Meta-analysis

In statistics, a meta-analysis refers to methods focused on contrasting and combining results from different studies, in the hope of identifying 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. In its simplest form, this is normally by identification of a common measure of effect size, of which a weighted average might be the output of a meta-analysis. The weighting might be related to sample sizes within the individual studies. More generally there are other differences between the studies that need to be allowed for, but the general aim of a meta-analysis is to more powerfully estimate the true effect size as opposed to a less precise effect size derived in a single study under a given single set of assumptions and conditions.

Meta-analyses are often, but not always, important components of a systematic review procedure. For instance, a meta-analysis 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 "meta-analysis" 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.

History

Early usage of Meta-analysis dates back to the twelfth century in the ancient China, a famous philospher, Chu Hsi (朱熹, 1130~1200), built up his philosophical theory by summarizing a series of related literatures. He called this research methodology 'Theory of Systematic Rule'(道統論) (See reference http://ir.lib.ntnu.edu.tw/retrieve/52215/). While in the Western World, the historical roots of meta-analysis may be traced back to 17th century studies of astronomy, a paper published in 1904 by the statistician Karl Pearson in the British Medical Journal which collated data from several studies of typhoid inoculation is seen as the first time a meta-analytic approach was used to aggregate the outcomes of multiple clinical studies. The first meta-analysis of all conceptually identical experiments concerning a particular research issue, and conducted by independent researchers, has been identified as the 1940 book-length publication Extra-sensory perception after sixty years, authored by Duke University psychologists J. G. Pratt, J. B. Rhine, and associates. This encompassed a review of 145 reports on ESP experiments published from 1882 to 1939, and included an estimate of the influence of unpublished papers on the overall effect (the file-drawer problem). Although meta-analysis is widely used in epidemiology and evidence-based medicine today, a meta-analysis of a medical treatment was not published until 1955. In the 1970s, more sophisticated analytical techniques were introduced in educational research, starting with the work of Gene V. Glass, Frank L. Schmidt and John E. Hunter.

The term "meta-analysis" was coined by Gene V. Glass, who was the first modern statistician to formalize the use the term meta-analysis. He states "my major interest currently is in what we have come to call ...the meta-analysis of research. The term is a bit grand, but it is precise and apt ... Meta-analysis refers to the analysis of analyses". Although this led to him being widely recognized as the modern founder of the method, the methodology behind what he termed "meta-analysis" predates his work by several decades. The statistical theory surrounding meta-analysis was greatly advanced by the work of Nambury S. Raju, Larry V. Hedges, Harris Cooper, Ingram Olkin, John E. Hunter, Jacob Cohen, Thomas C. Chalmers, Robert Rosenthal and Frank L. Schmidt.

Advantages of meta-analysis

The advantages of meta-analysis (e.g. over classical literature reviews, simple overall means of effect sizes etc.) are that it:

• Shows whether the results are more varied than what is expected from the sample diversity,
• Allows derivation and statistical testing of overall factors and effect-size parameters in related studies,
• Is a generalization to the population of studies,
• Is able to control for between-study variation,
• Includes moderators to explain variation,
• Has higher statistical power to detect an effect than individual studies,
• Deals with information overload: the high number of articles published each year,
• Combines several studies and will therefore be less influenced by local biases than single studies will be, and
• Makes it possible to show whether a publication bias exists.

Pitfalls

A meta-analysis of several small studies does not predict the results of a single large study, especially in a field like medicine where results are truly unpredictable. Some have argued that a weakness of the method is that sources of bias are not controlled by the method: a good meta-analysis of badly designed studies will still result in bad statistics. This would mean that only methodologically sound studies should be included in a meta-analysis, a practice called 'best evidence synthesis'. Other meta-analysts would include weaker studies, and add a study-level 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

A funnelplot expected without the file drawer problem

A funnelplot expected with the file drawer problem

Another potential pitfall is the reliance on the available corpus of published studies, which may create exaggerated outcomes due to publication bias, as it is far harder to publish studies which show negative results. For any given research area, one cannot know how many studies have been conducted but never reported and the results filed away.

This file drawer problem results in the distribution of effect sizes that are biased, skewed or completely cut off, creating a serious base rate fallacy, in which the significance of the published studies is overestimated. For example, if there were fifty tests, and only ten got results, then the real outcome is only 20% as significant as it appears, except that the other 80% were not submitted for publishing, or thrown out by publishers as uninteresting. This should be seriously considered when interpreting the outcomes of a meta-analysis.

This can be visualized with a funnel plot which is a scatter plot of sample size and effect sizes. If no publication bias is present, one would expect that there is no relation between sample size and effect size. A negative relation between sample size and effect size would imply that studies that found significant effects were more likely to be published and/or to be submitted for publication. There are several procedures available that attempt to correct for the file drawer problem, once identified, such as guessing at the cut off part of the distribution of study effects.

