# Bayesian Meta-Analysis for Binary Data and Prior Distribution on Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Bayesian Binomial Model

#### 2.1. The Linking Distribution

#### 2.2. Clusters

#### 2.3. The Likelihood of $\theta $ for a Particular Partition

#### 2.4. The Likelihood of $\theta $ the Prior Distribution over the Partitions

- The Uniform prior.The first prior proposed is the uniform prior (U), which gives the same probability to every model, that is,$${\pi}^{U}(p,{\mathbf{r}}_{p}|k)=\frac{1}{{\mathcal{B}}_{k}},$$
- The Hierarchical Uniform Prior with 2 levels (HU2).As recommended by Casella et al. [13], a hierarchical uniform prior can be appropriate to take into account the different levels of complexity of the partitions. This prior distribution distinguishes two levels of complexity in the partitions. The first level is given by the number of clusters p in which the k samples are grouped. The second level will be given by the number of possible partitions of the k samples into p clusters. Let ${\mathcal{R}}_{p}$ represent this set of partitions into p clusters, which we call the cluster class. The number of partitions in ${\mathcal{R}}_{p}$ is given by the Stirling number of the second kind $\mathcal{S}(k,p)$ and can be written as$$\mathcal{S}(k,p)=\sum _{1\le {k}_{1}\le \dots \le {k}_{p}}\left({\displaystyle \genfrac{}{}{0pt}{}{{k}_{1}+\dots +{k}_{p}}{{k}_{1}\cdots {k}_{p}}}\right)\frac{1}{R({k}_{1},\dots ,{k}_{p})},$$$$\mathcal{S}(4,2)=\left({\displaystyle \genfrac{}{}{0pt}{}{4}{1,3}}\right)\frac{1}{R(1,3)}+\left({\displaystyle \genfrac{}{}{0pt}{}{4}{2,2}}\right)\frac{1}{R(2,2)}=\frac{4!}{1!3!}\frac{1}{1!1!}+\frac{4!}{2!2!}\frac{1}{2!}=4+3=7,$$The hierarchical uniform distribution for 2 levels will be given by the decomposition$${\pi}^{HU2}(p,{\mathbf{r}}_{p}|k)=\pi \left({\mathbf{r}}_{p}\right|p,k)\pi \left(p\right|k)=\frac{1}{\mathcal{S}(k,p)}\frac{1}{k}.$$Figure 3 shows the prior probabilities for each partition using the hierarchical uniform prior with 2 levels with 4 studies. Note that this hierarchical distribution assigns a higher prior probability to cases of homogeneity and heterogeneity.
- The Hierarchical Uniform Prior with 3 levels (HU3).Following Casella et al. [13] and Moreno et al. [10], the prior specification for $(p,{\mathbf{r}}_{p})$ can be decomposed in three levels:$${\pi}^{HU3}(p,{\mathbf{r}}_{p}|k)=\pi (p,{\mathbf{r}}_{p}|{\mathcal{R}}_{p;{k}_{1},\dots ,{k}_{p}},k)\pi \left({\mathcal{R}}_{p;{k}_{1},\dots ,{k}_{p}}\right|p,k)\pi \left(p\right|k).$$Unlike the previous prior distribution, the hierarchical uniform prior with 3 levels considers the number of ways the integer k can be partitioned into p clusters. We will call it the number of configuration classes within each ${\mathcal{R}}_{p}$ and it will be denoted by $b(k,p)$. In our illustrative example with $k=4$, this value is equal to 1 for $p=1,3,4$ ($b(4,1)=b(4,3)=b(4,4)=1)$, and only for the cluster class $p=2$ there are two configuration classes, corresponding to the configurations $x|xxx$ and $xx|xx$, so $b(4,2)=2$.The hierarchical uniform prior with 3 levels is given by the expression$${\pi}^{HU3}(p,{\mathbf{r}}_{p}|k)=\frac{{k}_{1}!\xb7\dots \xb7{k}_{p}!}{k!}\frac{R({k}_{1},\dots ,{k}_{p})}{b(k,p)}\frac{1}{k}.$$Figure 4 shows the prior probabilities for each partition using the hierarchical uniform prior with 3 levels and 4 studies.
- The Hierarchical Poisson Prior with 3 levels (HP3).Casella et al. [13] argue that when analyzing a cluster problem of a sample size k, the extreme case of having k clusters should be given a priori a smaller probability than that given to any other case. Extending this argument for any k, it might be reasonable that the prior distribution on the number of clusters $\pi \left(p\right|k)$ might be a truncated Poisson distribution $\mathcal{P}\left(p\right|\lambda )$, where $\lambda $ is an unknown parameter. We can assume an intrinsic prior ${\pi}^{I}\left(\lambda \right|{\lambda}_{0}=1)$ for $\lambda $, constructed by testing the Poisson null hypothesis ${H}_{0}:\lambda ={\lambda}_{0}$ versus ${H}_{1}:\lambda \in {R}^{+}$ [17],$${\pi}^{I}\left(\lambda \right|{\lambda}_{0}=1)=\frac{{\lambda}^{-1/2}}{\mathsf{\Gamma}(1/2)}{e}^{-(\lambda +1)}{}_{0}{F}_{1}(1/2,\lambda ),$$$${\pi}^{I}\left(p\right|k)=\frac{{m}^{I}\left(p\right)}{{\sum}_{p=1}^{k}{m}^{I}\left(p\right)},p=1,\dots ,k,\phantom{\rule{1.em}{0ex}}{m}^{I}\left(p\right)={\int}_{0}^{+\infty}\frac{{\lambda}^{p}{e}^{-\lambda}}{p!}{\pi}^{I}\left(\lambda \right|{\lambda}_{0}=1)d\lambda .$$We cannot assume a Poisson distribution for the other two levels of the hierarchical structure because there is no a clear order in relation to complexity. For this reason, a uniform distribution is assumed for the other two levels. The hierarchical Poisson prior will be given by$${\pi}^{HP3}(p,{\mathbf{r}}_{p}|k)=\frac{{k}_{1}!\xb7\dots \xb7{k}_{p}!}{k!}\frac{R({k}_{1},\dots ,{k}_{p})}{b(k,p)}{\pi}^{I}\left(p\right|k).$$Figure 5 shows the prior probabilities for each partition using the hierarchical Poisson prior with 4 studies. The prior probability for the homogeneity cluster is more than four times higher than the prior probability of the heterogeneity case.

