Meta-regression models
Meta-regression covers a large class of models which can differ depending on the characterization of the data at one's disposal. There is generally no one-size-fits-all description for meta-regression models. Individual participant data, in particular, allow flexible modeling that reflects different types of response variable(s): continuous, count, proportion, and correlation. However, aggregate data are generally modeled as a normal linear regression ''y''''tk'' = x''tk''′''β'' + ''ε''''tk'' using the central limit theorem and variable transformation, where the subscript ''k'' indicates the ''k''th study or trial, ''t'' denotes the ''t''th treatment, ''y''''tk'' indicates the response endpoint for the ''k''th study's ''t''th arm, x''tk'' is the arm-level covariate vector, ''ε''''tk'' is the error term that is independently and identically distributed as a normal distribution. For example, a sample proportion ''p̂''''tk'' can be logit-transformed or arcsine-transformed prior to meta-regression modeling, i.e., ''y''''tk'' logit(''p̂''''tk'') or ''y''''tk'' arcsin(''p̂''''tk''). Likewise, Fisher's ''z''-transformation can be used for sample correlations, i.e., ''y''''tk'' arctanh(''r''''tk''). The most common summary statistic reported in a study is the sample mean and the sample standard deviation, in which case no transformation is needed. It is also possible to derive an aggregate-data model from an underlying individual-participant-data model. For example, if ''y''''itk'' is the binary response either zero or one where the additional subscript ''i'' indicates the ''i''th participant, the sample proportion ''p̂''''tk'' as the sample average of ''y''''itk'' for ''i'' = 1, 2, ..., ''n''''tk'' may not require any transformation if de Moivre–Laplace theorem is assumed to be at play. Note that if a meta-regression is study-level, as opposed to arm-level, there is no subscript ''t'' indicating the treatment assigned for the corresponding arm. One of the most important distinctions in meta-analysis models is whether to assume ''heterogeneity'' between studies. If a researcher assumes that studies are not ''heterogeneous'', it implies that the studies are only different due to sampling error with no material difference between studies, in which case no other source of variation would enter the model. On the other hand, if studies are heterogeneous, the additional source(s) of variation—aside from the sampling error represented by ''ε''''tk''—must be addressed. This ultimately translates to a choice between fixed-effect meta-regression and random-effect (rigorously speaking, mixed-effect) meta-regression.Fixed-effect meta-regression
Fixed-effect meta-regression reflects the belief that the studies involved lack substantial difference. An arm-level fixed-effect meta-regression is written as ''y''''tk'' x''tk''′''β'' + ''ɛ''''tk''. If only study-level summary statistics are available, the subscript ''t'' for treatment assignment can be dropped, yielding ''y''''k'' x''k''′''β'' + ''ɛ''''k''. The error term involves a variance term ''σ''''tk''2 (or ''σ''''k''2) which is not estimable unless the sample variance ''s''''tk''2 (or ''s''''k''2) is reported as well as ''y''''tk'' (or ''y''''k''). Most commonly, the model variance is assumed to be equal across arms and studies, in which case all subscripts are dropped, i.e., ''σ''2. If the between-study variation is nonnegligible, the parameter estimates will be biased, and the corresponding statistical inference cannot be generalized.Mixed-effect meta-regression
The terms ''random-effect meta-regression'' and ''mixed-effect meta-regression'' are equivalent. Although calling one a ''random-effect model'' signals the absence of fixed effects, which would technically disqualify it from being a regression model, one could argue that the modifier ''random-effect'' only adds to, not takes away from, what any regression model should include: fixed effects. Google Trends indicates that both terms enjoy similar levels of acceptance in publications as of July 24, 2021. Mixed-effect meta-regression includes a random-effect term in addition to the fixed effects, suggesting that the studies are heterogeneous. The random effects, denoted by w''tk''′''γ''''k'', represent the within-study variation. The full model then becomes ''y''''tk'' x''tk''′''β'' + w''tk''′''γ''''k'' + ''ε''''tk''. Random effects in meta-regression are intended to reflect the noisy treatment effects—unless assumed and modeled otherwise—which implies that the length of the corresponding coefficient vector ''γ''''k'' should be the same as the number of treatments included in the study. Restricting our attention to the narrow definition of meta-analysis including two treatments, ''γ''''k'' is two-dimensional, i.e., ''γ''''k'' (''γ''''1k'', ''γ''''2k'')′, for which the model is recast as ''y''''tk'' x''tk''′''β'' + ''γ''''tk'' + ''ε''''tk''. The advantage of writing the model in a matrix-vector notation is that the correlation between the treatments, Corr(''γ''''1k'', ''γ''''2k''), can be investigated. The random coefficient vector ''γ''''k'' is then a noisy realization of the real treatment effect denoted by ''γ''. The distribution of ''γ''''k'' is commonly assumed to be one in the location-scale family, most notably, aWhich model to choose
Meta-regression has been employed as a technique to derive improved parameter estimates that are of direct use to policymakers. Meta-regression provides a framework for replication and offers a sensitivity analysis for model specification.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. 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 fixed-effect regression model does not allow for within-study variation. The mixed effects model allows for within-study variation and between-study variation and is therefore taken as the most flexible model to choose in many applications. Although the heterogeneity assumption can be statistically tested and it is a widespread practice in many fields, if this test is followed by another set of regression analysis, the corresponding statistical inference is subject to what is called ''selective inference''. These heterogeneity tests also do not conclude that there is no heterogeneity even when they come out insignificant, and some researchers advise to opt for mixed-effect meta-regression at any rate.Applications
Meta-regression is a statistically rigorous approach toReferences
Further reading
* * * {{cite journal , author = Bonett DG , title = Meta-analytic interval estimation for standardized and unstandardized mean differences , journal = Psychological Methods , volume = 14 , issue = 3 , pages = 225–38 , year = 2009 , pmid = 19719359 , doi = 10.1037/a0016619 Meta-analysis Regression analysis