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

2. 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?

3. 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

4. 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

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

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

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

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

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

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

11. 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

12. 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?

13. 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

14. 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

15. 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

16. 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")

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