# -5.0854 20.1708 25.1495
## 8 Discussion  
This paper proposes a method for calculating overall covariance for regression coefficients in the presence of interactions in a two-stage meta-analysis. The approach utilizes the statistical relationship between the correlation coefficient and variances to estimate the covariance. By conducting separate meta-analyses, the variances and correlation coefficient can be obtained. While statistical software packages provide built-in capabilities for analyzing the overall main effect, interaction effects, and their variances, merging covariances in meta-analysis when interactions are present remains a challenge [16]. We present a transparent and straightforward method to synthesize covariance in meta-analysis.  
Detecting interactions is crucial in medical research, as interaction analyses can help determine whether an intervention is more effective for certain individuals [17; 18; 19]. In a two-stage meta-analysis, interactions between an exposure and covariates can be explored using meta-regression. However, the meta-regression method cannot study patient-level factors [20] and often suffers from low statistical power [21]. An alternative is dividing participants into subgroups (e.g., by age) and performing separate meta-analyses for each subgroup. While this avoids synthesizing interactions, it also tends to have low power [22].  
Multivariate meta-analysis is another approach for investigating interactions in meta-analysis, allowing for the synthesis of correlated effects. For instance, in hypertension trials, systolic and diastolic blood pressure outcomes can be pooled using this approach [6; 23]. In multivariate meta-analysis, the first-stage analysis estimates the coefficients (e.g., \(\beta\) coefficients) and their covariances, which serve as inputs for the second-stage analysis conducted using mixed-model regression [8; 24]. Compared to multivariate meta-analysis, our proposed method provides a narrower confidence interval for the overall effect.  
Our method has some limitations. It assumes a linear relationship between the two regression coefficients, which may not always hold. Additionally, the estimation of the correlation coefficient may be less reliable and biased when the number of included studies is small [25]. Furthermore, this method is likely only useful for two-stage IPD meta-analysis. For one-stage approaches, particularly in advanced modeling contexts, covariance can be calculated directly from the regression modeling.  
Meta-analysis has become a critical tool for synthesizing research findings across studies, enabling the identification of patterns and effect sizes in diverse fields. In future research in meta-analysis, particularly focusing on interaction effects and covariates, Bayesian hierarchical models can be developed to integrate interaction effects while accounting for study-level heterogeneity [26]. Traditional meta-regression models might be expanded to include individual level covariates as moderators, enabling the exploration of how study-level and participant-level covariates influence outcomes [27].  
In conclusion, we have introduced a simple and transparent method for calculating covariance and confidence intervals in meta-analysis when interactions are present.  
Conceptualization, E.L.; methodology, E.L.; software, E.L. and R.Y.L.; validation, R.Y.L. and E.L.; writing--original draft preparation, E.L.; writing--review and editing, E.L. and R.Y.L. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. Not applicable. Not applicable. Not applicable. Data Availability Statement: Dataset and all code for this paper are available at [[https://github.com/enwuliu/meta-analysis](https://github.com/enwuliu/meta-analysis)]([https://github.com/enwuliu/meta-analysis](https://github.com/enwuliu/meta-analysis)) (accessed on 9 December 2024). The authors declare no conflicts of interest.