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## Chapter 9 Comparing Two Groups

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**Chapter 9Comparing Two Groups**• Learn …. How to Compare Two Groups On a Categorical or Quantitative Outcome Using Confidence Intervals and Significance Tests**Bivariate Analyses**• The outcome variable is theresponse variable • The binary variable that specifies the groups is theexplanatory variable**Bivariate Analyses**• Statistical methods analyze how the outcome on the response variabledepends on oris explained bythe value of the explanatory variable**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**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**Section 9.1**Categorical Response: How Can We Compare Two Proportions?**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)**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”**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?**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**Example: Aspirin, the Wonder Drug**Results displayed in a contingency table:**Example: Aspirin, the Wonder Drug**• What is the response variable? • What are the groups to compare?**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**Example: Aspirin, the Wonder Drug**• Estimate the difference between the two population parameters of interest**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**Example: Aspirin, the Wonder Drug**Sample Statistics:**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:**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 :**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**Confidence Interval Comparing Heart Attack Rates for Aspirin**and Placebo • 95% CI:**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**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**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**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**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**Significance Tests Comparing Population Proportions**1. Assumptions: • Categorical response variable for two groups • Independent random samples**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**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**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)**Significance Tests Comparing Population Proportions**3. The test statistic is:**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**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**Example: Is TV Watching Associated with Aggressive**Behavior?**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**Example: Is TV Watching Associated with Aggressive**Behavior?**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**In 2002, the median net worth was estimated as $89,000 for**white households and $6000 for black households. What is the response variable? • Net worth • Households: white or black**In 2002, the median net worth was estimated as $89,000 for**white households and $6000 for black households. What is the explanatory variable? • Net worth • Households: white or black**In 2002, the median net worth was estimated as $89,000 for**white households and $6000 for black households. • Identify the two groups that are the categories of the explanatory variable. • White and Black households • Net worth and households**In 2002, the median net worth was estimated as $89,000 for**white households and $6000 for black households. • The estimated medians were based on a sample of households. Were the samples of white households and black households independent samples or dependent samples? • Independent samples • Dependent samples**Section 9.2**Quantitative Response: How Can We Compare Two Means?**Comparing Means**• We can compare two groups on a quantitative response variable by comparing their means**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)**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**Example: Teenagers Hooked on Nicotine**• The study considered explanatory variables, such as gender, that might be associated with the HONC score**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**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:**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 :**Confidence Interval for the Difference between Two**Population Means • A 95% CI: • Software provides the t-score with right-tail probability of 0.025**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