Chapter 9 Comparing Two Groups

1. Chapter 9Comparing Two Groups • Learn …. How to Compare Two Groups On a Categorical or Quantitative Outcome Using Confidence Intervals and Significance Tests

2. Bivariate Analyses • The outcome variable is theresponse variable • The binary variable that specifies the groups is theexplanatory variable

3. Bivariate Analyses • Statistical methods analyze how the outcome on the response variabledepends on oris explained bythe value of the explanatory variable

4. Independent Samples • The observations in one sample are independent of those in the other sample • Example: Randomized experiments that randomly allocate subjects to two treatments • Example: An observational study that separates subjects into groups according to their value for an explanatory variable

5. Dependent Samples • Data are matched pairs – each subject in one sample is matched with a subject in the other sample • Example: set of married couples, the men being in one sample and the women in the other. • Example: Each subject is observed at two times, so the two samples have the same people

6. Section 9.1 Categorical Response: How Can We Compare Two Proportions?

7. Categorical Response Variable • Inferences compare groups in terms of their population proportions in a particular category • We can compare the groups by the difference in their population proportions: (p1 – p2)

8. Example: Aspirin, the Wonder Drug • Recent Titles of Newspaper Articles: • “Aspirin cuts deaths after heart attack” • “Aspirin could lower risk of ovarian cancer” • “New study finds a daily aspirin lowers the risk of colon cancer” • “Aspirin may lower the risk of Hodgkin’s”

9. Example: Aspirin, the Wonder Drug • The Physicians Health Study Research Group at Harvard Medical School • Five year randomized study • Does regular aspirin intake reduce deaths from heart disease?

10. Example: Aspirin, the Wonder Drug • Experiment: • Subjects were 22,071 male physicians • Every other day, study participants took either an aspirin or a placebo • The physicians were randomly assigned to the aspirin or to the placebo group • The study was double-blind: the physicians did not know which pill they were taking, nor did those who evaluated the results

11. Example: Aspirin, the Wonder Drug Results displayed in a contingency table:

12. Example: Aspirin, the Wonder Drug • What is the response variable? • What are the groups to compare?

13. Example: Aspirin, the Wonder Drug • The response variable is whether the subject had a heart attack, with categories ‘yes’ or ‘no’ • The groups to compare are: • Group 1: Physicians who took a placebo • Group 2: Physicians who took aspirin

14. Example: Aspirin, the Wonder Drug • Estimate the difference between the two population parameters of interest

15. Example: Aspirin, the Wonder Drug • p1: the proportion of the population who would have a heart attack if they participated in this experiment and took the placebo • p2: the proportion of the population who would have a heart attack if they participated in this experiment and took the aspirin

16. Example: Aspirin, the Wonder Drug Sample Statistics:

17. Example: Aspirin, the Wonder Drug • To make an inference about the difference of population proportions, (p1 – p2), we need to learn about the variability of the sampling distribution of:

18. Standard Error for Comparing Two Proportions • The difference, , is obtained from sample data • It will vary from sample to sample • This variation is the standard error of the sampling distribution of :

19. Confidence Interval for the Difference between Two Population Proportions • The z-score depends on the confidence level • This method requires: • Independent random samples for the two groups • Large enough sample sizes so that there are at least 10 “successes” and at least 10 “failures” in each group

20. Confidence Interval Comparing Heart Attack Rates for Aspirin and Placebo • 95% CI:

21. Confidence Interval Comparing Heart Attack Rates for Aspirin and Placebo • Since both endpoints of the confidence interval (0.005, 0.011) for (p1- p2) are positive, we infer that (p1- p2) is positive • Conclusion: The population proportion of heart attacks is larger when subjects take the placebo than when they take aspirin

22. Confidence Interval Comparing Heart Attack Rates for Aspirin and Placebo • The population difference (0.005, 0.011) is small • Even though it is a small difference, it may be important in public health terms • For example, a decrease of 0.01 over a 5 year period in the proportion of people suffering heart attacks would mean 2 million fewer people having heart attacks

23. Confidence Interval Comparing Heart Attack Rates for Aspirin and Placebo • The study used male doctors in the U.S • The inference applies to the U.S. population of male doctors • Before concluding that aspirin benefits a larger population, we’d want to see results of studies with more diverse groups

24. Interpreting a Confidence Interval for a Difference of Proportions • Check whether 0 falls in the CI • If so, it is plausible that the population proportions are equal • If all values in the CI for (p1- p2) are positive, you can infer that (p1- p2) >0 • If all values in the CI for (p1- p2) are negative, you can infer that (p1- p2) <0 • Which group is labeled ‘1’ and which is labeled ‘2’ is arbitrary

25. Interpreting a Confidence Interval for a Difference of Proportions • The magnitude of values in the confidence interval tells you how large any true difference is • If all values in the confidence interval are near 0, the true difference may be relatively small in practical terms

26. Significance Tests Comparing Population Proportions 1. Assumptions: • Categorical response variable for two groups • Independent random samples