#### 2.5. Bayesian Model Averaging in the Meta-Analysis

`Wolfram Mathematica`(see code in the supplementary material Section).

## 3. Simulated Data and Frequentist Validation

#### 3.1. Simulated Data

#### 3.2. Frequentist Evaluation

## 4. An Illustrative Example with Real Data

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

`Mathematica`codes implementing the simulate and real data experiment are available on the supplementary material Section.

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Hierarchical Uniform prior with 2 levels probabilities for the partitions with $k=4$ studies.

**Figure 4.**Hierarchical Uniform prior with 3 levels probabilities for the partitions with $k=4$ studies.

**Figure 5.**Hierarchical Poisson prior with 3 levels probabilities for the partitions with $k=4$ studies.

**Figure 6.**Frequentist validation for the case $k=3$ and true partitions ${\mathbf{r}}_{1}=(1,1,1),{\mathbf{r}}_{2}=(1,1,2)$ and ${\mathbf{r}}_{3}=(1,2,3).$ (

**Left column**) proportion of times the true partition is found as the most probable. (

**Right column**) mean of the posterior probability for the true partition when it is found as the most probable.

**Figure 7.**Frequentist validation for the case $k=3$ and true partitions ${\mathbf{r}}_{1}=(1,1,1),{\mathbf{r}}_{2}=(1,1,2)$ and ${\mathbf{r}}_{3}=(1,2,3).$ (

**Left column**) proportion of times the homogeneity case is found as the most probable. (

**Right column**) proportion of times the heterogeneity case is found as the most probable.

**Figure 8.**Frequentist validation for the case k = 5 and true partitions ${\mathbf{r}}_{1}=(1,1,1,1,1),{\mathbf{r}}_{2}=(1,1,1,2,2),{\mathbf{r}}_{3}=(1,1,2,2,3)$, and ${\mathbf{r}}_{5}=(1,2,3,4,5)$. (

**Left column**) proportion of times the true partition is found as the most probable. (

**Right column**) mean of the posterior probability for the true partition when it is found as the most probable.

**Figure 9.**Frequentist validation for the case k = 5 and true partitions ${\mathbf{r}}_{1}=(1,1,1,1,1),{\mathbf{r}}_{2}=(1,1,1,2,2),{\mathbf{r}}_{3}=(1,1,2,2,3)$, and ${\mathbf{r}}_{5}=(1,2,3,4,5)$. (

**Left column**) proportion of times the homogeneity case is found as the most probable. (

**Right column**) proportion of times the heterogeneity case is found as the most probable.

True Model (${\mathbf{r}}_{\mathit{p}}$) | Parameters (${\mathit{\theta}}_{\mathit{i}}$’s) | Sample Sizes (${\mathit{n}}_{\mathit{i}}$) |
---|---|---|

