# Meta-analysis

Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

## Overview

In statistics, a meta-analysis combines the results of several studies that address a set of related research hypotheses. The first meta-analysis was performed by Karl Pearson in 1904, in an attempt to overcome the problem of reduced statistical power in studies with small sample sizes; analyzing the results from a group of studies can allow more accurate data analysis.

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 online Oxford English Dictionary lists the first usage of the term in the statistical sense as 1976 by Glass. The statistical theory surrounding meta-analysis was greatly advanced by the work of Nambury S. Raju, Larry V. Hedges, Ingram Olkin, John E. Hunter, and Frank L. Schmidt.

## Uses in modern science

Because the results from different studies investigating different independent variables are measured on different scales, the dependent variable in a meta-analysis is some standardized measure of effect size. To describe the results of comparative experiments the usual effect size indicator is the standardized mean difference (d) which is the standard score equivalent to the difference between means, or an odds ratio if the outcome of the experiments is a dichotomous variable (success versus failure). A meta-analysis can be performed on studies that describe their findings in correlation coefficients, as for example, studies of the correlation between familial relationships and intelligence. In these cases, the correlation itself is the indicator of the effect size.

The method is not restricted to situations in which one or more variables is defined as "dependent." For example, a meta-analysis could be performed on a collection of studies each of which attempts to estimate the incidence of left-handedness in various groups of people.

Researchers should be aware that variations in sampling schemes can introduce heterogeneity to the result, which is the presence of more than one intercept in the solution. For instance, if some studies used 30mg of a drug, and others used 50mg, then we would plausibly expect two clusters to be present in the data, each varying around the mean of one dosage or the other. This can be modelled using a "random effects model."

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 free Excel-based calculator to perform Mantel Haenszel analysis is available at: http://www.pitt.edu/~super1/lecture/lec1171/014.htm.

They also have a free Excel-based Peto method calculator at: http://www.pitt.edu/~super1/lecture/lec1171/015.htm

Cochraine and other sources provide a useful discussion of the differences between these two approaches.

Question: Why not just add up all the results across studies ?

Note, however that Mantel Haenszel analysis and Peto analysis introduce their own biases and distortions of the data results.

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.

Modern meta-analysis does more than just combine the effect sizes of a set of studies. It can test if the studies' outcomes 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 leads to a shift of emphasis from single studies to multiple studies. It emphasises 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 meta-analysis are often shown in a forest plot.

## Trial quality

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. Robert Slavin has argued that only methodologically sound studies should be included in a meta-analysis, a practice he calls 'best evidence meta-analysis'. 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. Another weakness of the method is the heavy reliance on published studies, which may increase the effect as it is very hard to publish studies that show no significant results. This publication bias or "file-drawer effect" (where non-significant studies end up in the desk drawer instead of in the public domain) should be seriously considered when interpreting the outcomes of a meta-analysis. Because of the risk of publication bias, many meta-analyses now include a "failsafe N" statistic that calculates the number of studies with null results that would need to be added to the meta-analysis in order for an effect to no longer be reliable.

## Types of meta-analyses

### Network meta-analysis

A network meta-analysis[1] and Bayesian hierarchical models[2] pool studies in order to compare to treatments that have not been directly compared.[3][4] Network meta-analyses are commonly not well performed.[5] Network meta-analyses of both randomized controlled trials[6][7][8] and diagnostic test assessments[9] can have misleading results. Network meta-analyses have been conducted by the Cochrane Collaboration.[10][11]

Network meta-analyses can be conducted with Bugs and OpenBugs software.