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16: Risk Ratios

16: Risk Ratios. 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

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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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