Convert between Cohen’s d, r, odds ratios and η²
The studies in your review report their effects in different currencies. Convert them onto one scale so your synthesis compares like with like — and so you can set a realistic effect for your own power calculation.
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Why your literature review needs one effect-size currency
Reading across a field, you will find the same underlying relationship reported as a standardised mean difference in one paper, a correlation in another, and an odds ratio in a third. Written up side by side without conversion, they cannot be compared, and a synthesis that lists them uncritically is doing description rather than analysis.
For a two-group comparison these metrics are transformations of one another, so the conversion is straightforward. Cohen’s d acts as the hub:
- d ↔ r — r = d / √(d² + 4), and d = 2r / √(1 − r²).
- d ↔ OR — ln(OR) = d × π/√3, the Cox/Chinn logit method.
- r ↔ η² — for a two-group comparison, η² = r².
There is a second, more immediate use. Every power calculation needs an assumed effect size, and the most defensible source is prior work in your own field. If those papers report correlations and you are planning a group comparison, this is how you get from their number to yours — and it gives your proposal a citation to point at.
The conversions assume roughly equal group sizes, and for odds ratios a logistic latent variable. They are accurate enough for synthesis and reporting; when you hold the raw data, compute the effect size directly instead.
Frequently asked questions
Why would I need to convert effect sizes at all?
Two reasons, both common in doctoral work. First, synthesis: to say anything comparative about the studies in your review, their effects have to be on one scale. Second, planning: your power calculation needs an assumed effect, and the strongest justification is a published effect from your own field — which often arrives in the wrong metric.
How do I convert Cohen's d to a correlation?
For two roughly equal groups, r = d / √(d² + 4). Reverse it with d = 2r / √(1 − r²). So d = 0.5 corresponds to r ≈ 0.24.
How do I convert an odds ratio to Cohen's d?
d = ln(OR) × √3/π, which is approximately ln(OR) × 0.5513. Going the other way, OR = exp(d × π/√3), roughly exp(d × 1.814). This is the Cox logit method and it assumes an underlying continuous variable dichotomised into the two outcome categories.
Which effect size should I report in my own thesis?
Whichever your field expects, and always alongside a confidence interval. Cohen's d is conventional for group comparisons in psychology and education; odds ratios dominate epidemiology and clinical research; correlations suit relationship questions. If your discipline has a reporting standard, follow it — and report the raw difference too, because a standardised effect on its own hides whether the finding matters practically.
How exact are these conversions?
They are standard approximations, not identities. Equal group sizes are assumed, and the odds-ratio conversion additionally assumes a logistic latent variable. That is fine for meta-analysis and for narrative synthesis. It is not a substitute for computing the effect size from raw data when you have it.
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