# Consistency of the Estimator for the Common Mean in Fixed-Effect Meta-Analyses

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Fixed-Effect Meta-Analysis

## 3. Consistency

**Theorem**

**1:**

**Theorem**

**2:**

**Theorem**

**3:**

**Theorem**

**4:**

- $\alpha =-1$ giving a consistent estimator by ${\mathrm{lim}}_{n\to \infty}{\sum}_{i=1}^{n}{i}^{1}=\infty $
- $\alpha =0$ giving a consistent estimator by ${\mathrm{lim}}_{n\to \infty}{\sum}_{i=1}^{n}{i}^{0}=\infty $
- $\alpha =1$ giving a consistent estimator by ${\mathrm{lim}}_{n\to \infty}{\sum}_{i=1}^{n}{i}^{-1}=\infty $
- $\alpha =1.5$ giving an inconsistent estimator by ${\mathrm{lim}}_{n\to \infty}{\sum}_{i=1}^{n}{i}^{-1.5}=2.612\dots <\infty $
- $\alpha =2$ giving an inconsistent estimator by ${\mathrm{lim}}_{n\to \infty}{\sum}_{i=1}^{n}{i}^{-2}={\pi}^{2}/6=1.644\dots <\infty $
- $\alpha =3$ giving an inconsistent estimator by ${\mathrm{lim}}_{n\to \infty}{\sum}_{i=1}^{n}{i}^{-3}=1.202\dots <\infty $

## 4. Data Analysis

#### 4.1. Allergic Reaction Data

#### 4.2. Diabetes Data

#### 4.3. COVID-19 Data

## 5. Extension to Unknown Variances

**Theorem**

**5:**

## 6. Conclusions and Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Pooled Variance Estimator

## Appendix B

**Proof**

**of**

**Theorem**

**3.**

## Appendix C

**Proof**

**of**

**Theorem**

**4.**

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**Figure 1.**The estimates ${\widehat{\mu}}_{n}$ for $n=1,2,\dots ,1000$ 0, based on ${Y}_{i}~N(\mu ,{\sigma}_{i}^{2})$ with $\mu =1$ and ${\sigma}_{i}^{2}=c{i}^{\alpha}$ under $\alpha =-1$, $\alpha =0$, $\alpha =1$, $\alpha =1.5$, $\alpha =2$, and $\alpha =3$.

**Figure 2.**Observed value ${\sigma}_{i}^{2}$ vs. fitted value $\widehat{c}{i}^{\widehat{\alpha}}$ based on the allergic reaction data. The LSE $\left(\widehat{c},\widehat{\alpha}\right)=(0.0964,0.13)$ was obtained for the sequence ${\left\{\mathrm{log}\left({\sigma}_{i}^{2}\right)\right\}}_{i=1}^{5}$ against ${\left\{\mathrm{log}\left(i\right)\right\}}_{i=1}^{5}$.

**Figure 3.**Observed value ${\sigma}_{i}^{2}$ vs. fitted value $\widehat{c}{i}^{\widehat{\alpha}}$ based on the diabetes data. The LSE $\left(\widehat{c},\widehat{\alpha}\right)=\left(\mathrm{0.0009,1.63}\right)$ was obtained for the increasing sequence ${\left\{\mathrm{log}\left({\sigma}_{i}^{2}\right)\right\}}_{i=1}^{8}$ against ${\left\{\mathrm{log}\left(i\right)\right\}}_{i=1}^{8}$.

**Figure 4.**Observed value ${\sigma}_{i}^{2}$ vs. fitted value $\widehat{c}{i}^{\widehat{\alpha}}$ based on the COVID-19 data. The LSE $\left(\widehat{c},\widehat{\alpha}\right)=(1.69,-1.91)$ were obtained for the decreasing sequence ${\left\{\mathrm{l}\mathrm{o}\mathrm{g}\left({\sigma}_{i}^{2}\right)\right\}}_{i=1}^{11}$ against ${\left\{\mathrm{log}\left(i\right)\right\}}_{i=1}^{11}$.

