Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 10 Comparing Two Groups Section 10.1 Categorical Response: Comparing Two Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 3 Methods for comparing two groups are special cases of bivariate statistical methods. There are two variables:  The outcome variable on which comparisons are made is the response variable.  The binary variable that specifies the groups is the explanatory variable. Statistical methods analyze how the outcome on the response variable depends on or is explained by the value of the explanatory variable. Bivariate Analyses: A Response Variable and a Binary Explanatory Variable

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 4 Most comparisons of groups use independent samples from the groups: 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 and Independent Samples

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 5 Dependent samples result when the 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 subject. Dependent and Independent Samples

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 6 For a 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: Categorical Response: Comparing Two Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 7 Experiment:  Subjects were 25,570 male physicians.  Every other day for five years, 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

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 8 Results displayed in a contingency table: Table 10.1 Whether or Not Subject Died of Cancer, for Placebo and Aspirin Treatment Groups Example: Aspirin, the Wonder Drug

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 9 What is the response variable? The response variable is whether the subject died from cancer, with categories ‘yes’ or ‘no’. What are the groups to compare? The groups to compare are: Group 1: Physicians who took a placebo Group 2: Physicians who took aspirin Example: Aspirin, the Wonder Drug

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 10 Estimate the difference between the two population parameters of interest. The proportion of the population who would have died if they participated in this experiment and took the placebo. The proportion of the population who would not have died if they participated in this experiment and took the aspirin. Example: Aspirin, the Wonder Drug

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 11 The sample proportions of death from cancer were: for the in the placebo group and: for the in the aspirin group. Example: Aspirin, the Wonder Drug

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 12 To make an inference about the difference of population proportions,, we need to learn about the variability of the sampling distribution of: Example: Aspirin, the Wonder Drug

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 13  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 and it is Example: Aspirin, the Wonder Drug

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 14 The z-score depends on the confidence level. This method requires:  Categorical response variable for two groups.  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 for the Difference Between Two Population Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 15 A 95% confidence interval for is: Example: Comparing Death Rates for Aspirin and Placebo

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 16 Since both endpoints of the confidence interval for are positive, we infer that is positive. Conclusion: The population proportion of cancer deaths is larger when subjects take the placebo than when they take aspirin. Table 10.2 MINITAB Output for Confidence Interval Comparing Proportions Example: Comparing Death Rates for Aspirin and Placebo

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 17 The population difference 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. Example: Comparing Death Rates for Aspirin and Placebo

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 18 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, it is important to replicate the study to see if results are similar or different for populations used in this study and other populations. Example: Comparing Death Rates for Aspirin and Placebo

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 19  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 are positive, you can infer that. If all values in the CI for are negative, you can infer that, or.  Which group is labeled ‘1’ and which is labeled ‘2’ is arbitrary. SUMMARY: Interpreting a Confidence Interval for a Difference of Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 20 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. SUMMARY: Interpreting a Confidence Interval for a Difference of Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc Assumptions:  Categorical response variable for two groups.  Independent random samples, either from random sampling or a randomized experiment.  and are large enough that there are at least five successes and five failures in each group if using a two-sided alternative. SUMMARY: Two-Sided Significance Test for Comparing Two Population Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc Hypotheses: The null hypothesis is the hypothesis of no difference or no effect:, that is, The alternative hypothesis is the hypothesis of interest to the investigator. (two-sided test) (one-sided test) SUMMARY: Two-Sided Significance Test for Comparing Two Population Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc Test Statistic: Pooled Estimate  Under the presumption that, we estimate the common value of and by the proportion of the total sample in the category of interest  This pooled estimate is calculated by combining the number of successes in the two groups and dividing by the combined sample size. SUMMARY: Two-Sided Significance Test for Comparing Two Population Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc The test statistic is: where is the pooled estimate SUMMARY: Two-Sided Significance Test for Comparing Two Population Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc P-value: Two-tail probability from standard normal distribution (Table A) of values even more extreme than observed z test statistic presuming the null hypothesis is true. 5. Conclusion: Smaller P-values give stronger evidence against and supporting. Interpret the P-value in context. If a decision is needed, reject if P-value significance level, (such as 0.05). SUMMARY: Two-Sided Significance Test for Comparing Two Population Proportions

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 26 A study considered whether greater levels of television watching by teenagers were associated with a greater likelihood of aggressive behavior. Researchers sampled 707 families in two counties in New York state and made follow-up observations over 17 years. Table 10.3 on the following slide shows results about whether a sampled teenager later conducted any aggressive act against another person, according to a self-report by that person or by his or her parent. Example: TV Watching and Aggressive Behavior

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 27 Table 10.3 TV Watching by Teenagers and Later Aggressive Acts Example: TV Watching and Aggressive Behavior

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 28 Define Group 1 as those who watched less than 1 hour of TV per day, on the average, as teenagers.  Define Group 2 as those who averaged at least 1 hour of TV per day, as teenagers.  = population proportion committing aggressive acts for the lower level of TV watching.  = population proportion committing aggressive acts for the higher level of TV watching. Example: TV Watching and Aggressive Behavior

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 29 Test the Hypotheses: against using a significance level of 0.05 Standard Error: Test Statistic: Example: TV Watching and Aggressive Behavior

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 30 Example: TV Watching and Aggressive Behavior

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 31 Conclusion:  Since the P-value is less than 0.05, we reject.  We support, and 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. Example: TV Watching and Aggressive Behavior