A metaanalysis is a statistical analysis that combines the results of multiple scientific studies.
The basic tenet behind metaanalyses is that there is a common truth behind all conceptually similar scientific studies, but which has been measured with a certain error within individual studies. The aim then is to use approaches from statistics to derive a pooled estimate closest to the unknown common truth based on how this error is perceived.
In essence, all existing methods yield a weighted average from the results of the individual studies and what differs is the manner in which these weights are allocated and also the manner in which the uncertainty is computed around the point estimate thus generated.
In addition to providing an estimate of the unknown common truth, metaanalysis has the capacity to contrast results from different studies and identify patterns among study results, sources of disagreement among those results, or other interesting relationships that may come to light in the context of multiple studies.
Metaanalysis can be thought of as “conducting research about previous research.” Metaanalysis can only proceed if we are able to identify a common statistical measure that is shared among studies, called the effect size, which has a standard error so that we can proceed with computing a weighted average of that common measure. Such weighting usually takes into consideration the sample sizes of the individual studies, although it can also include other factors, such as study quality.
A key benefit of this approach is the aggregation of information leading to a higher statistical power and more robust point estimate than is possible from the measure derived from any individual study. However, in performing a metaanalysis, an investigator must make choices which can affect the results, including deciding how to search for studies, selecting studies based on a set of objective criteria, dealing with incomplete data, analyzing the data, and accounting for or choosing not to account for publication bias.
Metaanalyses are often, but not always, important components of a systematic review procedure. For instance, a metaanalysis may be conducted on several clinical trials of a medical treatment, in an effort to obtain a better understanding of how well the treatment works. Here it is convenient to follow the terminology used by the Cochrane Collaboration, and use “metaanalysis” to refer to statistical methods of combining evidence, leaving other aspects of ‘research synthesis’ or ’evidence synthesis’, such as combining information from qualitative studies, for the more general context of systematic reviews.
Advantages Of Metaanalysis
Conceptually, a metaanalysis uses a statistical approach to combine the results from multiple studies in an effort to increase power (over individual studies), improve estimates of the size of the effect and/or to resolve uncertainty when reports disagree. A metaanalysis is a statistical overview of the results from one or more systematic review. Basically, it produces a weighted average of the included study results and this approach has several advantages:

Results can be generalized to a larger population

The precision and accuracy of estimates can be improved as more data is used. This, in turn, may increase the statistical power to detect an effect

Inconsistency of results across studies can be quantified and analyzed. For instance, does inconsistency arise from sampling error, or are study results (partially) influenced by betweenstudy heterogeneity

