Meta-regression is a meta-analysis that uses regression analysis to combine, compare, and synthesize research findings from multiple studies while adjusting for the effects of available covariates on a response variable. A meta-regression analysis aims to reconcile conflicting studies or corroborate consistent ones; a meta-regression analysis is therefore characterized by the collated studies and their corresponding data sets—whether the response variable is study-level (or equivalently aggregate) data or individual participant data (or individual patient data in medicine). A data set is aggregate when it consists of summary statistics such as the sample mean, effect size, or odds ratio. On the other hand, individual participant data are in a sense raw in that all observations are reported with no abridgment and therefore no information loss. Aggregate data are easily compiled through internet search engines and therefore not expensive. However, individual participant data are usually confidential and are only accessible within the group or organization that performed the studies. Although meta-analysis for observational data is also under extensive research, the literature largely centers around combining randomized controlled trials (RCTs). In RCTs, a study typically includes a trial that consists of arms. An arm refers to a group of participants who received the same therapy, intervention, or treatment. A meta-analysis with some or all studies having more than two arms is called network meta-analysis, indirect meta-analysis, or a multiple treatment comparison. Despite also being an umbrella term, meta-analysis sometimes implies that all included studies have strictly two arms each—same two treatments in all trials—to distinguish itself from network meta-analysis. A meta-regression can be classified in the same way—meta-regression and network meta-regression—depending on the number of distinct treatments in the regression analysis. Meta-analysis (and meta-regression) is often placed at the top of the evidence hierarchy provided that the analysis consists of individual participant data of randomized controlled clinical trials. Meta-regression plays a critical role in accounting for covariate effects, especially in the presence of categorical variables that can be used for subgroup analysis.
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 ytk = xtk′β + εtk using the central limit theorem and variable transformation, where the subscript k indicates the kth study or trial, t denotes the tth treatment, ytk indicates the response endpoint for the kth study's tth arm, xtk is the arm-level covariate vector, εtk is the error term that is independently and identically distributed as a normal distribution. For example, a sample proportion p̂tk can be logit-transformed or arcsine-transformed prior to meta-regression modeling, i.e., ytk = logit(p̂tk) or ytk = arcsin(p̂tk). Likewise, Fisher's z-transformation can be used for sample correlations, i.e., ytk = arctanh(rtk). 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 yitk is the binary response either zero or one where the additional subscript i indicates the ith participant, the sample proportion p̂tk as the sample average of yitk for i = 1, 2, ..., ntk 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 εtk—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 ytk = xtk′β + ɛtk. If only study-level summary statistics are available, the subscript t for treatment assignment can be dropped, yielding yk = xk′β + ɛk. The error term involves a variance term σtk2 (or σk2) which is not estimable unless the sample variance stk2 (or sk2) is reported as well as ytk (or yk). Most commonly, the model variance is assumed to be equal across arms and studies, in which case all subscripts are dropped, i.e., σ2. If the between-study variation is nonnegligible, the parameter estimates will be biased, and the corresponding statistical inference cannot be generalized.
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