When odds ratio approximates relative risk

OR vs. RR - Tale of Two Ratios

• Relative Risk (RR): Compares the risk of an outcome in an exposed group to the risk in an unexposed group. The primary measure for cohort studies.

  • Formula: $RR = [a/(a+b)] / [c/(c+d)]$

• Odds Ratio (OR): Compares the odds of prior exposure in the disease group to the odds of prior exposure in the control group. The primary measure for case-control studies.

  • Formula: $OR = (a/c) / (b/d) = ad/bc$

• When OR ≈ RR: The OR approximates RR when the disease prevalence is low (the "rare disease assumption").
  • With low prevalence, the number of cases (a and c) is negligible compared to the non-cases (b and d).

⭐ The Odds Ratio is the primary measure of association in case-control studies, as incidence (and thus Relative Risk) cannot be directly calculated.

The Approximation - When Odds Meet Risk

The Odds Ratio (OR) is a good approximation of the Relative Risk (RR) under the rare disease assumption.

• Core Principle: When a disease has a low prevalence (or incidence in case-control studies), the OR will closely estimate the RR.
• Threshold: The approximation holds well when disease prevalence is <10%.

📌 When the disease is RARE, the OR and RR are RELATIVELY the same.

The mathematical basis is that for a rare disease, the number of individuals with the disease (a and c) is negligible compared to those without it.

• Thus, a+b ≈ b and c+d ≈ d.
• This simplifies the RR formula: $RR = [a/(a+b)] / [c/(c+d)] ≈ (a/b) / (c/d) = OR$.

⭐ The OR always overestimates the RR when the RR is >1, and underestimates the RR when the RR is <1 (i.e., it's always further from the null value of 1).

Divergence Details - When OR Overstates

• Primary Condition: OR significantly overestimates RR when the disease is not rare (i.e., common).
• Mechanism of Divergence:
  • The OR and RR diverge as the disease prevalence ↑.
  • The magnitude of this overestimation increases as the prevalence moves away from 0.
• Direction of Overestimation: The OR is always further from the null value of 1.0 than the RR.
  • If RR > 1, then OR > RR.
  • If RR < 1, then OR < RR.

⭐ For common diseases, the OR can significantly exaggerate the true measure of association, potentially leading to misinterpretation of a risk factor's importance.

High-Yield Points - ⚡ Biggest Takeaways

• The Odds Ratio (OR) is a good approximation of the Relative Risk (RR) when the disease is rare.
• This is often called the rare disease assumption.
• "Rare" generally means a disease prevalence or incidence of <10%.
• The OR consistently overestimates the RR, and this divergence worsens as prevalence increases.
• This approximation is crucial for interpreting case-control studies, where RR cannot be calculated directly.