Methods for detecting publication bias have been controversial as they typically have low power for detection of bias, but also may create false positives under some circumstances. For instance small study effects, wherein methodological differences between smaller and larger studies exist, may cause differences in effect sizes between studies that resemble publication bias. However, small study effects may be just as problematic for the interpretation of meta-analyses, and the imperative is on meta-analytic authors to investigate potential sources of bias. A Tandem Method for analyzing publication bias has been suggested for cutting down false positive error problems, and suggesting that 25% of meta-analyses in the psychological sciences may have publication bias. However low power problems likely remain at issue, and estimations of publication bias may remain lower than the true amount.

Most discussions of publication bias focus on journal practices in which publication rates of statistically significant finds are higher than for non-significant findings. However questionable researcher practices, such as reworking statistical models until significance is achieved may also promote a bias toward statistically significant findings allowing high bias for researchers to confirm their own beliefs. Given that, unlike journal practices, questionable researcher practices aren't necessarily sample size dependent, and thus unlikely to demonstrate on the funnel plot and thus go undetected by most publication bias detection methods currently in use.

Other weaknesses are Simpson's paradox (two smaller studies may point in one direction, and the combination study in the opposite direction); the coding of an effect is subjective; the decision to include or reject a particular study is subjective; there are two different ways to measure effect: correlation or standardized mean difference; the interpretation of effect size is purely arbitrary; it has not been determined if the statistically most accurate method for combining results is the fixed, random or quality effect models; and, for medicine, the underlying risk in each studied group is of significant importance, and there is no universally agreed-upon way to weight the risk.

Dangers of agenda-driven bias

The most severe weakness and abuse of meta-analysis often occurs when the person or persons doing the meta-analysis have an economic, social, or political agenda such as the passage or defeat of legislation. Those persons with these types of agenda have a high likelihood to abuse meta-analysis due to personal bias. For example, researchers favorable to the author's agenda are likely to have their studies cherry picked while those not favorable will be ignored or labeled as "not credible". In addition, the favored authors may themselves be biased or paid to produce results that support their overall political, social, or economic goals in ways such as selecting small favorable data sets and not incorporating larger unfavorable data sets. The influence of such biases on the results of a meta-analysis is possible because the methodology of meta-analysis is highly malleable.

A 2011 study done to disclose possible conflicts of interests in underlying research studies used for medical meta-analyses reviewed 29 meta-analyses and found that conflicts of interests in the studies underlying the meta-analyses were rarely disclosed. The 29 meta-analyses included 11 from general medicine journals; 15 from specialty medicine journals, and three from the Cochrane Database of Systematic Reviews. The 29 meta-analyses reviewed an aggregate of 509 randomized controlled trials (RCTs). Of these, 318 RCTs reported funding sources with 219 (69%) industry funded. Of the 509 RCTs, 132 reported author conflict of interest disclosures, with 91 studies (69%) disclosing industry financial ties with one or more authors. The information was, however, seldom reflected in the meta-analyses. Only two (7%) reported RCT funding sources and none reported RCT author-industry ties. The authors concluded “without acknowledgment of COI due to industry funding or author industry financial ties from RCTs included in meta-analyses, readers’ understanding and appraisal of the evidence from the meta-analysis may be compromised.”

Steps in a meta-analysis

1. Formulation of the problem

2. Search of literature

3. Selection of studies ('incorporation criteria')

• Based on quality criteria, e.g. the requirement of randomization and blinding in a clinical trial
• Selection of specific studies on a well-specified subject, e.g. the treatment of breast cancer.
• Decide whether unpublished studies are included to avoid publication bias (file drawer problem)

4. Decide which dependent variables or summary measures are allowed. For instance:

• Differences (discrete data)
• Means (continuous data)
• Hedges' g is a popular summary measure for continuous data that is standardized in order to eliminate scale differences, but it incorporates an index of variation between groups:

$\delta ={\frac {\mu _{t}-\mu _{c}}{\sigma }},$ in which $\mu _{t}$ is the treatment mean, $\mu _{c}$ is the control mean, $\sigma ^{2}$ the pooled variance.

5. Model selection (see next paragraph)

For reporting guidelines, see the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) statement

Meta-analysis models

1. Fixed effect model

The fixed effect model provides a weighted average of the study estimates, the weights being the inverse of the variance of the study estimate. Thus larger studies get larger weights than smaller studies and if the studies within the meta-analysis are dominated by a very large study, it receives essentially all the weight and smaller studies are ignored. This is not so bad if study effect sizes differ only by sampling error, but once heterogeneity is present, then this must be accounted for by the model and one of the other models below should be utilized

2. Random effects model

A common model used to synthesize heterogenous research is the random effects model of meta-analysis. This is simply the weighted average of the effect sizes of a group of studies. The weight that is applied in this process of weighted averaging with a random effects meta-analysis is achieved in two steps:

1. Step 1: inverse variance weighting
2. Step 2: Un-weighting of this inverse variance weighting by applying a random effects variance component (REVC) that is simply derived from the extent of variability of the effect sizes of the underlying studies.

