---
title: "OR <-> RR Conversion & Effect Size Transformations"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{OR <-> RR Conversion & Effect Size Transformations}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

## 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.