1. Chapter 17 Comparing Two Proportions
4/16/2017
4/16/2017
Chapter 17 Comparing Two Proportions
April 17
Basic Biostat 1

2. In Chapter 17: 17.1 Data [17.2 Risk Difference] [17.3 Hypothesis Test]
17.4 Risk Ratio
[17.5 Systematic Sources of Error]
[17.6 Power and Sample Size]

3. Data conditions Binary response variables (“success/failure”)
Binary explanatory variable
Notation:

4. Incidence proportion ≡ average risk
Sample Proportions
Incidence proportion, exposed group:
Incidence proportion, non-exposed group:
Incidence proportion ≡ average risk

5. Example: WHI Estrogen Trial
Group 1
n1 = 8506
Estrogen
Treatment
Compare risks
of index disease
Random
Assignment
Group 2
n2 = 8102
Placebo

6. 2-by-2 Table Disease+ Disease− Total Exposure + a1 b1 n1 Exposure − a2m1 m2 N
Risk, non-exposed
Risk, exposed

7. WHI Data D+ D− Total E+ 751 7755 8506 E- 623 7479 8102 1374 15234
16608
Compare these risks

8. Proportion Difference (Risk Difference)
Quantifies excess risk in absolute terms

9. In large samples, the sampling distribution of the risk difference is approximately Normal

10. (1 – α)100% CI for p1 – p2
Plus-four method

11. Estrogen Trial, 95% CI for p1−p2
Data: a1 = 751, n1 = 8506, a2 = 623, n2 = 8102

12. Excess risk of between 0.3% and 2.0% (in absolute terms)
95% CI for p1−p2
Excess risk of between 0.3% and 2.0% (in absolute terms)

13. 95% CI for p1 – p2
Plus-four method similar to Wilson’s score method. Output from WinPepi > Compare 2 program:

14. §17.3 Hypothesis Test C. P-value D. Interpret  evidence against H0
A. H0: p1 = p2 (equivalently H0: RR = 1)
B. Test statistic (three options)
z (large samples)
Chi-square (large samples, next chapter)
Fisher’s exact (any size sample)
C. P-value
D. Interpret  evidence against H0

15. z Test A. H0: p1 = p2 vs.Ha:p1 ≠ p2 (two-sided) B.
C. One-sided P = Pr(Z ≥ |zstat|) Two-sided P = 2 × one-sided P

16. z Test Example
A. H0: p1 = p2 against vs. Ha:p1 ≠ p2
B. Test statistic

17. One-sided P
= Pr(Z ≥ 2.66)
= .0039
Two-sided P = 2 × = .0078
The evidence against H0 is v. significant
 proportions (average risks) differ significantly

18. z Test: Notes z statistic Numerator = observed difference
Denominator = standard error when p1 = p2
A continuity correction can be optionally applied (p. 382)
Equivalent to the chi-square test of association (HS 267)
Avoid z tests in small samples; use exact binomial procedure (HS 267)

19. Fisher’s Exact Test All purpose test for testing H0: p1 = p2
Based on exact binomial probabilities
Calculation intensive, but easy with modern software
Comes in original and Mid-Probability corrected forms

20. Example: Fisher’s Test
Data. The incidence of colonic necrosis in an exposed group is 2 of 117. The incidence in a non-exposed group is 0 of 862.
Ask: Is this difference statistically significant?
Hypothesis statements. Under the null hypothesis, there is no difference in risks in the two populations. Thus: H0: p1 = p2
Ha: p1 > p2 (one-sided) or Ha: p1 ≠ p2 (two-sided)

21. Fisher’s Test, Example B. Test statistic  none per se
C. P-value. Use WinPepi > Compare2.exe > A.
D. Interpret. P-value = .014  strong (“significant”) evidence against H0 D+ D− E+ 2
115 E−
862

22. §17.4 Proportion Ratio (Relative Risk)
Compare incidences from the two groups in form of a RATIO
Quantifies effect of the exposure in relative terms
Relative Risk
Parameter
Relative Risk Estimator
(“RR hat”)

23. Example: RR (WHI Data) + − Total Estrogen + 751 7755 8506 Estrogen −
623
7479
8102

24. Interpretation When p1 = p2, RR = 1  indicating “no association”
RR > 1  positive association
RR < 1  negative association
The RR indicates how much the exposure multiplies the risk over the baseline risk of the non-exposed group
RR of 1.15 suggests risk in exposed group is “1.15 times” that of non-exposed group
Baseline RR is 1!
Thus, an RR of 1.15 is 0.15 (15%) above the baseline

25. Confidence Interval for the RR
To derive information about the precision of the estimate, calculate a (1– α)100% CI for the RR with this formula:
ln ≡ natural log, base e

26. 90% CI for RR, WHI D+ D−
Total E+
751
7755
8506 E−
623
7479
8102

27. WinPepi > Compare2.exe > Program BD+ D−
Total E+
751
7755
8506
E −
623
7479
8102
See prior slide for hand calculations

28. Confidence Interval for the RR
Interpretation similar to other confidence intervals
Interval intends to capture the parameter (in this case the RR parameter)
Confidence level refers to confidence in the procedure
CI length quantifies the precision of the estimate

29. §17.5 Systematic Error CIs and P-values address random error only
In observational studies, systematic errors are more important than random error
Consider three types of systematic errors:
Confounding
Information bias
Selection bias

30. Confounding
Confounding = mixing together of the effects of the explanatory variable with the extraneous factors.
Example:
WHI trial found 15% increase in risk in estrogen exposed group.
Earlier observational studies found 40% lower in estrogen exposed groups.
Plausible explanation: Confounding by extraneous lifestyles factors in observational studies

31. Information Bias
Information bias - mismeasurement (misclassification) leading to overestimation or underestimation in risk
Nondifferential misclassification (occurs to the same extent in the groups)  tends to bias results toward the null or have no effect
Differential misclassification (one groups experiences a greater degree of misclassification than the other)  bias can be in either direction.

32. Nondifferential & Differential Misclassification - Examples

33. Selection Bias
Selection bias ≡ systematic error related to manner in which study participants are selected
Example. If we shoot an arrow into the broad side of a barn and draw a bull’s-eye where it had landed, have we identified anything that is nonrandom?

34. Sample Size & Power for Comparing Proportions
Three approaches:
n needed to estimate given effect with margin of error m (not covered in Ch 17)
n needed to test H0 at given α and power
Power of test of H0 under given conditions

35. Sample Size Requirements for Comparing Proportions
Depends on:
r ≡ sample size ratio = n2 / n1
1−β ≡ power (acceptable type II error rate)
α ≡ significance level (type I error rate)
p1 ≡ expected proportion, group 1
p2 ≡ expected proportion in group 2
Instead of specifying p1 and p2, we can specify p2 and the expected RR

36. Calculation Formulas on pp. 396 – 402 (complex)
In practice  use WinPEPI > Compare2.exe > Sample size

37. WinPepi > Compare2 > S1