Blazon Research Resources · Meta-analysis
How to convert OR, RR and HR for meta-analysis
Why ratio measures are analysed on the log scale
OR, RR and HR are bounded below by zero, have 1 as the null value and can extend to infinity. That scale is asymmetric. Taking natural logs maps the null value 1 to 0 and makes reciprocal effects symmetric: ln(0.5) ≈ −0.693 and ln(2) ≈ +0.693.
Cochrane recommends analysing ratio measures on the natural-log scale and notes that generic inverse-variance meta-analysis typically uses the log effect and its SE as inputs. Cochrane Handbook, Chapter 6
Convert a ratio to its log effect
For any positive ratio R:
log effect = ln(R)
Examples
- OR = 2 → ln(OR) ≈ 0.693
- RR = 1 → ln(RR) = 0
- HR = 0.8 → ln(HR) ≈ −0.223
Calculate SE from the confidence interval
For a two-sided normal/Wald confidence interval reported on the ratio scale:
SE[ln(R)] = [ln(upper) − ln(lower)] / (2 × z*)
For a 95% CI, z* ≈ 1.959964.
Cochrane describes this process explicitly for ratio effect measures and emphasizes that the SE refers to the log of the ratio measure. Cochrane Handbook, Chapter 6
Example: hazard ratio
Suppose HR = 0.80 with a 95% CI of 0.65 to 0.98.
ln(HR) = ln(0.80) ≈ −0.2231
SE ≈ [ln(0.98) − ln(0.65)] / (2 × 1.959964)
The resulting log(HR) and SE can be used with generic inverse-variance meta-analysis when the underlying assumptions are appropriate.
Why OR, RR and HR must not be mixed up
These measures share the same log-transformation mathematics but describe different estimands. The converter does not treat OR, RR and HR as interchangeable and does not convert one into another. Cochrane specifically warns that OR and RR differ, particularly when events are common. Cochrane Handbook, Chapter 6
What about asymmetric confidence intervals?
Published ratio estimates and confidence limits are often rounded. Statistical Converter therefore assesses consistency on the natural-log scale while accounting for the precision with which the values were reported. If the observed asymmetry cannot plausibly be explained by rounding, the tool displays a warning rather than silently assuming a symmetric Wald interval.
Hazard ratios and time-to-event data
Hazard ratios arise from survival/time-to-event analyses. Cochrane notes that Cox models can provide log(HR) and its SE directly, and that when an HR is reported with a confidence interval the SE can be reconstructed using the log-scale approach. Cochrane Handbook, Chapter 6
Frequently asked questions
Why do meta-analyses use log odds ratios and log risk ratios?
The natural-log transformation makes reciprocal ratio effects symmetric around zero and gives a scale suitable for standard inverse-variance methods. Cochrane Handbook, Chapter 6
How do I convert an OR to log OR?
Take the natural logarithm: log(OR) = ln(OR).
How do I get SE from an OR confidence interval?
For a suitable 95% Wald CI, use [ln(upper) − ln(lower)] / (2 × 1.959964). The same structure applies to RR and HR.
Can I convert an odds ratio directly to a risk ratio?
Not from the OR alone. OR and RR are different effect measures and depend differently on the underlying risks. Cochrane Handbook, Chapter 6
Can an HR be analysed like a continuous mean?
No. Time-to-event data require survival-analysis methods; mean survival times generally cannot substitute for methods that properly handle censoring. Cochrane Handbook, Chapter 6
References
- Higgins JPT, Li T, Deeks JJ, eds. Chapter 6: Choosing effect measures and computing estimates of effect. In: Cochrane Handbook for Systematic Reviews of Interventions, version 6.5. Cochrane; 2024. Cochrane Handbook, Chapter 6