OR <-> RR Conversion & Effect Size Transformations

Overview

Network Meta-Analysis (NMA) requires all treatment effects on a common scale. However, trials report results as Odds Ratios, Relative Risks, or Standardised Mean Differences depending on the outcome type. ParCC provides bidirectional conversions to unify these metrics before pooling.

Tutorial: Preparing Data for an NMA in Depression

The Scenario

You are conducting an NMA comparing three antidepressants. Your systematic review found:

• Trial A (Drug vs Placebo): OR = 1.85 for “Response” (>=50% reduction in HAM-D). Baseline response in placebo arm = 30%.
• Trial B (Drug vs Placebo): RR = 1.42 for “Response”.
• Trial C (Drug vs Placebo): Reports a continuous outcome: SMD = 0.45 (Cohen’s d) on HAM-D score.

To pool these in a single NMA, you need all three on the same scale.

Step 1: Convert OR to RR (Zhang & Yu Method)

The Zhang & Yu (1998) formula accounts for baseline risk:

$$RR = \frac{OR}{1 - p_0 + p_0 \times OR}$$

where $p_0$ is the baseline risk in the control group.

In ParCC:

1. Navigate to Convert > Rate <-> Probability > OR <-> RR tab.
2. Select direction: OR -> RR.
3. Input OR = 1.85, Baseline Risk = 0.30.
4. Result: RR ~ 1.42.

Why This Matters

If the outcome were rare (<10%), OR ~ RR and conversion wouldn’t matter. But with a 30% baseline risk, the OR of 1.85 overstates the effect compared to the RR of 1.42. Failing to convert would bias the NMA.

Step 2: Convert SMD to log(OR) (Chinn Method)

The Chinn (2000) approximation uses the logistic distribution:

$$\ln(OR) = SMD \times \frac{\pi}{\sqrt{3}} \approx SMD \times 1.8138$$

In ParCC:

1. Switch to the Effect Size Conversions tab.
2. Select direction: SMD -> log(OR).
3. Input SMD = 0.45.
4. Result: log(OR) = 0.816, i.e. OR ~ 2.26.

Step 3: Convert log(OR) to log(RR)

To bring Trial C onto the RR scale (matching Trials A and B):

$$\ln(RR) = \ln\left(\frac{e^{\ln(OR)}}{1 - p_0 + p_0 \times e^{\ln(OR)}}\right)$$

ParCC chains the Chinn and Zhang & Yu methods automatically.

When to Use These Conversions

Scenario	Conversion	Method
NMA mixing binary effect measures	OR -> RR or RR -> OR	Zhang & Yu (1998)
NMA mixing binary + continuous outcomes	SMD -> log(OR)	Chinn (2000)
Clinical interpretation of OR	OR -> RR	Zhang & Yu – RR is more intuitive
Checking the rare-disease approximation	Compare OR and RR at your baseline risk	If they diverge >10%, convert explicitly

The Rare-Disease Approximation

When the baseline risk is very low ($p_0 < 0.10$), OR ~ RR mathematically. ParCC displays a note when this approximation holds. For common outcomes (>10%), always convert explicitly.

References

1. Zhang J, Yu KF. What’s the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA. 1998;280(19):1690-1691.
2. Chinn S. A simple method for converting an odds ratio to effect size for use in meta-analysis. Statistics in Medicine. 2000;19(22):3127-3131.
3. Cochrane Handbook for Systematic Reviews of Interventions, Chapter 12: Synthesizing and presenting findings using other methods.