9.3 Multiple Meta-Regression

Previously, we only considered the scenario in which we use one predictor \(\beta_1x_1\) in our meta-regression. When we add more than one predictor, we’re using multiple meta-regression. In multiple meta-regression we use several moderators (variables) to predict the outcome (effect sizes). When we look back at the general meta-regression formula we defined before, we actually see that the formula already provides us with this feature through the \(\beta_nx_{pk}\) part. Here, the parameter \(p\) denotes that we can include \(p\) more predictors/variables into our meta-regression, making it a multiple meta-regression.

Imagine, for example, that we expect the effect size to be determined by sex, but also by the type of outcome variable that was measured, outcomecode. We could include both predictors in a mutliple meta-regression as follows:

m_multi <- rma(yi = d,
               vi = vi,
               mods = ~sex+outcomecode,
               data = df)
m_multi

## 
## Mixed-Effects Model (k = 56; tau^2 estimator: REML)
## 
## tau^2 (estimated amount of residual heterogeneity):     0.0595 (SE = 0.0190)
## tau (square root of estimated tau^2 value):             0.2440
## I^2 (residual heterogeneity / unaccounted variability): 67.54%
## H^2 (unaccounted variability / sampling variability):   3.08
## R^2 (amount of heterogeneity accounted for):            0.00%
## 
## Test for Residual Heterogeneity:
## QE(df = 51) = 148.3124, p-val < .0001
## 
## Test of Moderators (coefficients 2:5):
## QM(df = 4) = 2.2088, p-val = 0.6974
## 
## Model Results:
## 
##                               estimate      se     zval    pval    ci.lb 
## intrcpt                         0.0880  0.1491   0.5903  0.5550  -0.2042 
## sex                             0.0049  0.0035   1.4148  0.1571  -0.0019 
## outcomecodeLife Satisfaction   -0.0625  0.1543  -0.4051  0.6854  -0.3649 
## outcomecodeOther               -0.0318  0.1272  -0.2500  0.8026  -0.2811 
## outcomecodePN Affect           -0.0169  0.1121  -0.1510  0.8800  -0.2365 
##                                ci.ub 
## intrcpt                       0.3801    
## sex                           0.0118    
## outcomecodeLife Satisfaction  0.2399    
## outcomecodeOther              0.2175    
## outcomecodePN Affect          0.2027    
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We will discuss a few important pitfalls in multiple meta-regression and how we can build multiple meta-regression models which are robust and trustworthy. But first, let’s cover another important feature of multiple meta-regression: interactions.