27. Significance Tests Comparing Population Proportions Assumptions (continued): • Significance tests comparing proportions use the sample size guideline from confidence intervals: Each sample should have at least about 10 “successes” and 10 “failures” • Two–sided tests are robust against violations of this condition • At least 5 “successes” and 5 “failures” is adequate

28. Significance Tests Comparing Population Proportions 2. Hypotheses: • The null hypothesis is the hypothesis of no difference or no effect: H0: (p1- p2) =0 • Under the presumption that p1= p2, we create a pooled estimate of the common value of p1and p2 • This pooled estimate is

29. Significance Tests Comparing Population Proportions 2. Hypotheses (continued): Ha: (p1- p2) ≠ 0 (two-sided test) Ha: (p1- p2) < 0 (one-sided test) Ha: (p1- p2) > 0 (one-sided test)

30. Significance Tests Comparing Population Proportions 3. The test statistic is:

31. Significance Tests Comparing Population Proportions 4. P-value: Probability obtained from the standard normal table 5. Conclusion: Smaller P-values give stronger evidence against H0 and supporting Ha

32. Example: Is TV Watching Associated with Aggressive Behavior? • Various studies have examined a link between TV violence and aggressive behavior by those who watch a lot of TV • A study sampled 707 families in two counties in New York state and made follow-up observations over 17 years • The data shows levels of TV watching along with incidents of aggressive acts

33. Example: Is TV Watching Associated with Aggressive Behavior?

34. Example: Is TV Watching Associated with Aggressive Behavior? Test the Hypotheses: H0: (p1- p2) = 0 Ha: (p1- p2) ≠ 0 • Using a significance level of 0.05 • Group 1: less than 1 hr. of TV per day • Group 2: at least 1 hr. of TV per day

35. Example: Is TV Watching Associated with Aggressive Behavior?

36. Example: Is TV Watching Associated with Aggressive Behavior? • Conclusion: Since the P-value is less than 0.05, we reject H0 • We conclude that the population proportions of aggressive acts differ for the two groups • The sample values suggest that the population proportion is higher for the higher level of TV watching

37. Section 9.2 Quantitative Response: How Can We Compare Two Means?

38. Comparing Means • We can compare two groups on a quantitative response variable by comparing their means

39. Example: Teenagers Hooked on Nicotine • A 30-month study: • Evaluated the degree of addiction that teenagers form to nicotine • 332 students who had used nicotine were evaluated • The response variable was constructed using a questionnaire called the Hooked on Nicotine Checklist (HONC)

40. Example: Teenagers Hooked on Nicotine • The HONC score is the total number of questions to which a student answered “yes” during the study • The higher the score, the more hooked on nicotine a student is judged to be

41. Example: Teenagers Hooked on Nicotine • The study considered explanatory variables, such as gender, that might be associated with the HONC score

42. Example: Teenagers Hooked on Nicotine • How can we compare the sample HONC scores for females and males? • We estimate (µ1 - µ2) by (x1 - x2): 2.8 – 1.6 = 1.2 • On average, females answered “yes” to about one more question on the HONC scale than males did

43. Example: Teenagers Hooked on Nicotine • To make an inference about the difference between population means, (µ1 – µ2), we need to learn about the variability of the sampling distribution of:

44. Standard Error for Comparing Two Means • The difference, , is obtained from sample data. It will vary from sample to sample. • This variation is the standard error of the sampling distribution of :

45. Confidence Interval for the Difference between Two Population Means • A 95% CI: • Software provides the t-score with right-tail probability of 0.025

46. Confidence Interval for the Difference between Two Population Means • This method assumes: • Independent random samples from the two groups • An approximately normal population distribution for each group • this is mainly important for small sample sizes, and even then the method is robust to violations of this assumption

47. Example: Nicotine – How Much More Addicted Are Smokers than Ex-Smokers? • Data as summarized by HONC scores for the two groups: • Smokers: x1 = 5.9, s1 = 3.3, n1 = 75 • Ex-smokers:x2 = 1.0, s2 = 2.3, n2 = 257

48. Example: Nicotine – How Much More Addicted Are Smokers than Ex-Smokers? • Were the sample data for the two groups approximately normal? • Most likely not for Group 2 (based on the sample statistics): x2 = 1.0, s2 = 2.3) • Since the sample sizes are large, this lack of normality is not a problem

49. Example: Nicotine – How Much More Addicted Are Smokers than Ex-Smokers? • 95% CI for (µ1- µ2): • We can infer that the population mean for the smokers is between 4.1 higher and 5.7 higher than for the ex-smokers

50. How Can We Interpret a Confidence Interval for a Difference of Means? • Check whether 0 falls in the interval • When it does, 0 is a plausible value for (µ1 – µ2), meaning that it is possible that µ1 = µ2 • A confidence interval for (µ1 – µ2) that contains only positive numbers suggests that (µ1 – µ2) is positive • We then infer that µ1 is larger than µ2