Effect Sizes in Meta-Analysis

Putting studies on a common scale

Meta-analysis is an evidence-synthesis method that statistically combines findings from multiple independent studies. This pooling is only possible once each study's finding is expressed as a common effect-size metric. For continuous outcomes the standardized mean difference is used; for binary outcomes the odds ratio or risk ratio; and for association questions the correlation coefficient. Selecting the correct metric and converting all studies to it consistently is the essential prerequisite for a valid meta-analytic synthesis.

What Is an Effect Size?

An effect size is a standardized measure that expresses the practical and statistical magnitude of an intervention or association in a single number. Unlike scale-dependent values such as raw mean differences or frequencies, effect sizes are unit-free, which allows studies conducted with different instruments or across different populations to be compared and combined. In meta-analysis the effect size serves as the primary input to the synthesis rather than the raw data. Although Cohen proposed general reference points for small, medium, and large effects, these thresholds should not be applied uniformly across all research domains.

Core Metrics: Selection and Computation

For continuous outcomes the most widely used metric is the standardized mean difference. Cohen d divides the mean difference between two groups by the pooled standard deviation. Hedges g, which applies a correction factor for small samples, is generally preferred in meta-analysis. For binary outcomes the odds ratio or risk ratio is used; both are log-transformed before analysis and exponentiated back to the original scale for reporting. When an association between two variables is the focus, Pearson r is the metric of choice, but it must be stabilized via Fisher z transformation before averaging. For all metrics, computing the standard error or confidence interval is mandatory.

A Concrete Example: Pooling an Intervention Effect

A researcher wants to synthesize ten studies examining the effect of cognitive behavioural therapy on depressive symptoms. Although each study used a different symptom scale, all outcomes are continuous, so Hedges g can be computed. For each study the mean, standard deviation, and sample size are obtained, and then g and its corresponding standard error are calculated. A weighted average effect size is then derived through a fixed-effect or random-effects model. The resulting pooled g expresses how large an effect the therapy produces on average in standard units, and it can be interpreted without reference to any of the original scales.

Common Pitfalls and Good Practice Principles

One of the most frequent mistakes is attempting to pool different metric types without conversion; an odds ratio and a standardized mean difference cannot be placed in the same analysis directly. Another error is averaging Pearson r values without first applying Fisher z transformation, which leads to underestimation of correlations. Incorrect model selection caused by ignoring study heterogeneity is also common. Good practice requires following PRISMA guidelines, reporting effect-size conversion formulas with their sources, presenting confidence intervals and weights for each effect size, and interpreting heterogeneity statistics such as I-squared and tau-squared rather than ignoring them.

Key terms

Standardized Mean Difference (Cohen d / Hedges g): Metric for continuous outcomes dividing the mean difference by the pooled standard deviation.
Odds Ratio: Effect-size measure for binary outcomes comparing the ratio of odds between two groups.
Fisher z Transformation: Mathematical procedure converting Pearson r values to a normally distributed scale suitable for pooling.
Heterogeneity (I-squared): Statistic indicating what proportion of variance across studies exceeds chance-level sampling error.
Random-Effects Model: Meta-analytic model that assumes true effect sizes vary across studies beyond sampling error.