{\mathrm{RML}}$, ${\mathit{\delta}}^{\mathrm{RJS}}$, ${\mathit{\delta}}^{\mathrm{RJS}+}$, ${\mathit{\delta}}^{\mathrm{PT}}$, and ${\mathit{\delta}}^{\mathrm{GPT}}$. Comparison is based on Monte Carlo average:$\text{}\mathrm{TMSE}\equiv {\displaystyle \sum}_{r=1}^{\mathrm{10,000}}\left[{\displaystyle \sum}_{i=1}^{G}{\left({\delta}_{i}\left({\mathit{Y}}^{\left(r\right)}\right)-{\mu}_{i}\right)}^{2}\right]/\mathrm{10,000}$.

**Figure 4.**The LOWLESS 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.

**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).

Treatment Effect on SBP | SE | Treatment Effect on DBP | SE | |
---|---|---|---|---|

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 |

**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).

Covariate | $\mathit{Y}$ | ${\mathit{\delta}}^{\mathbf{JS}}$ | ${\mathit{\delta}}^{\mathbf{JS}+}$ | ${\mathit{\delta}}^{\mathbf{RML}}$ | ${\mathit{\delta}}^{\mathbf{RJS}}$ | ${\mathit{\delta}}^{\mathbf{RJS}+}$ | |
---|---|---|---|---|---|---|---|

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 |

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## Share and Cite

**MDPI and ACS Style**

Taketomi, N.; Michimae, H.; Chang, Y.-T.; Emura, T.
*meta.shrinkage*: An R Package for Meta-Analyses for Simultaneously Estimating Individual Means. *Algorithms* **2022**, *15*, 26.
https://doi.org/10.3390/a15010026

**AMA Style**

Taketomi N, Michimae H, Chang Y-T, Emura T.
*meta.shrinkage*: An R Package for Meta-Analyses for Simultaneously Estimating Individual Means. *Algorithms*. 2022; 15(1):26.
https://doi.org/10.3390/a15010026

**Chicago/Turabian Style**

Taketomi, Nanami, Hirofumi Michimae, Yuan-Tsung Chang, and Takeshi Emura.
2022. "*meta.shrinkage*: An R Package for Meta-Analyses for Simultaneously Estimating Individual Means" *Algorithms* 15, no. 1: 26.
https://doi.org/10.3390/a15010026