**Table 1.**The dataset on the allergic reaction of medical students who were exposed to formaldehyde during an anatomy course [23].

i (Year) | Sample Size (Male) | Mean (Male) | SD (Male) | Sample Size (Female) | Mean (Female) | SD (Female) | ${\mathit{Y}}_{\mathit{i}}=$MD | SE | ${\mathit{\sigma}}_{\mathit{i}}^{2}=$SE^{2} |
---|---|---|---|---|---|---|---|---|---|

1 (2015) | 74 | 5.09 | 1.59 | 42 | 5.26 | 1.65 | −0.17 | 0.3114 | 0.0970 |

2 (2016) | 79 | 4.84 | 1.62 | 37 | 5.42 | 1.69 | −0.58 | 0.3272 | 0.1071 |

3 (2018) | 79 | 4.81 | 1.54 | 34 | 5.26 | 1.69 | −0.45 | 0.3253 | 0.1058 |

4 (2019) | 74 | 4.86 | 1.62 | 33 | 5.61 | 1.58 | −0.75 | 0.3366 | 0.1133 |

5 (2022) | 73 | 4.79 | 1.68 | 37 | 5.35 | 1.83 | −0.56 | 0.3494 | 0.1221 |

**Table 2.**The summary of 8 studies that examined the mean difference (MD) between the intervention group and the control group for pregnant women with diabetes [24].

Study | Sample Size | ${\mathit{Y}}_{\mathit{i}}=\mathbf{M}\mathbf{D}$ (mmol/L) | SE | ${\mathit{\sigma}}_{\mathit{i}}^{2}=\mathbf{S}{\mathbf{E}}^{2}$ |
---|---|---|---|---|

Aslfalah 2020 | 60 | −0.70 | 0.0256 | 0.0007 |

Fei 2014 | 97 | −0.47 | 0.1224 | 0.0150 |

Hajimoosayi 2020 | 70 | −0.20 | 0.0816 | 0.0067 |

Jamilian 2018 | 40 | −0.40 | 0.1786 | 0.0319 |

Jamilian 2019 | 60 | −0.10 | 0.0765 | 0.0059 |

Jamilian 2020 | 51 | −0.33 | 0.0918 | 0.0084 |

Lindsay 2015 | 100 | 0.01 | 0.0867 | 0.0075 |

Ostadmohammadi 2019 | 54 | −0.20 | 0.1633 | 0.0267 |

**Table 3.**Summary of the 11 studies (shown by the author and publication year) on COVID-19 patients for examining the effects of hypertension on mortality.

Study | Sample Size | ${\mathit{Y}}_{\mathit{i}}=$log (RR) | SE | ${\mathit{\sigma}}_{\mathit{i}}^{2}=$SE^{2} |
---|---|---|---|---|

Akbari 2020 | 440 | 0.6881 | 0.6732 | 0.4532 |

Bai 2000 | 127 | 0.5933 | 0.2754 | 0.0758 |

Cao 2020 | 102 | 1.1756 | 0.2821 | 0.0796 |

Chen 2020 | 123 | 0.5365 | 0.2493 | 0.0621 |

Chen T 2020 | 274 | 0.6780 | 0.1713 | 0.0294 |

Fu 2020 | 200 | 0.5878 | 0.3302 | 0.1090 |

Grasselli 2020 | 1591 | 0.4637 | 0.0956 | 0.0091 |

Li 2020 | 102 | 0.5247 | 0.3272 | 0.1071 |

Luo 2020 | 403 | 1.2326 | 0.1489 | 0.0222 |

Yuan 2020 | 27 | 2.8904 | 1.4263 | 2.0344 |

Zhou 2020 | 191 | 1.1378 | 0.2097 | 0.0440 |

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**MDPI and ACS Style**

Taketomi, N.; Emura, T.
Consistency of the Estimator for the Common Mean in Fixed-Effect Meta-Analyses. *Axioms* **2023**, *12*, 503.
https://doi.org/10.3390/axioms12050503

**AMA Style**

Taketomi N, Emura T.
Consistency of the Estimator for the Common Mean in Fixed-Effect Meta-Analyses. *Axioms*. 2023; 12(5):503.
https://doi.org/10.3390/axioms12050503

**Chicago/Turabian Style**

Taketomi, Nanami, and Takeshi Emura.
2023. "Consistency of the Estimator for the Common Mean in Fixed-Effect Meta-Analyses" *Axioms* 12, no. 5: 503.
https://doi.org/10.3390/axioms12050503