# What is Meta-Analysis?

By **
Megha Joshi
**

Scientific researchers tend to produce literature on the same topic either to replicate or extend prior studies or due to a lack of awareness of prior evidence (Hedges & Cooper, 2009). Results across studies tend to vary, even when researchers try to replicate studies, due to differences in sample characteristics, research designs, analytic strategies or sampling error (Hedges & Cooper, 2009).

Meta-analysis is a set of statistical techniques for synthesizing results from multiple primary studies on a common topic. Meta-analysis can be used to synthesize effect estimates from randomized or quasi-experimental studies, and correlations between variables from descriptive studies.

The three major goals of meta-analysis include:

summarizing effect size estimates across studies

characterizing variability in effect sizes across studies

explaining the variability in the effect sizes

The three goals of meta-analysis are discussed below.

## 1. Summarizing effect sizes

The first major goal of meta-analysis is to summarize effect size estimates across multiple studies to estimate the average effect of an intervention or the average measure of the relationship between two variables (Konstantopoulos & Hedges, 2019). Let \(m\) denote the number of studies in a meta-analysis, with each study contributing one effect size estimate, \(T_i\) for \(i = 1,…,m\). Below, let \(w_i\) indicate some general weight. Effect size estimates can be pooled as follows (Konstantopoulos & Hedges, 2019):

\[\hat{\mu} = \frac{\sum_{i= 1}^{m} w_i T_i}{\sum_{i= 1}^{m} w_i}\]

One way to pool effect size estimates is by weighing them by the inverse of their variance estimates; these weights denote the precision of the estimated effect sizes (Viechtbauer, 2007). The calculation of the inverse variance weights depends on certain assumptions which are discussed below.

### Common and Fixed Effects Models

One assumption is the identical parameters assumption underlying the common effect model. This assumption states that one true effect size underlies all of the studies (Rice et al., 2018). An alternative assumption is the independent parameters assumption underlying the fixed effects model. This assumption treats the set of studies in a meta-analysis as all of the studies in the population of interest (Rice et al., 2018). When using the common effect and fixed effects models, the inverse variance weights can be calculated as \(w_i = 1/ \hat{\sigma_{i}}^2\), where \(\hat{\sigma}_i^2\) denotes the sampling error in the estimation of the effect sizes (Konstantopoulos & Hedges, 2019).

### Random Effects Model

Unlike the fixed effects model, the random effects model treats the set of studies in a meta-analysis as a sample of all possible studies in the population of interest (Higgins et al., 2009; Konstantopoulos & Hedges, 2019). The variance of the effect sizes between studies is denoted by \(\tau^2\) (Higgins et al., 2009; Konstantopoulos & Hedges, 2019). When using the random effects model, the inverse variance weights can be calculated as the inverse of the sum of the two variance components, \(w_i = 1/(\hat{\tau}^2 + \hat{\sigma_{i}}^2)\), where \(\hat\tau^2\) is some estimate of the between-study variance (Konstantopoulos & Hedges, 2019).

## 2. Characterizing Variation in Effect Sizes

A commonly used statistic to characterize the variation in effect sizes is the \(I^2\) statistics. However, Borenstein et al. (2017) argued that \(I^2\) values do not really capture the variation in effects between studies. For more discussion on the trickiness in interpreting \(I^2\) values, see my blog post here.

An absolute measure of heterogeneity that can be used instead to quantify between-study heterogeneity is \(\tau^2\). It is a descriptive statistic that denotes the estimated variation in the true effects or, as Viechtbauer (2007) described it, the estimated variance of the random variable producing the true effect sizes.

## 3. Explaining Variation in Effects

In addition to pooling effect size estimates and characterizing variability in effect sizes, meta-analysts often want to examine what factors explain or are associated with the variability. Identifying answers to such questions can clarify whether an intervention is effective for groups of interest, and whether the intervention should be developed further to be more effective for target populations under relevant conditions.

To explain variability in the effect sizes, a meta-regression model is generally used. Let \(T_i\) denote effect size estimate \(i\), \(p\) denote the number of regression parameters, \(x_{i1}, … ,x_{i,p-1}\) denote a set of moderator values associated with effect size estimate \(i\), \(\beta_0, …, \beta_{p-1}\) denote a vector of regression coefficients, and \(\epsilon_i\) denote the error term (Konstantopoulos & Hedges, 2019). The meta-regression model can be written as follows:

\[ {T}_i = \beta_0 + x_{i1} \beta_1 \, + \, , … , \, + \, x_{i,p-1}\beta_{p-1} + {\epsilon}_i \]

A random effects weighted least squares meta-regression model can be used to estimate the parameters in the equation above. An intercept-only model can be used to estimate the overall average effect size. A statistically significant test of a regression coefficient for a moderator would indicate that the effects of a treatment or intervention depend on the level of the moderator—that the moderator explains statistically significant variation in the effect sizes.

## References

Borenstein, M., Higgins, J. P., Hedges, L. V., & Rothstein, H. R.
(2017). Basics of meta-analysis: I2 is not an absolute measure of
heterogeneity. *Research Synthesis Methods, 8* (1), 5–18.

Hedges, L. V., & Cooper, H. M. (2009). Research synthesis as a
scientific process. *The handbook of research synthesis and
meta-analysis*.

Higgins, J. P., Thompson, S. G., & Spiegelhalter, D. J. (2009). A
re-evaluation of random-effects meta-analysis. *Journal of the Royal
Statistical Society: Series A (Statistics in Society), 172* (1),
137–159.

Konstantopoulos, S., & Hedges, L. V. (2019). Statistically analyzing
effect sizes: Fixed-and random-effects models. *The Handbook of Research
Synthesis and Meta-Analysis*, 245–279.

Rice, K., Higgins, J. P., & Lumley, T. (2018). A re-evaluation of fixed
effect (s) meta-analysis. *Journal of the Royal Statistical Society:
Series A (Statistics in Society)*, *181* (1), 205–227.

Viechtbauer, W. (2007). Accounting for heterogeneity via random-effects
models and moderator analyses in meta-analysis. *Zeitschrift für
Psychologie/Journal of Psychology*, *215* (2), 104–121.