When you pool results across studies in a meta-analysis, you will almost certainly face a common problem: different studies report different effect sizes. One trial reports an odds ratio (OR), another reports a standardized mean difference (SMD), and a third gives you a Cohen's d. Before you can run a pooled analysis, everything must be on the same scale.
This guide walks you through every common conversion formula, explains when each effect size is appropriate, provides worked examples with real numbers, and covers practical issues like missing data, software implementation in R, and reporting requirements for journals and reviewers.
Try our free online effect size tool to run these conversions without manual computation.
Choosing the Right Effect Size for Your Data
Cohen's d is the most familiar standardized mean difference. It divides the difference between two means by a pooled standard deviation. It works well when sample sizes are reasonably large (typically above 20 per group). Cohen (1988) introduced the metric along with the widely cited benchmarks of 0.2, 0.5, and 0.8 for small, medium, and large effects in behavioral research.
Hedges' g is a corrected version of Cohen's d. When sample sizes are small, Cohen's d systematically overestimates the true effect size. Hedges (1981) introduced a correction factor (sometimes called J) that removes this small-sample bias. In most meta-analyses, Hedges' g is the preferred choice, and it is the default in the metafor R package developed by Viechtbauer (2010).
The odds ratio is used for binary outcomes. It expresses how much more likely the event is in one group compared to another. The Cochrane Handbook recommends the odds ratio for meta-analyses of clinical trials with dichotomous outcomes because it has desirable mathematical properties, including symmetry around the null.
The risk ratio (RR), also called the relative risk, is also for binary outcomes but is more intuitive for clinical interpretation. Clinicians often prefer risk ratios in treatment guidelines because they communicate the magnitude of benefit more directly than odds ratios.
The Pearson correlation coefficient (r) is a standardized measure of linear association between two continuous variables. It ranges from -1 to +1 and is commonly reported in observational studies and psychometric research.
The number needed to treat (NNT) is derived from the absolute risk difference. You can use our free NNT Calculator to move between risk difference and NNT.
When to Use Each Effect Size Metric: A Decision Flowchart
Choosing the right metric depends on three factors: the type of outcome data, the design of included studies, and the intended audience of your findings.
Binary outcomes (yes/no events). If all studies report event counts, use the odds ratio or risk ratio directly. The odds ratio is preferred when baseline event rates vary substantially across studies. The risk ratio is preferred when your audience is clinicians who need intuitive interpretation. If baseline risk is below 10%, the two metrics produce nearly identical results.
Continuous outcomes. If all studies report means and standard deviations for two groups, compute Hedges' g (corrected standardized mean difference). Use Cohen's d only if you have a specific reason and all sample sizes exceed 20 per group.
Correlational data. If studies report associations between two continuous variables, use Pearson r or convert to Fisher's z for pooling. Fisher's z transformation stabilizes the variance when correlations are far from zero.
Mixed outcome types. This is the most complex scenario. When some studies report binary outcomes (odds ratios) and others report continuous outcomes (mean differences), you must convert everything to a single metric. The standard approach, described by Borenstein et al. (2009), is to convert odds ratios to Cohen's d (or vice versa) using the formulas in this guide. Then apply the Hedges' g correction if needed.
Why Hedges' g Corrects Small-Sample Bias
Cohen's d is calculated as: d = (M1 - M2) / SD_pooled
Hedges' g applies a multiplicative correction factor J: g = d * J
Where J is approximated as: J = 1 - (3 / (4*df - 1)) and df = n1 + n2 - 2.
For large samples (n > 20 per group), J is very close to 1. For small samples (n = 5 per group), J can be around 0.88, meaning d overestimates the true effect by about 12%.
Worked example with real numbers. Suppose a study with n1 = 8 and n2 = 10 reports M1 = 24.3, M2 = 20.1, SD_pooled = 5.6. Cohen's d = (24.3 - 20.1) / 5.6 = 0.750. The degrees of freedom are 8 + 10 - 2 = 16. J = 1 - (3 / (4*16 - 1)) = 1 - (3/63) = 0.952. Hedges' g = 0.750 * 0.952 = 0.714. The correction shaved off about 5% of the effect, which matters when you pool many small studies.
