1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 25 Paired Samples and Blocks

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 3 Paired Data Data are paired when the observations are collected in pairs or the observations in one group are naturally related to observations in the other group. Paired data arise in a number of ways. Perhaps the most common is to compare subjects with themselves before and after a treatment. When pairs arise from an experiment, the pairing is a type of blocking. When they arise from an observational study, it is a form of matching.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 4 Paired Data (cont.) If you know the data are paired, you can (and must!) take advantage of it. To decide if the data are paired, consider how they were collected and what they mean (check the W’s). There is no test to determine whether the data are paired. Once we know the data are paired, we can examine the pairwise differences. Because it is the differences we care about, we treat them as if they were the data and ignore the original two sets of data.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 5 Paired Data (cont.) Now that we have only one set of data to consider, we can return to the simple one-sample t-test. Mechanically, a paired t-test is just a one-sample t-test for the mean of the pairwise differences. The sample size is the number of pairs.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 6 Assumptions and Conditions Paired Data Assumption: The data must be paired. Independence Assumption: The differences must be independent of each other. Check the: Randomization Condition Normal Population Assumption: We need to assume that the population of differences follows a Normal model. Nearly Normal Condition: Check this with a histogram or Normal probability plot of the differences.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 7 The Paired t-Test When the conditions are met, we are ready to test whether the paired differences differ significantly from zero. We test the hypothesis H 0 :  d =  0, where the d’s are the pairwise differences and  0 is almost always 0.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 8 The Paired t-Test (cont.) We use the statistic where n is the number of pairs. is the ordinary standard error for the mean applied to the differences. When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t-model on n – 1 degrees of freedom, so we can use that model to obtain a P-value.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 9 Confidence Intervals for Matched Pairs When the conditions are met, we are ready to find the confidence interval for the mean of the paired differences. The confidence interval is where the standard error of the mean difference is The critical value t* depends on the particular confidence level, C, that you specify and on the degrees of freedom, n – 1, which is based on the number of pairs, n.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 10 Example 1: Pulse Rates Once Again One indicator of physical fitness is resting pulse rate. Ten men volunteered to test an exercise device advertised on television by using it three times a week for 20 minutes. Their resting pulse rates were measured before the test began, and then again after six weeks. Results are shown in the table. Is there evidence that this kind of exercise can reduce resting pulse rates? How much?

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 11 Example 1: Table of Data Pulse Rates(bpm) SubjectBeforeAfter Allen73 Brandon8379 Carlos8581 David8786 Edwin9187 Franco9991 Graeme8784 Hans8583 Ivan8384 Jorge7976

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 12 Example 1: continued The before and after measurements are not independent – we CAN’T use a two sample t-test. When this happens we will look at the paired differences using a one sample t-test.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 13 Example 1: continued Hypothesis: The null hypothesis is that there will be no difference in the pulse rates before and after using the exercise device. The alternative hypothesis is that the exercise device will cause a decrease the resting pulse rate. The difference will be final pulse rate – beginning pulse rate.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 14 Example 1: continued Model: 1. The men are not selected randomly, but we assume that they are a representative sample of adult men. 2. Independence: The pulse rate of one man does not influence the pulse rate of another. 3. 10 is certainly less than 10% of all men. 4. Since n<30, we will do a boxplot. Since the graph of the paired differences is unimodal and approximately symmetric, the sampling distribution is nearly normal. Since all conditions are met, we will use a matched pairs t-test.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 15 Example 1: continued Mechanics:

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 16 Example 1: continued The P value of.0034 is much lower than our level of significance of.05. Therefore, using the matched pairs t-test, we reject the null hypothesis. There is strong evidence that this form of exercise can reduce resting pulse rates. 90% confidence interval: (-4.266,-1.334) I am 90% confident that mean decrease in the resting pulse rates is between 4.3 and 1.3 beats per minute.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 17 Blocking Consider estimating the mean difference in age between husbands and wives. The following display is worthless. It does no good to compare all the wives as a group with all the husbands—we care about the paired differences.

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 18 Blocking (cont.) In this case, we have paired data—each husband is paired with his respective wife. The display we are interested in is the difference in ages: Insert histogram from page 579 of the text.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 19 Blocking (cont.) Pairing removes the extra variation that we saw in the side-by-side boxplots and allows us to concentrate on the variation associated with the difference in age for each pair. A paired design is an example of blocking.

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 20 *The Sign Test Again? Because we have paired data, we’ve been using a simple t-test for the paired differences. This suggests that if we want a distribution-free method, a sign test on the paired differences testing whether the median of the differences is 0 is appropriate. The advantage of the sign test for matched pairs is that we don’t require the Nearly Normal Condition for the paired differences. However, if the assumptions of a paired t-test are met, the paired t-test is more powerful than the sign test.

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 21 What Can Go Wrong? Don’t use a two-sample t-test for paired data. Don’t use a paired-t method when the samples aren’t paired. Don’t forget outliers—the outliers we care about now are in the differences. Don’t look for the difference in side-by-side boxplots.

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 25- 22 What have we learned? Pairing can be a very effective strategy. Because pairing can help control variability between individual subjects, paired methods are usually more powerful than methods that compare individual groups. Analyzing data from matched pairs requires different inference procedures. Paired t-methods look at pairwise differences. We test hypotheses and generate confidence intervals based on these differences. We learned to Think about the design of the study that collected the data before we proceed with inference.