Chapter 17 Comparing Two Proportions 4/17/2017 Chapter 17 Comparing Two Proportions April 17 Basic Biostat

In Chapter 17: 17.1 Data 17.2 Proportion Difference (Risk Difference) 17.3 Hypothesis Test 17.4 Proportion Ratio (Risk Ratio) 17.5 Systematic Sources of Error 17.6 Power and Sample Size

§17.1 Data Two independent groups Binary response Epidemiologic jargon: Group 1 = “exposed group” and Group 2 = “nonexposed group” Count “successes” in each group and convert to proportions

Sample Proportions Proportion in the exposed group: Proportion in the nonexposed group:

§17.2 Proportion Difference (Risk Difference) The risk difference is the absolute difference in incidence proportions in the groups.

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

Confidence Interval, Risk Difference Plus-four confidence interval method for a difference in proportions. This method is accurate in samples as small as 5 per group.

95% CI for p1 – p2, Example WHI data a1 = 751, n1 = 8503, a2 = 623, n2 = 8102

95% CI for p1 – p2, Example The plus-four method is similar to Wilson’s score method. Here’s output from from WinPepi > Compare2.exe > Program B showing results from the traditional large-sample method and Wilson score CI for the illustrative example.

§17.3 Hypothesis Test We test the proportions for a significant (“nonrandom”) difference Two methods are covered in this chapter z test (large sample) Fisher’s exact procedure (small samples) A third method called the chi-square test is covered in the next chapter

z Test A. Hypotheses. H0: p1 = p2 against Ha:p1 ≠ p2 [One-sided: Ha: p1 > p2 or Ha: p1 < p2] B. Test statistic. C. P-value. Convert zstat to P-value [Table B or F]

conclude: highly significant

z Test: Notes z statistic dissection numerator is observed difference denominator is standard error when p1 = p2 A continuity correction can be optionally applied (p. 382) This z test is equivalent to the chi-square test of association (Chapter 18) In small samples (fewer than 5 successes expected in either group), avoid the z test and use the exact Fisher or Mid-P procedure

Fisher’s Exact Test (2-by-2) Before conducting Fisher’s test, data are rearranged to form a 2-by-2 table : Successes Failures Total Group 1 a1 b1 n1 Group 2 a2 b2 n2 m1 m2 N Recall that and

WHI Data, 2-by-2 Format + − Total Estrogen + 751 7755 8506 Estrogen − 623 7479 8102 1374 15234 16608

Fisher’s Exact Test, Procedure A. Hypotheses. H0: p1 = p2 vs. Ha: p1 ≠ p2 [one sided Ha: p1 > p2 or Ha: p1 < p2] B. Test statistic. Observed counts in 2-by-2 table C. P-value. Use computer program (e.g., WinPepi > Compare2.exe > Program B). The mathematical basis of the test is described on pp. 386–7.

Fisher’s Exact Test, Example The incidence of colonic necrosis in an exposed group is 2 of 117. The incidence in a non-exposed group is 0 of 862. Is this difference statistically significant? A. H0: p1 = p2 against Ha: p1 ≠ p2 B. Data. + − Total Group 1 2 115 117 Group 2 862 977 979

C. P = 0. 014 (WinPepi output shown here) C. P = 0.014 (WinPepi output shown here). The evidence against H0 is “significant.”

§17.4 Proportion (Risk) Ratio Let RR refer to an risk ratio or prevalence ratios Interpretation The RR is a risk multiplier, e.g., an RR of 2 suggests that the exposure doubles risk When p1 = p2 , RR = 1. This is the “baseline RR,” indicating no association.

RR Example, WHI Data + − Total Estrogen + 751 7755 8506 Estrogen − 623 7479 8102 The indicates a positive association; specifically, 15% higher risk (in relative terms) with exposure.

Note natural log scale of sampling distribution (1– α)100% CI for the RR Note natural log scale of sampling distribution

90% CI for RR, WHI Example + − Total Estrogen + 751 7755 8506 623 7479 8102

CI for RR, Computerized Results + − Total Estrogen + 751 7755 8506 Estrogen − 623 7479 8102 Output from WinPepi > Compare2.exe > Program B. See prior slide for hand calculations

§17.5 Systematic Error (Advanced Topic) In observational studies, systematic errors are often more important than random sampling error Three types of systematic error are considered: Confounding Information bias Selection bias

Confounding Confounding = the mixing together of the effects of the explanatory variable with the effects of “lurking” variables. Consider this example: The WHI estrogen experiment found increased morbidity and mortality in estrogen users Earlier, non-experimental studies found the opposite: lower morbidity and mortality in users Plausible explanation: In non-experimental studies, estrogen users (self-selected) were more likely to have “lurking” lifestyles factors that contributed to better health, i.e., confounding

Information Bias Information bias is due to the mismeasurement or misclassification of variables in the study. Misclassification may be nondifferential (occurs to the same extent in the groups) or differential (one groups experiences a greater degree of misclassification than the other) Nondifferential misclassification tends to bias results toward the null (or have no effect). Differential misclassification can bias results in either direction.

Nondifferential & Differential Misclassification - Examples

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

17.6 Power and Sample Size Power and sample sizes analysis for comparing proportions requires us to understand relationships between these factors: r ≡ sample size allocation ratio n1 / n2 1−β ≡ power (type II error) α ≡ significance level (type I error) p1 ≡ expected proportion, group 1 p2 ≡ expected proportion in group 2, or some measure of effect size, such as the expected RR

17.6 Power and Sample Size Because of the complexity of calculations (pp. 396 – 402), use software… Here’s WinPepi’s Compare2 Sample size menu.