Meta-regression

Meta-regression models

Generally, three types of models can be distinguished in the literature on meta-analysis: simple regression, fixed effect meta-regression and random effects meta-regression.

Simple regression

The model can be specified as

$y_{j}=\beta _{0}+\beta _{1}x_{1j}+\beta _{2}x_{2j}+\cdots +\varepsilon$

Where $y_{j}$ is the effect size in study $j$ and $\beta _{0}$ (intercept) the estimated overall effect size. The variables $x_{i.}(i=1\ldots k)$ specify different characteristics of the study, $\varepsilon$ specifies the between study variation. Note that this model does not allow specification of within study variation.

Fixed-effect meta-regression

Fixed-effect meta-regression assumes that the true effect size $\theta$ is normally distributed with ${\mathcal {N}}(\theta ,\sigma _{\theta })$ where $\sigma _{\theta }^{2}$ is the within-study variance of the effect size. A fixed-effect meta-regression model thus allows for within-study variability but not between-study variability because all studies have an identical expected fixed effect size $\theta$, i.e. $\varepsilon =0$.

$y_{j}=\beta _{0}+\beta _{1}x_{1j}+\beta _{2}x_{2j}+\cdots +\eta _{j}\,$

Here $\sigma _{\eta _{j}}^{2}$ is the variance of the effect size in study $j$. Fixed effect meta-regression ignores between study variation. As a result, parameter estimates are biased if between study variation can not be ignored. Furthermore, generalizations to the population are not possible.

Random effects meta-regression

Random effects meta-regression rests on the assumption that $\theta$ in ${\mathcal {N}}(\theta ,\sigma _{i})$ is a random variable following a (hyper-)distribution ${\mathcal {N}}(\theta ,\sigma _{\theta }).$ A random effects meta-regression is called a mixed effects model when moderators are added to the model.

$y_{j}=\beta _{0}+\beta _{1}x_{1j}+\beta _{2}x_{2j}+\cdots +\eta +\varepsilon _{j}\,$

Here $\sigma _{\varepsilon _{j}}^{2}$ is the variance of the effect size in study $j$. Between study variance $\sigma _{\eta }^{2}$ is estimated using common estimation procedures for random effects models (restricted maximum likelihood (REML) estimators).

Which model to choose

The simple regression model does not allow for within study variation, this yields in to significant results too easy. The fixed effects regression model does not allow for between study variation, this also yields in to significant results too easy. The random or mixed effects model allows for within study variation and between study variation and is therefore the most appropriate model to choose. Whether there is between study variation can be tested by testing whether the effect sizes are homogeneous. If the test shows that the effect sizes are not heterogeneous the fixed effects meta-regression might seem appropriate, however this test often does not have enough power to detect between study variation. Besides the lack of power of this test, you can reason that the fixed effects assumption of homogeneous effect sizes is rather weak, because it assumes that all studies are exactly the same. However you can assume that no two studies are exactly the same. To cope with the fact that each study is different (different sample; different time; different place; etc) a random or mixed effects model is always the appropriate model to choose and gives the most reliable results.