1October 15

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 Group 1 = “exposed” Group 2 = “non-exposed” Notation:

4 Sample Proportions Sample proportion (average risk), group 1: Sample proportion (average risk), group 2:

5 Example: WHI Estrogen Trial Random Assignment Group 1 n 1 = 8506 Group 2 n 2 = 8102 Estrogen Treatment Placebo Compare risks of index outcome* *Death, MI, breast cancer, etc.

6 2-by-2 Table SuccessesFailuresTotal Group 1a1a1 b1b1 n1n1 Group 2a2a2 b2b2 n2n2 Totalm1m1 m2m2 N

7 WHI Data D+D−Total E E Total

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 p 1 – p 2 Plus-four method

11 Estrogen Trial, 95% CI for p 1 −p 2 Data: a 1 = 751, n 1 = 8506, a 2 = 623, n 2 = 8102

12 95% CI for p 1 −p 2 Excess risk of between 0.3% and 2.0% (in absolute terms)

13 95% CI for p 1 – p 2 Plus-four method similar to Wilson’s score method. Output from WinPepi > Compare 2 program:

14 §17.3 Hypothesis Test A. H 0 : p 1 = p 2 (equivalently H 0 : 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 H 0

15 z Test A. H 0 : p 1 = p 2 vs.H a :p 1 ≠ p 2 (two-sided) B. C. One-sided P = Pr(Z ≥ |z stat |) Two-sided P = 2 × one-sided P

16 z Test Example A. H 0 : p 1 = p 2 against vs. H a :p 1 ≠ p 2 B. Test statistic

17 One-sided P = Pr(Z ≥ 2.66) =.0039 Two-sided P = 2 ×.0039 =.0078 The evidence against H 0 is v. significant  proportions (average risks) differ significantly

18 z Test: Notes z statistic –Numerator = observed difference –Denominator = standard error when p 1 = p 2 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 H 0 : p 1 = p 2 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? A.Hypothesis statements. Under the null hypothesis, there is no difference in risks in the two populations. Thus: H 0 : p 1 = p 2 H a : p 1 > p 2 (one-sided) or H a : p 1 ≠ p 2 (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 H 0 D+D− E+2115 E−0862

22 §17.4 Proportion Ratio (Relative Risk) “Relative risk” is used to refer to the RATIO of two proportions Also called “risk ratio”

23 Example: RR ( WHI Data) +−Total Estrogen Estrogen −

24 Interpretation The RR is a risk multiplier –RR of 1.15 suggests risk in exposed group is “1.15 times” that of non-exposed group –This is 0.15 (15%) above the relative baseline When p 1 = p 2, RR = 1. –Baseline RR is 1, indicating “no association” –RR of 1.15 represents a weak positive association

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

26 90% CI for RR, WHI D+D−Total E E−

27 WinPepi > Compare2.exe > Program B D+D−Total E E − 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: 1.n needed to estimate given effect with margin of error m (not covered in Ch 17) 2.n needed to test H 0 at given α and power 3.Power of test of H 0 under given conditions

35 Sample Size Requirements for Comparing Proportions Depends on: r ≡ sample size ratio = n 1 / n 2 1−β ≡ power (acceptable type II error rate) α ≡ significance level (type I error rate) p 1 ≡ expected proportion, group 1 p 2 ≡ expected proportion in group 2, or expected effect size (e.g., RR)

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

37 WinPepi > Compare2 > S1