# 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.