Meta-regression

The model can be specified as

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

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

Fixed-effect meta-regression assumes that the sampled effect size ${\displaystyle \theta }$ is normally distributed with ${\displaystyle {\mathcal {N}}(\theta ,\sigma _{\theta })}$ where ${\displaystyle \sigma _{\theta }^{2}}$ 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 ${\displaystyle \theta }$, i.e. ${\displaystyle \varepsilon =0}$.

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

Here ${\displaystyle \sigma _{\eta _{j}}^{2}}$ is the variance of the effect size in study ${\displaystyle j}$. 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 rests on the assumption that ${\displaystyle \theta }$ in ${\displaystyle {\mathcal {N}}(\theta ,\sigma _{i})}$ is a random variable following a (hyper-)distribution ${\displaystyle {\mathcal {N}}(\theta ,\sigma _{\theta }).}$ A random effects meta-regression is called a mixed effects model when moderators are added to the model.

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

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

Meta-regression has been employed as a technique to derive improved parameter estimates that are of direct use to policy makers. Meta-regression provides a framework for replication and offers a sensitivity analysis for model specification.[2] There are a number of strategies for identifying and coding empirical observational data. Meta-regression models can be extended for modeling within-study dependence, excess heterogeneity and publication selection.[2] The simple regression model does not allow for within study variation. The fixed effects regression model does not allow for between study variation. The random or mixed effects model allows for within study variation and between study variation and is therefore the most appropriate model to choose in many applications. Whether there is between study variation (excess heterogeneity) can be tested under the assumption that effect sizes are homogeneous or have a tendency to a central mean. If the test shows that the effect sizes have excess heterogeneity, the random effects meta-regression model may be most appropriate.