Fixed vs Random Effects
Conceptual & Mathematical Differences
The Two Models
Meta-analysis is a Weighted Average. \[ \bar{ES} = \frac{\sum (W_i \times ES_i)}{\sum W_i} \]
The difference lies entirely in how we define the Weight (\(W_i\)).
1. Fixed Effect Model
Assumption: There is ONE true Population Effect Size (\(\theta\) is equal for all studies). * Differences are purely due to Sampling Error (\(\epsilon\)). * Inference Goal: To make a Conditional Inference about just these included studies (Hedges & Vevea, 1998). * Generalizability: Findings are not generalizable beyond the included set.
\[ W_{FE} = \frac{1}{Variance_i} \]
2. Random Effects Model
Assumption: There is a DISTRIBUTION of True Effect Sizes (\(\theta_i\) varies). * Differences are due to Sampling Error (\(\epsilon\)) PLUS True Heterogeneity (\(\tau^2\)). * Inference Goal: To make an Unconditional Inference about the universe of such studies. * Generalizability: Findings are generalizable to future studies.
\[ W_{RE} = \frac{1}{Variance_i + \tau^2} \]
- \(\tau^2\) (Tau-Squared): The Between-Study Variance.
- If \(\tau^2\) is large, the weights become more equal (Small studies get more weight than in FE).
- Confidence Intervals are Wider.
Which to Use?
βFixed effects models are appropriate only ifβ¦ we want to make a statement about just these studiesβ¦ Random effects models are appropriate if we want to make a statement about the universe of such studies.β β Hedges & Vevea (1998)
In Psychology/Medicine, we almost ALWAYS use Random Effects. Why? Because patient populations, protocols, and settings always vary. There is no single βTrueβ effect.
3. Interactive Weighting Demo
See how \(\tau^2\) changes the weights.