Risk ratios 12/6/2018 16: Risk Ratios 12/6/2018 Risk ratios StatPrimer

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) Disease + Disease - Total Exposed + a b n1 Exposed - c d n2 m1 m2 N 12/6/2018 Risk ratios

Illustrative Example (Jolson et al., 1992) Exposure = generic drug (yes/no) Disease = adverse drug reaction (yes/no) Disease + Disease - Total Exposed + 11 14 25 Exposed - 3 31 34 45 59 12/6/2018 Risk ratios

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 12/6/2018 Risk ratios

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) 12/6/2018 Risk ratios

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 12/6/2018 Risk ratios

Testing the Risk Ratio H0: RR = 1 (“no association”) Test statistics Risk ratios 12/6/2018 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 12/6/2018 Risk ratios StatPrimer

Exact test: Example E = Post-op exposure of Kayexalate in kidney patients D = Gangrene of intestine Dataset = kxnecro.sav Observed D+ D− Total E+ 2 115 117 E− 862 977 979 Expected D+ D− Total E+ 0.24 116.67 117 E− 1.76 860.24 862 2 977 979 Note: two table cells with expected counts of less than 5 → avoid chi-square → use an exact procedure (by computer) → next slide 12/6/2018 Risk ratios

Exact tests: OpenEpi computation Either Fisher’s or the Mid-P are acceptable 12/6/2018 Risk ratios

Multiple Levels of Exposure With multiple levels of exposure, break up the table. Compare each exposure level to the least exposed group. Smoke+ Smoke− 1 (High school graduate) 12 38 Break this 3-by-2 into these 3 2-by-2s Highest degree Smoke+ Smoke− 1 (High school graduate) 12 38 2 (Associated degree) 18 67 3 (Some college) 27 95 2 (Associated degree) 18 67 1 (High school graduate) 12 38 3 (Some college) 27 95 1 (High school graduate) 12 38 12/6/2018 Risk ratios

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 12/6/2018 Risk ratios

Simpson’s Paradox: Example Consider a trial at two clinics. Overall we find: Success Failure Total Treatment 1095 9005 10,100 Control 5050 5950 11,000 6145 14,955 21,100 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 succeed 5050 / 11,100 = 46% of control group succeed Relative incidence of success = 11% / 46% = 0.25 Treatment appears harmful 12/6/2018 Risk ratios

Simpson’s Paradox: Example Clinic 1 Success Failure Total Treatment 1000 9000 10,000 Control 50 950 1050 9950 11000 1000 / 10,000 = 10% of treatment group showed success 50 / 1000 = 5% of the control group showed success The relative incidence (RR) of success = 2, in favor of the treatment 12/6/2018 Risk ratios

Simpson’s Paradox: Example Clinic 2 Success Failure Total Treatment 95 5 100 Control 5000 10,000 5095 5005 10,100 95 / 100 = 95% of treatment group showed success 5000 / 10,000 = 50% of the control group showed success The relative incidence of success is almost 2, in favor of the treatment 12/6/2018 Risk ratios

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 12/6/2018 Risk ratios

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 12/6/2018 Risk ratios

Sample Size Requirements Example a = .05 two-sided 1 – b = .90 n2/n1 = 1 p2 = .10 RR = 2.0 N for regular chi-square 12/6/2018 Risk ratios