Moderator Analysis
Explaining the Variance
Why Moderators?
When we find significant Heterogeneity (\(I^2 > 0\)), our job is to explain Why. Moderator analysis looks for study characteristics that predict Effect Size.
2. Interpreting Categorical Output
When you run a categorical moderator analysis (e.g., macat or summary(mod)$Model), here is what the output means:
| Term | Definition |
|---|---|
estimate |
\(\hat{\mu}\) = Mean ES for each moderator factor level. |
se |
Standard Error of the ES. |
var |
\(\sigma^2\) = Variance of the ES. |
ci.l / ci.u |
95% Confidence Interval. |
p.h |
p-value for Heterogeneity within that group. |
I2 |
% of True Between-Study Heterogeneity. |
2.1 Model Logic: \(Q\)-Total Decomposition
\[ Q_{total} = Q_{within} (Q_w) + Q_{between} (Q_b) \]
| Term | Definition |
|---|---|
Qw |
Measure of error in the model (Within-group variation). |
p.w |
Homogeneity p-value. |
Qb |
Measure of MODEL FIT (Between-group variation). |
p.b |
Significance of the Moderator (Does the grouping matter?). |
1. Categorical Moderators (Subgroup Analysis)
Analogous to ANOVA. We group studies (e.g., “CBT” vs “IPT”).
- Within-Group Homogeneity (\(Q_{w}\)): Do studies within the group agree?
- Between-Group Heterogeneity (\(Q_{b}\)): Do the groups differ?
\[ Q_{total} = Q_{within} + Q_{between} \]
Significant \(Q_{b}\) indicates the moderator is significant.
2. Continuous Moderators (Meta-Regression)
Analogous to Linear Regression. We predict Effect Size (\(Y\)) from a continuous variable \(X\) (e.g., Dosage, Duration, Year).
\[ ES_i = \beta_0 + \beta_1(X_i) + \epsilon_i + \zeta_i \]
- \(\beta_1\): The slope. (How much does ES change for 1 unit of X?)
- \(\zeta_i\): Residual between-study variance (unexplained heterogeneity).
Method of Estimation
We use Method of Moments or REML (Restricted Maximum Likelihood) to estimate the model weights.
Warning: Statistical Power for meta-regression is usually low. You need many studies per covariate (Rule of Thumb: \(k \ge 10\) per predictor).