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Chapter 17 Comparing Two Proportions

Chapter 17 Comparing Two Proportions. Data conditions. Binary response variables (“success/failure”) Binary explanatory variable Group 1 = “exposed” Group 2 = “non-exposed” Notation:. Sample Proportions. Sample proportion (average risk), group 1:.

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Chapter 17 Comparing Two Proportions

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  1. Chapter 17Comparing Two Proportions

  2. Data conditions • Binary response variables (“success/failure”) • Binary explanatory variable Group 1 = “exposed” Group 2 = “non-exposed” • Notation:

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

  4. Example: WHI Estrogen Trial Group 1 n1 = 8506 Estrogen Treatment Compare risks of index outcome* Random Assignment Group 2 n2 = 8102 Placebo *Death, MI, breast cancer, etc.

  5. 2-by-2 Table

  6. WHI Data

  7. §17.3 Hypothesis Test 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

  8. 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

  9. 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)

  10. 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

  11. §17.4 Proportion Ratio (Relative Risk) • “Relative risk” is used to refer to the RATIO of two proportions • Also called “risk ratio”

  12. Example: RR (WHI Data)

  13. 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 p1 = p2, RR = 1. • Baseline RR is 1, indicating “no association” • RR of 1.15 represents a weak positive association

  14. 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

  15. 90% CI for RR, WHI

  16. WinPepi > Compare2.exe > Program B See prior slide for hand calculations

  17. 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

  18. §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

  19. 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

  20. 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.

  21. Nondifferential & Differential Misclassification - Examples

  22. 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?

  23. Sample Size & Powerfor 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

  24. Sample Size Requirements for Comparing Proportions Depends on: r ≡ sample size ratio = n1 / n2 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, or expected effect size (e.g., RR)

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

  26. WinPepi > Compare2 > S1

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