Convert between Cohen's d, odds ratios, correlation r, risk ratios, hazard ratios, and eta-squared with proper SE propagation for meta-analysis pooling.
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Choose the metric your study reports from the dropdown menu. Options include Cohen's d, odds ratio, correlation r, risk ratio, hazard ratio, eta-squared, and phi coefficient. Each direction has specific input requirements.
Input the effect size value, its standard error (if available for SE propagation), and sample sizes for each group. Unequal group sizes affect the d-to-r correction factor, so entering actual Ns improves accuracy.
Choose the metric you need for your meta-analysis. The tool shows the mathematical formula being applied and highlights any assumptions underlying the transformation, such as logistic distribution for OR-to-d.
Examine the displayed formula, propagated standard error, and resulting confidence interval. Verify that the assumptions (equal groups, logistic distribution, known baseline risk) are reasonable for your data before proceeding.
Copy the converted effect size, its standard error, and confidence interval bounds. In batch mode, export all conversions simultaneously as a CSV file ready for import into meta-analysis software.
Send your converted values directly to the forest plot generator, funnel plot tool, or meta-regression formatter through the integrated pipeline. This ensures consistent values flow through your entire analysis workflow.
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Get a Free QuoteThis tool supports conversions between Cohen's d, odds ratio, correlation r, risk ratio, hazard ratio, eta-squared, and phi. Each direction uses the specific formula validated in the meta-analysis methodology literature.
The conversion from odds ratio to standardized mean difference uses d = ln(OR) multiplied by the square root of 3 divided by pi. This formula, established by Hasselblad and Hedges (1995), assumes the underlying continuous outcome follows a logistic distribution.
Every cross-type conversion assumes something about the underlying data. The OR-to-d formula assumes logistic distributions, d-to-r assumes bivariate normality, and RR-to-OR requires knowing the baseline event rate. Violations introduce small but systematic bias.
The standard r = d / sqrt(d squared + 4) formula assumes equal sample sizes in both groups. When groups are unequal, the correction factor a = (n1 + n2) squared / (n1 times n2) replaces 4, producing a more accurate conversion.
Cohen's original benchmarks (d = 0.2 small, 0.5 medium, 0.8 large) were intended for behavioral sciences. In clinical medicine or education, typical effect sizes may be smaller or larger. Always interpret converted values in the context of your specific research domain.
Transparency requires reporting both the original effect size as published and the converted value used in your meta-analysis. This allows readers to verify the conversion and assess whether the underlying assumptions are reasonable for that particular study.
A systematic review often identifies studies that answer the same clinical question yet report their findings in incompatible formats. One randomized trial may express its result as a standardized mean difference, another as an odds ratio from a dichotomized outcome, and a third as a correlation coefficient. The effect size conversion framework described by Borenstein, Hedges, Higgins, and Rothstein (2009, Chapter 7) provides the mathematical machinery for these cross-type conversions while preserving the associated uncertainty through proper standard error propagation.
The most frequently needed conversion is between Cohen's d and the odds ratio. The relationship OR = exp(d × π / √3) derives from assuming the underlying continuous variable follows a logistic distribution (Chinn, 2000). This formula was originally validated by Hasselblad and Hedges (1995), who demonstrated its accuracy across a range of effect sizes commonly encountered in clinical research. Converting between SMD and correlation uses r = d / √(d² + a), where a = (n1 + n2)² / (n1 × n2) accounts for unequal group sizes. When groups are equal, a simplifies to 4.
The risk ratio to odds ratio conversion requires knowing the baseline risk p0 in the control group (Zhang & Yu, 1998). The formula OR = RR × (1 - p0) / (1 - RR × p0) is exact but depends on a reliable estimate of the control event rate. When baseline risk is unavailable, some researchers use the approximation OR ≈ RR for rare events (less than 10%), though this breaks down for common outcomes. For time-to-event data, the hazard ratio approximation HR ≈ OR^0.69 (Grant et al., 2014) provides a pragmatic bridge when studies report only logistic regression odds ratios rather than Cox proportional hazards models.
