Confidence intervals for OR and RR

OR vs. RR - Tale of Two Ratios

• Relative Risk (RR): Compares the probability of an outcome in an exposed group to the probability in an unexposed group. The preferred measure for cohort studies.
  • Formula: $RR = \frac{a/(a+b)}{c/(c+d)}$
• Odds Ratio (OR): Compares the odds of exposure among those with the disease (cases) to the odds of exposure among those without it (controls). The primary measure for case-control studies.
  • Formula: $OR = \frac{a/c}{b/d} = \frac{ad}{bc}$

Confidence Intervals & Significance

Because the distributions of OR and RR are skewed, confidence intervals (CIs) are calculated using their natural logarithm (ln), which is normally distributed.

• General Formula: $exp(ln(Ratio) \pm Z \times SE_{ln(Ratio)})$
  • For a 95% CI, the Z-score is 1.96.
• The Null Value is 1.0:
  • If the CI contains 1.0 (e.g., 0.9 to 2.5), the result is NOT statistically significant.
  • If the CI does NOT contain 1.0 (e.g., 1.5 to 3.0), the result IS statistically significant.

⭐ In case-control studies, you cannot calculate incidence, so you must use the Odds Ratio. Relative Risk is not a valid measure for case-control study designs.

💡 Pearl: When disease prevalence is low (<10%), the OR closely approximates the RR. This is known as the rare disease assumption.

Confidence Intervals - The Certainty Zone

• A Confidence Interval (CI) provides a range of plausible values for the true population Odds Ratio (OR) or Relative Risk (RR). It quantifies the uncertainty surrounding our sample estimate.
• The 95% CI is the standard used: we are 95% confident that the true population value lies within this interval.

Interpreting the CI for OR & RR:

• The critical step is checking if the interval contains the null value of 1.0 (which signifies no effect).
• CI Contains 1.0:
  • Example: RR = 1.8 (95% CI: 0.9-2.7)
  • The association is NOT statistically significant.
  • The p-value is ≥ 0.05.
• CI Does NOT Contain 1.0:
  • The association IS statistically significant.
  • Entirely > 1.0: Indicates increased risk or odds.
    • Example: OR = 2.5 (95% CI: 1.5-4.1)
  • Entirely < 1.0: Indicates decreased risk or odds (a protective effect).
    • Example: RR = 0.6 (95% CI: 0.4-0.9)

Precision & CI Width:

• Narrow CI: High precision, suggesting a more certain estimate (often from a large sample size).
• Wide CI: Low precision, suggesting more uncertainty (often from a small sample size).

⭐ The p-value Connection: If the 95% CI for an OR or RR does not cross the null value of 1.0, the corresponding p-value will be < 0.05. If the CI does cross 1.0, the p-value will be ≥ 0.05.

📌 Mnemonic: If the number one is in the CI run, the significance is none.

High-Yield Points - ⚡ Biggest Takeaways

• A Confidence Interval (CI) for an OR or RR that does not include 1.0 indicates a statistically significant result.
• If the CI contains 1.0, the result is not statistically significant, meaning there is no association.
• An entire CI > 1.0 suggests a significant increase in risk.
• An entire CI < 1.0 suggests a significant decrease in risk (protective effect).
• Narrower CIs indicate greater precision, typically from larger sample sizes.