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

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]

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

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

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

2-by-2 Table Disease+ Disease− Total Exposure + a1 b1 n1 Exposure − a2 m1 m2 N Risk, non-exposed Risk, exposed

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

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

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

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

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

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)

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

§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

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

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

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

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)

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

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)

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

§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”)

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

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

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

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

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

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

§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

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

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.

Nondifferential & Differential Misclassification - Examples

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?

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

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

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

WinPepi > Compare2 > S1