Fixed vs Random Effects
The Philosophical Divide
One True Effect or Many?
Do all your studies estimate the exact same number (Fixed Effect)? Or do they estimate a distribution of numbers (Random Effects)?
1. The Weighting War
In meta-analysis, we weigh studies by their precision (inverse variance). But the formula changes based on your model.
- Fixed Effect: \(W_i = \frac{1}{v_i}\)
- Random Effects: \(W_i^* = \frac{1}{v_i + \tau^2}\)
Where \(\tau^2\) (Tau-Squared) is the Between-Study Variance.
Interactive Simulator: The Impact of Tau-Squared
What happens when \(\tau^2\) gets big?
- Small studies gain more influence.
- Large studies lose their βdominanceβ.
- The weights become more equal (flat).
Move the slider below to increase Heterogeneity (\(\tau^2\)) and watch the weights shift.
Notice that when \(\tau^2 = 0\) (Fixed Effect), the Huge Study gets nearly 100% of the weight. As \(\tau^2\) increases, the Small Study starts to matter more. This is why Random Effects models have wider Confidence Intervals!
2. When to Use Which?
The field has moved decisively toward Random Effects.
βWe almost never have reason to believe the true effect size is identical across all psychological studies.β β Borenstein et al.
Unless you are replicating exact physics experiments, assume Heterogeneity. Use Random Effects.