This means that the greater this variability in effect sizes (otherwise known as heterogeneity), the greater the un-weighting and this can reach a point when the random effects meta-analysis result becomes simply the un-weighted average effect size across the studies. At the other extreme, when all effect sizes are similar (or variability does not exceed sampling error), no REVC is applied and the random effects meta-analysis defaults to simply a fixed effect meta-analysis (only inverse variance weighting).

The extent of this reversal is solely dependent on two factors:

1. Heterogeneity of precision
2. Heterogeneity of effect size

Since there is no reason to automatically assume that a larger variability in study sizes or effect sizes automatically indicates a faulty larger study or more reliable smaller studies, the re-distribution of weights under this model bears no relationship to what these studies have to offer. Indeed, there is no reason why the results of a meta-analysis should be associated with this method of reversal of the inverse variance weighting process of the included studies. As such, the changes in weight introduced by this model (to each study) results in a pooled estimate that can have no possible interpretation and, thus, bears no relationship with what the studies actually have to offer.

To compound the problem further, some statisticians are proposing that we take an estimate that has no meaning and compute a prediction interval around it. This is akin to taking a random guess at the effectiveness of a therapy and under the false belief that it is meaningful try to expand on its interpretation. Unfortunately, there is no statistical manipulation that can replace commonsense. While heterogeneity might be due to underlying true differences in study effects, it is more than likely that such differences are brought about by systematic error. The best we can do in terms of addressing heterogeneity is to look up the list of studies and attempt to un-weight (from inverse variance) based on differences in evidence of bias rather than effect size or precision that are consequences of these failures.

The most widely used method to estimate and account for heterogeneity is the DerSimonian-Laird (DL) approach. More recently the iterative and computationally intensive restricted maximum likelihood (REML) approach emerged and is catching up. However, a comparison between these two (and more) models demonstrated that there is little to gain and DL is quite adequate in most scenarios.

3. Quality effects model

Some researchers introduce a new approach to adjustment for inter-study variability by incorporating a relevant component (quality) that differs between studies in addition to the weight based on the intra-study differences that is used in any fixed effects meta-analysis model. The strength of the quality effects meta-analysis is that it allows available methodological evidence to be used over subjective random probability, and thereby helps to close the damaging gap which has opened up between methodology and statistics in clinical research. To do this a correction for the quality adjusted weight of the ith study called taui is introduced. This is a composite based on the quality of other studies except the study under consideration and is utilized to re-distribute quality adjusted weights based on the quality adjusted weights of other studies. In other words, if study i is of good quality and other studies are of poor quality, a proportion of their quality adjusted weights is mathematically redistributed to study i giving it more weight towards the overall effect size. As studies increase in quality, re-distribution becomes progressively less and ceases when all studies are of perfect quality. This model thus replaces the untenable interpretations that abound in the literature and a software is available to explore this method further

Meta-Regression

Main article: Meta-regression

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 regression techniques.

Applications in modern science

Modern statistical meta-analysis does more than just combine the effect sizes of a set of studies. It can test if the outcomes of studies show more variation than the variation that is expected because of sampling different research participants. If that is the case, study characteristics such as measurement instrument used, population sampled, or aspects of the studies' design are coded. These characteristics are then used as predictor variables to analyze the excess variation in the effect sizes. Some methodological weaknesses in studies can be corrected statistically. For example, it is possible to correct effect sizes or correlations for the downward bias due to measurement error or restriction on score ranges.

Meta-analysis can be done with single-subject design as well as group research designs. This is important because much of the research on low incidents populations has been done with single-subject research designs. Considerable dispute exists for the most appropriate meta-analytic technique for single subject research.

Meta-analysis 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 "meta-analytic thinking". The results of a meta-analysis are often shown in a forest plot.

Results from studies are combined using different approaches. One approach frequently used in meta-analysis 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 and the Peto method.

A recent approach to studying the influence that weighting schemes can have on results has been proposed through the construct of gravity, which is a special case of combinatorial meta-analysis.

Signed differential mapping is a statistical technique for meta-analyzing 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 meta-analysis of such expression profiles was performed to derive novel conclusions and to validate the known findings.