Hypothesis testing can be applied on summary estimates

Moderators can be included to explain variation between studies

The presence of publication bias can be investigated
Disadvantages
A metaanalysis of several small studies does not predict the results of a single large study. Some have argued that a weakness of the method is that sources of bias are not controlled by the method: a good metaanalysis of badly designed studies will still result in bad statistics. This would mean that only methodologically sound studies should be included in a metaanalysis, a practice called ‘best evidence synthesis’.
Other metaanalysts would include weaker studies, and add a studylevel predictor variable that reflects the methodological quality of the studies to examine the effect of study quality on the effect size. However, others have argued that a better approach is to preserve information about the variance in the study sample, casting as wide a net as possible, and that methodological selection criteria introduce unwanted subjectivity, defeating the purpose of the approach.
Publication Bias: The File Drawer Problem
Another potential pitfall is the reliance on the available body of published studies, which may create exaggerated outcomes due to publication bias, as studies which show negative results or insignificant results are less likely to be published.
For example, pharmaceutical companies have been known to hide negative studies and researchers may have overlooked unpublished studies such as dissertation studies or conference abstracts that did not reach publication. This is not easily solved, as one cannot know how many studies have gone unreported.
This file drawer problem (characterized by negative or nonsignificant results being tucked away in a cabinet), can result in a biased distribution of effect sizes thus creating a serious base rate fallacy, in which the significance of the published studies is overestimated, as other studies were either not submitted for publication or were rejected. This should be seriously considered when interpreting the outcomes of a metaanalysis.
The distribution of effect sizes can be visualized with a funnel plot which (in its most common version) is a scatter plot of standard error versus the effect size. It makes use of the fact that the smaller studies (thus larger standard errors) have more scatter of the magnitude of effect (being less precise) while the larger studies have less scatter and form the tip of the funnel. If many negative studies were not published, the remaining positive studies give rise to a funnel plot in which the base is skewed to one side (asymmetry of the funnel plot).
In contrast, when there is no publication bias, the effect of the smaller studies has no reason to be skewed to one side and so a symmetric funnel plot results. This also means that if no publication bias is present, there would be no relationship between standard error and effect size.
A negative or positive relation between standard error and effect size would imply that smaller studies that found effects in one direction only were more likely to be published and/or to be submitted for publication.
Apart from the visual funnel plot, statistical methods for detecting publication bias have also been proposed. These are controversial because they typically have low power for detection of bias, but also may create false positives under some circumstances. For instance small study effects (biased smaller studies), wherein methodological differences between smaller and larger studies exist, may cause asymmetry in effect sizes that resembles publication bias.
However, small study effects may be just as problematic for the interpretation of metaanalyses, and the imperative is on metaanalytic authors to investigate potential sources of bias.
A Tandem Method for analyzing publication bias has been suggested for cutting down false positive error problems. This Tandem method consists of three stages. Firstly, one calculates Orwin’s failsafe N, to check how many studies should be added in order to reduce the test statistic to a trivial size.
If this number of studies is larger than the number of studies used in the metaanalysis, it is a sign that there is no publication bias, as in that case, one needs a lot of studies to reduce the effect size. Secondly, one can do an Egger’s regression test, which tests whether the funnel plot is symmetrical.
As mentioned before: a symmetrical funnel plot is a sign that there is no publication bias, as the effect size and sample size are not dependent. Thirdly, one can do the trimandfill method, which imputes data if the funnel plot is asymmetrical.
The problem of publication bias is not trivial as it is suggested that 25% of metaanalyses in the psychological sciences may have suffered from publication bias. However, low power of existing tests and problems with the visual appearance of the funnel plot remain an issue, and estimates of publication bias may remain lower than what truly exists.
Most discussions of publication bias focus on journal practices favoring publication of statistically significant findings. However, questionable research practices, such as reworking statistical models until significance is achieved, may also favor statistically significant findings in support of researchers’ hypotheses.
Problems Related To The Statistical Approach
Other weaknesses are that it has not been determined if the statistically most accurate method for combining results is the fixed, IVhet, random or quality effect models, though the criticism against the random effects model is mounting because of the perception that the new random effects (used in metaanalysis) are essentially formal devices to facilitate smoothing or shrinkage and prediction may be impossible or illadvised.
The main problem with the random effects approach is that it uses the classic statistical thought of generating a “compromise estimator” that makes the weights close to the naturally weighted estimator if heterogeneity across studies is large but close to the inverse variance weighted estimator if the between study heterogeneity is small. However, what has been ignored is the distinction between the model we choose to analyze a given dataset, and the mechanism by which the data came into being.
A random effect can be present in either of these roles, but the two roles are quite distinct. There’s no reason to think the analysis model and datageneration mechanism (model) are similar in form, but many subfields of statistics have developed the habit of assuming, for theory and simulations, that the datageneration mechanism (model) is identical to the analysis model we choose (or would like others to choose).
As a hypothesized mechanisms for producing the data, the random effect model for metaanalysis is silly and it is more appropriate to think of this model as a superficial description and something we choose as an analytical tool – but this choice for metaanalysis may not work because the study effects are a fixed feature of the respective metaanalysis and the probability distribution is only a descriptive tool.
Applications
Modern statistical metaanalysis does more than just combine the effect sizes of a set of studies using a weighted average. It can test if the outcomes of studies show more variation than the variation that is expected because of the sampling of different numbers of research participants.
Additionally, study characteristics such as measurement instrument used, population sampled, or aspects of the studies’ design can be coded and used to reduce variance of the estimator. Thus some methodological weaknesses in studies can be corrected statistically. Other uses of metaanalytic methods include the development of clinical prediction models, where metaanalysis may be used to combine data from different research centers, or even to aggregate existing prediction models.
Metaanalysis can be done with singlesubject design as well as group research designs. This is important because much research has been done with singlesubject research designs. Considerable dispute exists for the most appropriate metaanalytic technique for single subject research.
Metaanalysis leads to a shift of emphasis from single studies to multiple studies. It emphasizes the practical importance of the effect size instead of the statistical significance of individual studies. This shift in thinking has been termed “metaanalytic thinking”. The results of a metaanalysis are often shown in a forest plot.
Results from studies are combined using different approaches. One approach frequently used in metaanalysis in health care research is termed ‘inverse variance method’. The average effect size across all studies is computed as a weighted mean, whereby the weights are equal to the inverse variance of each studies’ effect estimator. Larger studies and studies with less random variation are given greater weight than smaller studies. Other common approaches include the Mantel–Haenszel method.
Signed differential mapping is a statistical technique for metaanalyzing studies on differences in brain activity or structure which used neuroimaging techniques such as fMRI, VBM or PET.
Different high throughput techniques such as microarrays have been used to understand Gene expression. MicroRNA expression profiles have been used to identify differentially expressed microRNAs in particular cell or tissue type or disease conditions or to check the effect of a treatment. A metaanalysis of such expression profiles was performed to derive novel conclusions and to validate the known findings.
Hunter, John E; Schmidt, Frank L (1990)
Methods of MetaAnalysis: Correcting Error and Bias in Research Findings
SAGE Publications
Light & Pillemer (1984)
Summing up: The science of reviewing research
Harvard University Press