Meta-Essentials applies the inverse variance weighting method with, in the random effects model, an additive between-studies variance component based on the DerSimonian-Laird estimator (DerSimonian & Laird, 1986). Note that in Workbook 2 ‘Differences between independent groups - binary data.xlsx’ you can choose between three weighting methods. The confidence intervals are estimated using the weighted variance method for random effects models, see Sánchez-Meca and Marín-Martínez (2008). Therefore, the confidence and prediction intervals of the combined effect size calculated by Meta-Essentials might be different from one calculated by another meta-analysis program. Moreover, we also use the Student’s t-distribution to calculate the confidence interval of the individual study effect sizes (not done by most other meta-analysis tools).
For a discussion of the methods applied in the Publication Bias Analysis sheet, their application and how they should be interpreted, see Sterne, Gavaghanb, and Egger (2000) and Anzures-Cabrera and Higgins (2010). Specifically for the Trim and Fill plot, Meta-Essentials uses an iterative procedure for trimming the set of studies from the right (or left), re-estimate a combined effect size, and finally filling the plot with symmetric results on the other side of the mean. Meta-Essentials runs three iterations of the procedure, which is shown to be sufficient for many real-life cases (Duval & Tweedie, 2000a).
Anzures-Cabrera, J., & Higgins, J. P. T. P. T. (2010). Graphical displays for meta-analysis: An overview with suggestions for practice. Research Synthesis Methods, 1(1), 66-80. dx.doi.org/10.1002/jrsm.6
DerSimonian, R., & Laird, N. (1986). Meta-analysis in clinical trials. Controlled Clinical Trials, 7(3), 177-188. dx.doi.org/10.1016/0197-2456(86)90046-2
Duval, S., & Tweedie, R. (2000a). A nonparametric "trim and fill" method of accounting for publication bias in meta-analysis. Journal of the American Statistical Association, 95(449), 89-98. dx.doi.org/10.1080/01621459.2000.10473905
Higgins, J. P. T., & Thompson, S. G. (2002). Quantifying heterogeneity in a meta‐analysis. Statistics in Medicine, 21(11), 1539-1558. dx.doi.org/10.1037/1082-989x.3.4.486
Sánchez-Meca, J., & Marín-Martínez, F. (2008). Confidence intervals for the overall effect size in random-effects meta-analysis. Psychological Methods, 13(1), 31-48. dx.doi.org/10.1037/1082-989x.13.1.31
Sterne, J. A., Gavaghan, D., & Egger, M. (2000). Publication and related bias in meta-analysis: Power of statistical tests and prevalence in the literature. Journal of Clinical Epidemiology, 53(11), 1119-1129. dx.doi.org/10.1016/s0895-4356(00)00242-0