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'''Meta-regression''' is a tool used in [[meta-analysis]] to examine the impact of [[moderator variable]]s on study effect size using [[Regression analysis|regression]]-based techniques. Meta-regression is more effective at this task than are standard meta-analytic techniques.
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== 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
 
: <math>y_j=\beta_0+ \beta_1 x_{1j}+\beta_2 x_{2j}+\cdots+\varepsilon</math>
 
Where <math>y_j</math> is the effect size in study <math>j</math> and <math>\beta_0</math> (intercept) the estimated overall effect size. The variables <math>x_{i.} (i=1\ldots k)</math> specify different characteristics of the study, <math>\varepsilon</math> 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 <math>\theta</math> is normally distributed with <math>\mathcal{N}(\theta,\sigma_\theta)</math>  where <math>\sigma_\theta^2</math> 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 <math>\theta</math>, i.e. <math>\varepsilon=0</math>.
 
: <math>y_j=\beta_0+\beta_1 x_{1j}+\beta_2 x_{2j}+\cdots+\eta_j \, </math>
 
Here <math>\sigma^2_{\eta_j}</math> is the variance of the effect size in study <math>j</math>.
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 <math>\theta</math> in <math>\mathcal {N}(\theta,\sigma_i)</math> is a random variable following a (hyper-)distribution <math>\mathcal{N}(\theta,\sigma_\theta).</math> A random effects meta-regression is called a '''mixed effects model''' when moderators are added to the model.
 
: <math>y_j=\beta_0+\beta_1 x_{1j}+\beta_2 x_{2j}+\cdots+\eta+\varepsilon_j \, </math>
 
Here <math>\sigma^2_{\varepsilon_j}</math> is the variance of the effect size in study <math>j</math>. Between study variance <math>\sigma^2_{\eta}</math> 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.<ref name=Stanley_Jarrell1989>T.D. Stanley and Stephen B. Jarrell, (1989). Meta-regression analysis: A quantitative method of literature surveys. ''Journal of Economic Surveys'', 19(3) 299-308.</ref> 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.<ref name=Stanley_Jarrell1989 /> 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.<ref name=Stanley_Doucouliagos2012>T.D. Stanley and Hristos Doucouliagos (2009). ''Meta-regression Analysis in Economics and Business'', New York: Routledge.</ref> Meta-regression analyses have been seen in studies of price and income elasticities of various commodities and taxes,<ref name=Stanley_Doucouliagos2012 /> productivity spillovers on multinational companies,<ref name=Gorg_Strobl2001 >H. Gorg and Eric Strobl (2001). Multinational companies and productivity spillovers: A meta-analysis. ''The Economic Journal'', 111(475) 723-739.</ref> and calculations on the value of a statistical life (VSL).<ref name=Bellavance2009>F. Bellavance, Georges Dionne, and Martin Lebeau (2009). The value of a statistical life: A meta-analysis with a mixed effects regression model, ''Journal of Health Economics'', 28(2) 444-464.</ref> 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.<ref name=Stanley_Doucouliagos2012 />
 
In energy conservation, meta-regression analysis has been used to evaluate behavioral information strategies in the residential electricity sector.<ref name=Delmas2013>M.A. Delmas, Miriam Fischlein and Omar I. Asensio (2013). Information strategies and energy conservation behavior: A meta-analysis of experimental studies 1975-2012. ''Energy Policy'', 61, 729-739.</ref> 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.<ref name=Bel2010>G. Bel, Xavier Fageda and Mildred E. Warner (2010). Is private production of public services cheaper than public production? A meta-regression analysis of solid waste and water services. ''Journal of Policy Analysis and Management.'' 29(3), 553-577.</ref> 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==
{{reflist}}
<references />
 
==Further reading==
*{{cite pmid|12111920}}
*{{cite book|title= Meta-Regression Analysis: Issues of Publication Bias in Economics
|first1=Colin |last1=Roberts |first2= T. D. |last2= Stanley
|isbn=978-1-4051-3799-7
|year=2005|publisher=Wiley-Blackwell}}
 
[[Category:Meta-analysis]]
[[Category:Regression analysis]]

Latest revision as of 20:26, 21 September 2014

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