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16: Risk Ratios

Learn about risk ratios, confidence intervals, exact tests, Simpson’s Paradox, sample size requirements, and more in medical research. Explore illustrative examples and methods to gauge risks and associations effectively.

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16: Risk Ratios

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  1. 16: Risk Ratios Risk ratios

  2. Comparing proportions as a ratio • Incidence proportion (risk) = proportion experiencing an event over time • Prevalence = proportions with a condition at a time • Relative risk = the ratio of two risks • Prevalence ratio will equal the risk ratio when • Average duration of disease same in groups • Disease is rare (risk < 5%) • Disease does not influence exposure • 2-by-2 display (right) Risk ratios

  3. Illustrative Example (Jolson et al., 1992) Exposure = generic drug (yes/no) Disease = adverse drug reaction (yes/no) Risk ratios

  4. Interpretation of Risk Ratio • Risk multiplier • e.g., risk ratio of 5 implies 5× risk with exposure • Percent relative increase in risk • Baseline risk ratio is 1 (indicating no difference in risk) • Percent relative increase in risk = (RR – 1) × 100% • e.g., a RR of 5 indicates a (5 – 1) × 100% = 400% increase in risk (in relative terms) • Risk ratios less than 1 imply a benefit e.g., a risk ratio of 0.75 indicates a 25% decrease in risk Risk ratios

  5. 95% Confidence Interval for the RR Method • Convert RR^ to natural log (ln) scale • Calculate SE (right) • 95% CI for ln(RR) = ln(RR^) ± (1.96)(SE) • 95% CI for RR = take anti-logs of above limits Illustrative example (Jolson et al., 1992) • ln(RR^) = ln(4.99) = 1.607 • SE = 0.5964 (right) • 95% CI for ln(RR)= 1.607 ± (1.96)(0.5964) = 1.607 ± 1.169 = (0.4381, 2.779) • 95% CI for RR = e(0.4381, 2.779) = (1.55 , 16.1) Check or do work with computer (e.g., SPSS, www. OpenEpi.com, WinPepi) Risk ratios

  6. Confidence Interval • Locates parameter with “wiggle room” • We are 95% confident RR is between 1.55 and 16.1 • Confidence interval width quantifies precision of the estimate • Wide  imprecise estimate • Narrow  precise estimate Risk ratios

  7. Testing the Risk Ratio • H0: RR = 1 (“no association”) • Test statistics • z method (HS 167) • Chi-square method (last week) • Fisher’s or Mid-P exact (computer only) • P value • Conclusion – evidence against the claim of H0 Risk ratios

  8. Exact test: Example • E = Post-op exposure of Kayexalate in kidney patients • D = Gangrene of intestine • Dataset = kxnecro.sav Note: two table cells with expected counts of less than 5 → avoid chi-square → use an exact procedure (by computer) → next slide Risk ratios

  9. Exact tests: OpenEpi computation Either Fisher’s or the Mid-P are acceptable Risk ratios

  10. Multiple Levels of Exposure With multiple levels of exposure, break up the table.Compare each exposure level to the least exposed group. Break this 3-by-2 into these 3 2-by-2s Risk ratios

  11. Simpson’s Paradox (Extreme confounding) • Confounding  a form of bias in which a lurking variable creates a spurious association between variables • Simpson’s paradox  an extreme confounding in which the lurking variable creates a reversal in the direction of an association Risk ratios

  12. Simpson’s Paradox: Example Consider a trial at two clinics. Overall we find: Or is there a lurking variable that explains the association? To evaluate this, split applications according to the lurking variable “clinic 1095 / 10,100 = 11% of treatment group succeed5050 / 11,100 = 46% of control group succeed Relative incidence of success = 11% / 46% = 0.25 Treatment appears harmful Risk ratios

  13. Simpson’s Paradox: Example Clinic 1 1000 / 10,000 = 10% of treatment group showed success50 / 1000 = 5% of the control group showed success The relative incidence (RR) of success = 2, in favor of the treatment Risk ratios

  14. Simpson’s Paradox: Example Clinic 2 95 / 100 = 95% of treatment group showed success5000 / 10,000 = 50% of the control group showed success The relative incidence of success is almost 2, in favor of the treatment Risk ratios

  15. Simpson’s Paradox: Example • Within each clinic, a higher percentage of the treatment group experienced success • The treatment is effective • This is an example of Simpson’s Paradox. • When the lurking variable (clinic) was ignored, the data suggest the treatment is harmful* • When the clinic is considered, the association is reversed. * Clinic 1 treated refractory (more severe) cases Risk ratios

  16. Sample Size Requirements (Delay coverage) • “Inputs” needed to determine required sample size • Significance level (a) • Power (1 – b) • Minimal detectable risk ratio (RR) • Sample size ratio (e.g., n2/n1) • Maximum efficiency comes when n2 = n1 • Expected proportion in non-exposed group (p2) • Plug assumptions into formula or computer program Risk ratios

  17. Sample Size Requirements Example a = .05 two-sided 1 – b = .90 n2/n1 = 1 p2 = .10 RR = 2.0 N for regular chi-square Risk ratios

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