$k=3$ | ||

$1,1,1$ | $(0.5,0.5,0.5)$ | $(10,30,100,300)$ |

$1,1,2$ | $(0.5,0.5,0.2)$ | $(10,30,100,300)$ |

$1,2,3$ | $(0.7,0.5,0.2)$ | $(10,30,100,300)$ |

$k=5$ | ||

$1,1,1,1,1$ | $(0.5,0.5,0.5,0.5,0.5)$ | $(10,30,100,300)$ |

$1,1,1,2,2$ | $(0.5,0.5,0.5,0.2,0.2)$ | $(10,30,100,300)$ |

$1,1,2,2,3$ | $(0.7,0.7,0.5,0.5,0.2)$ | $(10,30,100,300)$ |

$1,2,3,4,5$ | $(0.9,0.7,0.5,0.3,0.1)$ | $(10,30,100,300)$ |

**Table 2.**Data in Stanworth et al. [19].

Study | Treatment | |
---|---|---|

Events | Total | |

Herzig 1977 (${x}_{1}$) | 1 | 16 |

Higby 1975 (${x}_{2}$) | 2 | 17 |

Scali 1978 (${x}_{3}$) | 0 | 13 |

Vogler 1977 (${x}_{4}$) | 7 | 17 |

Prior # 1: Uniform | Prior # 2: HU with 2 levels | Prior # 3: HU with 3 Levels | Prior # 4: HP with 3 Levels | ||||
---|---|---|---|---|---|---|---|

Cluster Model | Post. Prob. | Cluster Model | Post. Prob. | Cluster Model | Post. Prob. | Cluster Model | Post. Prob. |

${x}_{1}{x}_{2}{x}_{3}|{x}_{4}$ | 0.19 | ${x}_{1}|{x}_{2}\left|{x}_{3}\right|{x}_{4}$ | 0.37 | ${x}_{1}|{x}_{2}\left|{x}_{3}\right|{x}_{4}$ | 0.37 | ${x}_{1}|{x}_{2}\left|{x}_{3}\right|{x}_{4}$ | 0.21 |

${x}_{1}{x}_{3}\left|{x}_{2}\right|{x}_{4}$ | 0.14 | ${x}_{1}{x}_{2}{x}_{3}|{x}_{4}$ | 0.11 | ${x}_{1}{x}_{2}{x}_{3}|{x}_{4}$ | 0.10 | ${x}_{1}{x}_{2}{x}_{3}{x}_{4}$ | 0.18 |

${x}_{1}{x}_{2}\left|{x}_{3}\right|{x}_{4}$ | 0.11 | ${x}_{1}{x}_{3}\left|{x}_{2}\right|{x}_{4}$ | 0.09 | ${x}_{1}{x}_{3}\left|{x}_{2}\right|{x}_{4}$ | 0.09 | ${x}_{1}{x}_{2}{x}_{3}|{x}_{4}$ | 0.14 |

${x}_{1}\left|{x}_{2}{x}_{3}\right|{x}_{4}$ | 0.11 | ${x}_{1}{x}_{2}{x}_{3}{x}_{4}$ | 0.08 | ${x}_{1}{x}_{2}{x}_{3}{x}_{4}$ | 0.08 | ${x}_{1}{x}_{3}\left|{x}_{2}\right|{x}_{4}$ | 0.8 |

${x}_{1}|{x}_{2}\left|{x}_{3}\right|{x}_{4}$ | 0.09 | ${x}_{1}{x}_{2}\left|{x}_{3}\right|{x}_{4}$ | 0.08 | ${x}_{1}{x}_{2}\left|{x}_{3}\right|{x}_{4}$ | 0.08 | ${x}_{1}{x}_{3}|{x}_{2}{x}_{4}$ | 0.08 |

the rest | <0.09 | the rest | <0.07 | the rest | <0.07 | the rest | <0.07 |

BMA estimates of the meta-parameter $\theta $ | |||||||

Posterior mean: 0.164 | Posterior mean: 0.159 | Posterior mean: 0.159 | Posterior mean: 0.163 | ||||

$95\%$ HDI: 0.050–0.317 | $95\%$ HDI: 0.049–0.312 | $95\%$ HDI: 0.049–0.311 | $95\%$ HDI: 0.048–0.327 |

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**MDPI and ACS Style**

Negrín-Hernández, M.-A.; Martel-Escobar, M.; Vázquez-Polo, F.-J. Bayesian Meta-Analysis for Binary Data and Prior Distribution on Models. *Int. J. Environ. Res. Public Health* **2021**, *18*, 809.
https://doi.org/10.3390/ijerph18020809

**AMA Style**

Negrín-Hernández M-A, Martel-Escobar M, Vázquez-Polo F-J. Bayesian Meta-Analysis for Binary Data and Prior Distribution on Models. *International Journal of Environmental Research and Public Health*. 2021; 18(2):809.
https://doi.org/10.3390/ijerph18020809

**Chicago/Turabian Style**

Negrín-Hernández, Miguel-Angel, María Martel-Escobar, and Francisco-José Vázquez-Polo. 2021. "Bayesian Meta-Analysis for Binary Data and Prior Distribution on Models" *International Journal of Environmental Research and Public Health* 18, no. 2: 809.
https://doi.org/10.3390/ijerph18020809