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 the most common conversion formulas, explains when each effect size is appropriate, and shows worked examples so you can apply these conversions immediately.
Try our free Effect Size Calculator 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).
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' g applies a correction factor (sometimes called J) that removes this small-sample bias. In most meta-analyses, Hedges' g is the preferred choice.
The odds ratio is used for binary outcomes. It expresses how much more likely the event is in one group compared to another.
The risk ratio (RR), also called the relative risk, is also for binary outcomes but is more intuitive.
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.
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%.
Rule of thumb: Use Hedges' g by default in meta-analyses.
Converting Between Effect Sizes: The Formulas
Cohen's d to Hedges' g
g = d * [1 - (3 / (4*(n1+n2-2) - 1))]
Odds Ratio to Cohen's d
The standard formula (Hasselblad and Hedges, 1995): 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.
Cohen's d to Odds Ratio
OR = exp(d * pi / sqrt(3)) = exp(d * 1.8138)
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.
Point-Biserial Correlation r to Cohen's d
d = 2r / sqrt(1 - r^2) and the reverse: r = d / sqrt(d^2 + 4)
Benchmarks and Interpretation
Cohen's (1988) benchmarks: Small (d = 0.2), Medium (d = 0.5), Large (d = 0.8). These were intended as rough guides for psychology research, not universal standards. Always interpret effect sizes in context.
For visual pooling and forest plot generation, try our free Forest Plot Generator. For planning future studies, use the Sample Size Calculator.
Common Mistakes When Converting Effect Sizes
Using OR as if it were RR. When baseline risk is above 20%, treating an OR as an RR leads to substantial overestimation.
Ignoring direction. An OR below 1 means the event is less likely in the treatment group. A negative log produces a negative d.
Forgetting sample sizes for the J correction. If you convert d to g without the correct sample sizes, your correction will be wrong.
Mixing continuous and binary outcomes. Converting OR to d is appropriate when the underlying construct is continuous and the binary outcome was created by dichotomizing a latent variable.
Key Takeaways
- Use Hedges' g instead of Cohen's d in meta-analyses, especially when any included studies have small samples.
- Convert odds ratios to Cohen's d using the formula: d = ln(OR) * 0.5513.
- When converting between OR and RR, you must know the baseline event rate in the control group.
- Cohen's benchmarks (0.2, 0.5, 0.8) are starting points, not universal rules. Interpret in context.
- Always track the direction (sign) of effects when converting.
- Use software or online calculators to reduce arithmetic errors in conversion chains.
FAQ
When should I use Hedges' g instead of Cohen's d?
Use Hedges' g whenever any of your included studies have small samples (fewer than 20 per group). For large samples the two values are nearly identical, so defaulting to Hedges' g costs you nothing and protects against bias.
Can I convert an odds ratio to a mean difference?
Yes, using d = ln(OR) * 0.5513. This conversion assumes the binary outcome was created by dichotomizing an underlying continuous, normally distributed latent variable.
What is the difference between SMD and Cohen's d?
In most meta-analysis software, SMD refers to the same quantity as Cohen's d or Hedges' g. The software typically uses Hedges' g by default when you select SMD.
How do I convert a correlation coefficient to Cohen's d?
Use d = 2r / sqrt(1 - r^2). For example, r = 0.30 gives d = 0.60 / sqrt(0.91) = 0.629.
Why does the OR overestimate RR when baseline risk is high?
The odds ratio equals RR only when the outcome is rare (baseline risk below 10%). As baseline risk increases, the OR inflates relative to the RR.
Is it acceptable to pool different effect size types in one meta-analysis?
You should convert all effect sizes to a single metric before pooling. When possible, contact study authors to obtain the alternative effect size directly. If conversion is unavoidable, conduct a sensitivity analysis.
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