Rule of thumb: Use Hedges' g by default in meta-analyses. The Cochrane Handbook and the PRISMA 2020 reporting guidelines both expect standardized mean differences to be bias-corrected.
Converting Between Effect Sizes: The Formulas
Cohen's d to Hedges' g
g = d * [1 - (3 / (4*(n1+n2-2) - 1))]
Worked example: d = 0.62, n1 = 15, n2 = 18. df = 31. J = 1 - (3 / (4*31 - 1)) = 1 - (3/123) = 0.976. g = 0.62 * 0.976 = 0.605.
Odds Ratio to Cohen's d
The standard formula from Hasselblad and Hedges (1995), widely endorsed by Borenstein et al. (2009): d = ln(OR) * (sqrt(3) / pi), which simplifies to d = ln(OR) * 0.5513
Worked example: An OR of 2.5 gives: ln(2.5) = 0.916, d = 0.916 * 0.5513 = 0.505. So OR = 2.5 corresponds to approximately d = 0.50, a medium effect.
The variance of d converted from an OR is: Var(d) = Var(ln(OR)) * (3 / pi^2). You need this variance for the inverse-variance weighting in your meta-analysis.
Cohen's d to Odds Ratio
OR = exp(d * pi / sqrt(3)) = exp(d * 1.8138)
Worked example: d = 0.40, OR = exp(0.40 * 1.8138) = exp(0.726) = 2.066. A "small-to-medium" standardized mean difference corresponds to roughly doubling the odds.
Risk Ratio to Odds Ratio
When you have a risk ratio and need an OR, you need the baseline risk (p0): OR = RR * (1 - p0) / (1 - RR * p0)
When baseline risk is low (below 10%), OR and RR are approximately equal.
Worked example: RR = 1.8, baseline risk p0 = 0.30. OR = 1.8 * (1 - 0.30) / (1 - 1.8 * 0.30) = 1.8 * 0.70 / (1 - 0.54) = 1.26 / 0.46 = 2.739. Notice how the OR (2.74) is substantially larger than the RR (1.80) when baseline risk is 30%. This is why conflating the two metrics leads to overestimation.
Log Odds Ratio Conversions
Many meta-analysis software packages work on the log odds ratio (ln(OR)) scale because it has better statistical properties: the sampling distribution is approximately normal, and the confidence interval is symmetric.
OR to log OR: ln(OR) = natural logarithm of the odds ratio. Example: OR = 3.2, ln(OR) = 1.163.
Log OR to OR: OR = exp(ln(OR)). Example: ln(OR) = -0.45, OR = exp(-0.45) = 0.638.
Log OR to Cohen's d: d = ln(OR) * 0.5513. This is identical to the OR-to-d formula because d = ln(OR) * (sqrt(3)/pi).
Standard error of log OR: SE(ln(OR)) = sqrt(1/a + 1/b + 1/c + 1/d), where a, b, c, d are the cells of the 2x2 table.
Pearson Correlation (r) to Cohen's d
The correlation coefficient r is common in observational studies and psychometric research. Converting between r and d is essential when your meta-analysis includes both experimental and correlational studies.
r to d: d = 2r / sqrt(1 - r^2)
Worked example: r = 0.35. d = 2(0.35) / sqrt(1 - 0.1225) = 0.70 / sqrt(0.8775) = 0.70 / 0.937 = 0.747.
d to r: r = d / sqrt(d^2 + 4)
Worked example: d = 0.50. r = 0.50 / sqrt(0.25 + 4) = 0.50 / sqrt(4.25) = 0.50 / 2.062 = 0.243.
Use our free Correlation and Effect Size Converter to run these calculations instantly.
Fisher's z transformation. When pooling correlations in a meta-analysis, first transform each r to Fisher's z: z = 0.5 * ln((1+r)/(1-r)). Pool the z values, then back-transform the pooled z to r. This is necessary because the variance of r depends on the population correlation, while the variance of Fisher's z is approximately 1/(n-3) regardless of the correlation magnitude.
