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Meta-regression is a tool used in meta-analysis to examine the impact of moderator variables on study effect size using regression-based techniques. Meta-regression is more effective at this task than are standard meta-analytic techniques.

Meta-regression models

Meta-regression analysis (MRA) is a quantitative method of conducting literature surveys. Meta-regression has gained popularity in social, behavioral and economic sciences. Important applications have focused on qualifying estimates of policy-relevant parameters, testing economic theories, explaining heterogeneity, and qualifying potential biases. 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 sampled 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

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. 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. 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 fixed effects meta-regression model may be most appropriate.

Applications

Meta-regression is an objective and statistically rigorous approach to systematic reviews. Recent applications include quantitative reviews of the empirical literature in economics, business, energy and water policy. Meta-regression analyses have been seen in studies of price and income elasticities of various commodities and taxes, productivity spillovers on multinational companies, and calculations on the value of a statistical life (VSL). Other recent meta-regression analyses have focused on qualifying elasticities derived from demand functions. Examples include own price elasticities for alcohol, tobacco, water and energy.

In energy conservation, meta-regression analysis has been used to evaluate behavioral information strategies in the residential electricity sector. In water policy analysis, meta-regression has been used to evaluate cost savings estimates due to privatization of local government services for water distribution and solid waste collection. Meta-regression is an increasingly popular tool to evaluate the available evidence in cost-benefit analysis studies of a policy or program spread across multiple studies.

References

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Further reading

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