1. Objectives (Chapter 20) Comparing two proportions  Comparing 2 independent samples  Confidence interval for 2 proportion  Large sample method  Plus four method  Test of statistical significance  Treatment and risk reduction

2. Comparing 2 independent samples We often need to compare 2 treatments with 2 independent samples. For large enough samples, the sampling distribution of is approximately Normal. However, neither p 1 nor p 2 are known.

3. Large-sample CI for 2 proportions For 2 independent SRSs of sizes n 1 and n 2 with sample proportions of successes p ̂ 1 and p ̂ 2 respectively, an approximate level C confidence interval for p 1 – p 2 is Use this method when the number of successes and the number of failures are each at least 10 in each sample. C is the area under the standard Normal curve between -z* and z*.

4. Cholesterol and heart attacks How much does the cholesterol lowering drug Gemfibrozil help reduce the risk of heart attack? We compare the incidence of heart attack over a 5 year period for 2 random samples of middle-aged men taking either the drug or a placebo. So the 90% CI is (.0414-.0273) ± 1.645*.0057 =.014 ±.009 = (.005,.023) We are 90% confident that the percent of middle-age men who suffer a heart attack is.5 to 2.3 percentage points lower when taking the cholesterol- lowering drug than when taking a placebo. Standard error of the difference p ̂ 1 – p ̂ 2 : H. attackn p̂p̂ Drug5620512.73% Placebo8420304.14%

5. “Plus four” CI for 2 proportions The “plus 4” method again produces more accurate confidence intervals. We act as if we had 4 additional observations: 1 success and 1 failure in each of the 2 samples. The new combined sample size is n 1 + n 2 + 4 and the proportions of successes are: An approximate level C confidence interval is: Use this when C is at least 90% and both sample sizes are at least 5.

6. Researchers compared oral health in 46 young adult males wearing a tongue piercing (TP) and a control group of 46 young adult males without tongue piercing. They found that 38 individuals in the TP group and 26 in the control group had enamel cracks. We want to estimate with 95% confidence the difference between the proportions of individuals with enamel cracks among young adult males with and without TP. One count is too low for the large sample method, so we use the plus-four method. We are 95% confident that the percent of young adult males with a tongue piercing who have enamel cracks is about 7 to 43 percentage points greater than the percent with enamel cracks among young adult males without a tongue piercing.

7. We test: H 0 : p 1 = p 2 = p If H 0 is true, we are sampling twice from the same population and we can pool the information from both samples to estimate p. Test of significance Appropriate when all counts (successes and failures in each sample) are 5 or more. =0

8. Gastric Freezing Gastric freezing was once a treatment for ulcers. Patients would swallow a deflated balloon with tubes to cool the stomach for an hour in hope of reducing acid production and relieving ulcer pain. The treatment was shown to be safe and significantly reducing ulcer pain and was widely used for years. A randomized comparative experiment later compared the outcome of gastric freezing with that of a placebo: 28 of the 82 patients subjected to gastric freezing improved, while 30 of the 78 in the control group improved. H 0 : p gf = p placebo H a : p gf > p placebo

9. Results: 28 of the 82 patients subjected to gastric freezing improved 30 of the 78 patients in the control group improved The P-value is greater than ??%. (Refer to the graph, no computation needed)  Gastric freezing was not significantly better than a placebo (P-value > 0.1), and this treatment was abandoned. H 0 : p gf = p placebo H a : p gf > p placebo

10. Treatment and risk reduction In the health sciences, we often compare a given health risk in the treatment group with the same risk in the control group. One measure of this is the Relative Risk Reduction (RRR) which indicates how better off you would be relative to receiving a placebo or control treatment.

11. Cholesterol and heart attacks H. attackNo H.A.n p̂p̂ Drug5619952051 2.73% Placebo8419462030 4.14% How much does the cholesterol-lowering drug Gemfibrozil help reduce the risk of heart attack? We compare the incidence of heart attack over a 5-year period for two random samples of middle-aged men taking either the drug or a placebo. The drug Gemfibrozil reduces the risk of a heart attack in middle-aged men by about 34% over a 5-year period of continuous treatment, compared with middle- aged men taking a placebo (RRR = 34%). That is, the risk of a heart attack over that period is 34% smaller in the Gemfibrozil group than in the placebo group.

12. The Absolute Risk Reduction (ARR) is simply the absolute difference in outcome rates between the control and treatment groups: The Number Needed to Treat (NNT) is the number of patients that would need to be treated to prevent one additional negative outcome. NNT = 1 / ARR ARR and NNT are better indicators of treatment efficacy than RRR.

13. H. attackNo H.A.n p̂p̂ Drug5619952051 2.73% Placebo8419462030 4.14% The group taking Gemfibrozil had a rate of heart attack 1.4 percentage point lower than that of the placebo group (ARR = 1.4%). On average, we need to treat 71 men for 5 years with Gemfibrozil to avoid 1 heart attack (NNT = 71).