1. Comparing the Means of Two Dependent Populations

2. How are dependent samples created? Pre-test vs. Post-test, Before treatment vs. After treatment (i.e. subjects = blocks) Comparing different treatments using the same subjects, e.g. pain relievers used on the same subjects (again subjects = blocks) Matched subjects in the two populations according to some criteria, e.g. matched patients on basis of age, race, gender, socioeconomic status, weight, height, existing health conditions, etc… (need to be careful here)

3. Example 1: Captopril and Systolic Blood Pressure Research Question: Is there evidence that patients will experience a mean decrease in systolic blood pressure of more than 10 mmHg? Experiment: Measure the blood pressure of 10 patients before and after taking Captopril. Our interest is on the measured changes in blood pressure and whether or not we believe that those changes have a mean greater than 10 mmHg.

4. Example 1: Captopril and Systolic Blood Pressure SubjectSystolic BeforeSystolic AfterDifference = Before - After 12102019 21691654 318716621 41601573 516714720 617614531 718516817 820618026 917314726 1014613610 1117415123 1220116833 1319817919 1414812919 1515413123

5. Example 1: Captopril and Systolic Blood Pressure SubjectSystolic BeforeSystolic AfterDifference = Before - After 12102019 21691654 318716621 41601573 516714720 617614531 718516817 820618026 917314726 1014613610 1117415123 1220116833 1319817919 1414812919 1515413123 Do these differences indicate the mean the change in blood pressure for all patients who take captopril exceeding 10 mmHg? This is exactly the same as a single population test of the mean. Do these sampled differences come from a population whose mean is greater than 10?

6. Example 1: Captopril and Systolic Blood Pressure SubjectSystolic BeforeSystolic AfterDifference = Before - After 12102019 21691654 318716621 41601573 516714720 617614531 718516817 820618026 917314726 1014613610 1117415123 1220116833 1319817919 1414812919 1515413123

7. Hypotheses for Paired T-test Null hypothesis:H 0 :  D =  1 -  2 =  Alternative hypotheses:H A :  D =  1 -  2   H A :  D =  1 -  2 >  H A :  D =  1 -  2 <  Does the mean difference of the population,  D, differ from some hypothesized difference  ?  typically is equal to 0, i.e. we are looking for a difference between the population means.

8. The Paired t-Test Statistic Assumptions: there are n meaningfully matched pairs the paired differences are normally distributed, this can be relaxed when n is large. Test Statistic: The test statistic, which follows a t-distribution with df = n - 1 degrees of freedom, gives us our p-value:

9. Quantifying Effect Size Confidence Intervals t-value comes from a t-distribution with df = n - 1 Effect Size (d) Small effect d = 0.10 Medium effect d = 0.25 Large effect d = 0.40 Power and sample size issues are the same as for the single population mean case.

10. Example 1: Captopril and Systolic Blood Pressure STEP 1) State Hypotheses  D = mean change in blood pressure after taking Captopril H o :  D =  Before -  After < 10 mmHg (mean decrease is less than 10 units) H A :  D =  Before -  After > 10 mmHg (mean decrease is more than 10 units)

11. Example 1: Captopril and Systolic Blood Pressure STEP 2) Determine Test Criteria Choose  Test Statistic STEP 3) Collect Data and Compute Test Statistic

12. Example 1: Captopril and Systolic Blood Pressure STEP 4) Compute p-value STEP 5) Make Decision and Interpret We have strong evidence to suggest that the systolic blood pressure will decrease by more than 10 mmHg on average after taking Captopril (p <.0001).  t = 5.516 P-value approx. = 0

13. Example 1: Captopril and Systolic Blood Pressure STEP 6) Quantify Significant Effects In this example we imposed clinical importance by lookin specifically for an increase of at least 10 mmHg. We should still construct CI for the true mean change in systolic blood pressure, as this gives a range of plausible values for the average decrease will be.

14. Example 1: Captopril and Systolic Blood Pressure STEP 6) Quantify Significant Effects 95% CI for  d Conclusion: We estimate that the mean systolic blood pressure decrease for patients taking Captopril is between 13.9 and 23.9 mmHg with 95% confidence.

15. Example 1: Captopril and Systolic Blood Pressure STEP 6) Quantify Significant Effects For effect size we can look at the mean change in systolic blood pressure in terms of difference from 0 or 10 mmHg. Effect Size Effect Size (compared to 0) (compared to 10) Both effects are obviously very large!!!

16. Power and Sample Size (same as the single population mean case) For an observed difference of 8.933 and SD = 9.03 we have Power =.94 or 94%.