See also

• Newcastle–Ottawa scale
• Reporting bias
• Review journal
• Secondary research
• Study heterogeneity
• Systematic review

References

• Cooper, H. & Hedges, L.V. (1994). The Handbook of Research Synthesis. New York: Russell Sage.
• Cornell, J. E. & Mulrow, C. D. (1999). Meta-analysis. In: H. J. Adèr & G. J. Mellenbergh (Eds). Research Methodology in the social, behavioral and life sciences (pp. 285–323). London: Sage.
• Norman S.-L. T. (1999). "Tutorial in Biostatistics. Meta-Analysis: Formulating, Evaluating, Combining, and Reporting". Statistics in Medicine. 18 (3): 321–359. doi:10.1002/(SICI)1097-0258(19990215)18:3<321::AID-SIM28>3.0.CO;2-P. PMID 10070677.
• Sutton, A.J., Jones, D.R., Abrams, K.R., Sheldon, T.A., & Song, F. (2000). Methods for Meta-analysis in Medical Research. London: John Wiley. ISBN 0-471-49066-0
• Higgins JPT, Green S (editors). Cochrane Handbook for Systematic Reviews of Interventions Version 5.0.1 [updated September 2008]. The Cochrane Collaboration, 2008. Available from www.cochrane-handbook.org

Further reading

• Thompson, Simon G; Pocock, Stuart J (2 November 1991). "Can meta-analysis be trusted?" (PDF). The Lancet. 338 (8775): 1127–1130. doi:10.1016/0140-6736(91)91975-Z. PMID 1682553. Retrieved 17 June 2011.. Explores two contrasting views: does meta-analysis provide "objective, quantitative methods for combining evidence from separate but similar studies" or merely "statistical tricks which make unjustified assumptions in producing oversimplified generalisations out of a complex of disparate studies"?
• Wilson, D. B., & Lipsey, M. W. (2001). Practical meta-analysis. Thousand Oaks: Sage publications. ISBN 0-7619-2168-0
• O'Rourke, K. (2007) Just the history from the combining of information: investigating and synthesizing what is possibly common in clinical observations or studies via likelihood. Oxford: University of Oxford, Department of Statistics. Gives technical background material and details on the "An historical perspective on meta-analysis" paper cited in the references.
• Owen, A. B. (2009). "Karl Pearson's meta-analysis revisited". Annals of Statistics, 37 (6B), 3867–3892. Supplementary report.
• Ellis, Paul D. (2010). The Essential Guide to Effect Sizes: An Introduction to Statistical Power, Meta-Analysis and the Interpretation of Research Results. United Kingdom: Cambridge University Press. ISBN 0-521-14246-6
• Bonett, D.G. (2012). Replication-extension studies, Current Directions in Psychology, 21, 409-412.
• Bonett, D.G. (2010). Varying coefficient meta-analysis methods for alpha reliability, Psychological Methods, 15, 368–385.
• Bonett, D.G. (2009). Meta-analytic interval estimation for standardized and unstandardized mean differences, Psychological Methods, 14, 225–238.
• Bonett, D.G. (2008). Meta-analytic interval estimation for bivariate correlations, Psychological Methods, 13, 173–189.
• Stegenga, Jacob (2011). "Is meta-analysis the platinum standard of evidence?". Studies in History and Philosophy of Biological and Biomedical Sciences. 42 (4): 497–507. doi:10.1016/j.shpsc.2011.07.003.

External links

• Cochrane Handbook for Systematic Reviews of Interventions
• Effect Size and Meta-Analysis (ERIC Digest)
• Meta-Analysis at 25 (Gene V Glass)
• Meta-Analysis in Educational Research (ERIC Digest)
• Meta-Analysis: Methods of Accumulating Results Across Research Domains (article by Larry Lyons)
• Meta-analysis (Psychwiki.com article)
• EffectSizeFAQ.com
• Meta-Analysis in Economics (Reading list)
• Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) Statement, "an evidence-based minimum set of items for reporting in systematic reviews and meta-analyses."

Software

• MetaXL software page
• Effect Size Calculators Calculate d and r from a variety of statistics.
• ClinTools (commercial)
• Comprehensive Meta-Analysis (commercial)
• MIX 2.0 Professional Excel addin with Ribbon interface for meta-analysis and effect size conversions in Excel (free and commercial versions).
• What meta-analysis features are available in Stata? (free add-ons to commercial package)
• The Meta-Analysis Calculator free on-line tool for conducting a meta-analysis
• Metastat (Free)
• Meta-Analyst Free Windows-based tool for Meta-Analysis of binary, continuous and diagnostic data
• Revman A free software for meta-analysis and preparation of cochrane protocols and review available from the Cochrane Collaboration
• Metafor-project A free software package to conduct meta-analyses in R.
• Calculation of fixed and random effects in R source code for performing univariate and multivariate meta-analyses in R, and for calculating several statistics of heterogeneity.
• Macros in SPSS Free Macros to conduct meta-analyses in SPSS.
• compute.es: Compute Effect Sizes (R package).
• MAd GUI User friendly graphical user interface package to conduct meta-analysis in R (Free).