Converting eta-squared to Cohen's d uses the relationship d = 2√(η² / (1 - η²)), which assumes a two-group comparison. For multi-group ANOVA results, partial eta-squared requires additional care because it reflects variance explained relative to the residual rather than the total. Researchers should verify that the eta-squared value corresponds to a specific pairwise contrast before converting to d for inclusion in a meta-analysis.
Proper SE propagation is critical because every converted effect enters a meta-analysis weighted by the inverse of its variance. This tool propagates SEs using exact formulas where available and the delta method for nonlinear transformations. For the d-to-OR conversion, SE_ln(OR) = SE_d × π/√3. For d-to-r, the delta method yields SE_r ≈ √[(1 - r²)² × SE_d² / a]. Incorrect SE propagation leads to incorrect weights, which in turn bias the pooled estimate toward improperly converted studies.
After conversion, results can be exported directly to the forest plot generator for visual synthesis or the meta-regression formatter for moderator analysis. The effect size calculator handles the upstream step of computing d, OR, or r from raw study data before conversion is needed. For binary outcomes that require visualization beyond pooled estimates, the L'Abbe plot generator provides a two-dimensional scatter view of treatment versus control event rates.
Best practice in reporting requires transparency about every conversion applied. The Cochrane Handbook (Higgins et al., 2023) recommends documenting which studies required conversion, what formula was used, and what assumptions were made (e.g., baseline risk for RR-to-OR, equal groups for d-to-r). Sensitivity analyses comparing results with and without converted studies can assess whether the conversion assumptions materially affect the pooled estimate.
Meta-analyses often combine studies that report different metrics. One trial may report an odds ratio while another reports Cohen's d. To pool them in a single forest plot, you must express all effects on a common scale. Conversion formulas from Borenstein et al. (2009) allow this without re-analyzing the original data.
The formula OR = exp(d × π / √3) assumes the underlying continuous variable follows a logistic distribution in each group. Under normality the constant changes slightly (to √2 instead of π/√3), but the logistic approximation is standard in the Cochrane Handbook and produces virtually identical results in practice (Chinn, 2000).
The correction factor a = (n1 + n2)² / (n1 × n2) accounts for unequal group sizes. When groups are equal (n1 = n2), it simplifies to a = 4. Failing to adjust for group imbalance slightly biases the resulting correlation, so entering your sample sizes is recommended.
Not directly. You first need to convert the risk ratio to an odds ratio using OR = RR × (1 - p0) / (1 - RR × p0), where p0 is the baseline risk. Then convert the OR to d. This tool chains both steps automatically when you select the RR-to-OR direction and then switch to OR-to-d.
Each conversion has a corresponding SE transformation. For d-to-OR: SE_ln(OR) = SE_d × π/√3. For d-to-r, the delta method is applied: SE_r ≈ √[(1 - r²)² × SE_d² / a]. These propagated SEs allow correct confidence interval construction and downstream inverse-variance weighting.
The HR ≈ OR^0.69 approximation (Grant et al., 2014) works best when event rates are moderate (15-60%). For rare events (less than 10%), the OR already approximates the RR and therefore the HR. For common events (greater than 60%), the approximation may introduce meaningful error and should be used cautiously.
Before converting, compute your raw effect sizes with the effect size calculator, which handles means and SDs, 2x2 tables, and test statistics. After converting to a common metric, visualize your pooled results with the forest plot generator or examine potential moderators using the meta-regression formatter. For time-to-event data, the survival curve digitizer extracts hazard ratios directly from Kaplan-Meier figures without needing the OR approximation.
Reviewed by
Dr. Sarah Mitchell holds a PhD in Biostatistics from Johns Hopkins Bloomberg School of Public Health and has over 15 years of experience in systematic review methodology and meta-analysis. She has authored or co-authored 40+ peer-reviewed publications in journals including the Journal of Clinical Epidemiology, BMC Medical Research Methodology, and Research Synthesis Methods. A former Cochrane Review Group statistician and current editorial board member of Systematic Reviews, Dr. Mitchell has supervised 200+ evidence synthesis projects across clinical medicine, public health, and social sciences. She reviews all Research Gold tools to ensure statistical accuracy and compliance with Cochrane Handbook and PRISMA 2020 standards.
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