1. Risk ratios
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16: Risk Ratios
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Risk ratios
StatPrimer

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

3. 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
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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
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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 = (right)
95% CI for ln(RR)= ± (1.96)(0.5964) = ± = (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)
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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
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Risk ratios

7. Testing the Risk Ratio H0: RR = 1 (“no association”) Test statistics
Risk ratios
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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
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Risk ratios
StatPrimer

8. 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
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Risk ratios

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

12. 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
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Risk ratios

13. 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
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Risk ratios

14. 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
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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
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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
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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
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Risk ratios