- [[random-effects meta-analysis]], [[meta-analyze regression coefficients]] # Idea A fixed-effects meta-analysis assumes that a single "true" effect exists, which is common to all observed studies (thus also known as "common effects" meta-analysis). $y_i \mid \theta_i \sim N\left(\theta_i, v_i\right)$ $\hat{\theta}_k=\theta+\epsilon_k$ Deviations of individual studies from this "true" effect represent only random variation due to sampling error. Observed effect sizes may vary from study to study, but the variation is only because of the sampling error. Thus, the study weights are calculated taking into account only within-study variance (i.e., sampling error), and the pooled meta-analytic estimate is interpreted as the best estimate of the common underlying effect. It assumes no heterogeneity between studies and will consider within-study sampling error as the only source of variance. While all observed effect sizes are estimators of the true effect, some are better than others. When we pool the effects in our meta-analysis, we should therefore give effect sizes with a higher precision (i.e. a smaller standard error) a greater weight. If we want to calculate the pooled effect size under the fixed-effect model, we therefore simply use a weighted average of all studies. # References - [Fixed-Effects and Random-Effects Models in Meta-Analysis — misc-models • metafor](https://wviechtb.github.io/metafor/reference/misc-models.html) - differences between fixed effects, random effects, and equal effects - [How to Perform a Meta-Regression | Columbia Public Health](https://www.publichealth.columbia.edu/research/population-health-methods/meta-regression) - [Chapter 4 Pooling Effect Sizes | Doing Meta-Analysis in R](https://bookdown.org/MathiasHarrer/Doing_Meta_Analysis_in_R/pooling-es.html)