9.4 Interactions

So far, in our multiple meta-regression model, we only considered the case where we have multiple predictor variables \(x_1,x_2, ... x_p\), and along with their predictor estimates \(\beta_p\), add them together to calculate our estimate of the true effect size \(\hat \theta_k\) for each study \(k\). In multiple meta-regression models, however, we can not only model such additive relationships. We can also model so-called interactions. Interactions mean that the relationship between one predictor variable (e.g., \(x_1\)) and the estimated effect size is different for different values of another predictor variable (e.g. \(x_2\)).

Imagine a scenario where we want to model two predictors and their relationship to the effect size: the sex (\(x_1\)) of participants and the donor type (\(x_2\)) of participants in a study (either Anxious or Typical).

As we described before, we can now imagine a meta-regression model in which we combine these two predictors \(x_1\) and \(x_2\) and assume an additive relationship. We can do this by simply adding them:

\[\theta_k = \beta_0 + \beta_1x_{1k} + \beta_2x_{2k} + \epsilon_k + \zeta_k\]

We could now ask ourselves if the relationship of participants’ sex depends on the type of participants (Anxious or Typical; \(x_2\)). For example, maybe sex predicts effect size more strongly in Anxious populations? To examine such questions, we can add an interaction term to our meta-regression model. This interaction term lets predictions of \(x_1\) vary for different values of \(x_2\) (and vice versa). We can denote this additional interactional relationship in our model by introducing a third predictor, \(\beta_3\), which captures this interaction \(x_{1k}x_{2k}\) we want to test in our model:

\[\theta_k = \beta_0 + \beta_1x_{1k} + \beta_2x_{2k} + \beta_3x_{1k}x_{2k}+ \epsilon_k + \zeta_k\]

The R-syntax for this interaction is:

m_int <- rma(yi = d,
             vi = vi,
             mods = ~sex*donorcode,
             data = df)
m_int

## 
## Mixed-Effects Model (k = 56; tau^2 estimator: REML)
## 
## tau^2 (estimated amount of residual heterogeneity):     0.0537 (SE = 0.0174)
## tau (square root of estimated tau^2 value):             0.2318
## I^2 (residual heterogeneity / unaccounted variability): 66.54%
## H^2 (unaccounted variability / sampling variability):   2.99
## R^2 (amount of heterogeneity accounted for):            5.79%
## 
## Test for Residual Heterogeneity:
## QE(df = 52) = 144.6429, p-val < .0001
## 
## Test of Moderators (coefficients 2:4):
## QM(df = 3) = 5.0950, p-val = 0.1650
## 
## Model Results:
## 
##                       estimate      se     zval    pval     ci.lb    ci.ub 
## intrcpt                -5.4291  3.8294  -1.4177  0.1563  -12.9345   2.0764 
## sex                     0.2016  0.1399   1.4402  0.1498   -0.0727   0.4759 
## donorcodeTypical        5.5351  3.8316   1.4446  0.1486   -1.9747  13.0450 
## sex:donorcodeTypical   -0.1974  0.1400  -1.4101  0.1585   -0.4718   0.0770 
##  
## intrcpt 
## sex 
## donorcodeTypical 
## sex:donorcodeTypical 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1