Confidence Interval for Relative Risk Stat 301 – Day 23 Confidence Interval for Relative Risk

Announcements Review HW 5 commentary (1, 2, 3, 5) Especially problem 5 Graded project 1 returned Similar rubric for project 2 Introduction Data Collection Analysis (Descriptive and Inferential) Conclusions Presentation Still submit proposals?

Chapter 2 – Comparing 2 proportions Independent random samples Simulation: sample 1 from binomial, sample 2 from binomial with same value for p With large sample sizes can use normal approximation (confidence interval) Randomized experiment Simulation: Fix successes and failures, randomly shuffle which group they go to Exact p-value: Fisher’s Exact Test

Chapter 2 – Comparing 2 proportions Statistic: Difference in conditional proportions Can be a bit misleading when the proportions are small Want to take the “baseline risk” into account New statistic: Relative risk RR = cond. proportion in group 1 (larger) cond. proportion in group 2 Group 1’s risk is XX times higher or (XX-1)x100% percent higher than Group 2’s risk If RR < 1, (1-XX)x100% is the percentage decrease

Relative Risk So how do we decide whether we have an unusual value for Relative Risk under the null hypothesis? Simulation Exact Normal approximation

Relative Risk - Simulation rhyper(1, M=210, N-M=4958, n=2584)  106 Vaccinated control Total Influenza 106 No symptoms Statistic Count: 106 Diff: .00077 Rel risk: 1.019 Vaccinated control Total Influenza 106 104 210 No symptoms 4958 2584 5168

Confidence interval Null distribution of relative risks: Centered at 1 but not the most normal looking distribution

Take the natural log Will produce a normal distribution, mean 0, preserves ordering of values But now need a standard deviation…

Theoretical result The standard error of the log-relative risk statistics: sqrt(1/62 – 1/2584 + 1/148 – 1/2584) = .1487 Vs. simulation

(t) We are 95% confident that the long-run probability of developing influenza with the “control” vaccine can be anywhere between 1.779 to 3.203 times higher than with the quadrivalent vaccine, for children ages 3 to 8 similar to the ones in the study.

Investigation 2.10 replacement Wynder and Graham (a) Relative risk of lung cancer, comparing regular smokers to non-smokers (583/1159)/(22/226) = 5.17 Smokers are 5.17 times more likely to get LC

Investigation 2.10 Wynder and Graham (c) What is the estimate of the baseline rate of lung cancer from this table? Does that seem to be a reasonable estimate to you? How is this related to the design of the study?

Investigation 2.10 Wynder and Graham (d) Relative risk of control, comparing non smokers to regular smokers (204/226)/(576/1159) = 1.82 Nonsmokers are 1.82 times more likely to not be LC patient

Odds Ratio (e) Odds of LC regular smokers: 583/576 Odds of LC non-smokers: 22/204 Odds ratio: (583/576)/(22/204) = 9.385 Odds of having lung cancer are 9.4 times higher for the regular smokers

Odds Ratio (f) Odds of control non-smokers: 204/22 Odds of control regular smokers: 576/583 Odds ratio: (204/22)/(576/583)= 9.385 Odds of not having lung cancer are 9.4 times higher for the non-smokers

Odds Ratio vs. Relative Risk May give different results from relative risk Odds ratio is invariant to choice of success/failure and also to choice of EV/RV Much more appropriate for “case control” studies were the response variable is fixed by design Will this change the p-value? How find a confidence interval? fisher.test(matrix(c(204, 22, 576, 583), nrow=2), alt="two.sided")

To Do Investigation 2.10A Review questions for Exam 2 in PolyLearn