7 A clinical trial is run to evaluate the efficacy of a new medication to relieve pain in patients undergoing total knee replacement surgery. In the trial, patients are randomly assigned to receive either the new medication or the standard medication. After receiving the assigned medication, patients are asked to report their pain on a scale of 0-100 with higher scores indicative of more pain. Data on the primary outcome are shown below. New Medication Standard Medication Mean Pain Score 30.31 53.85 Sample Size 60 60 Standard Deviation of Pain Score 7.52 7.44 Because procedures can be more complicated in older patients, the investigators are concerned about confounding by age. For analysis, patients are classified into two age groups, less than 65 and 65 years of age and older. The data are shown below. Age < 65 Years New Medication Standard Medication Total: Age < 65 Years Age 65+ Years New Medication Standard Medication Total: Age 65+ Sample Size 40 25 65 Mean Pain Score 25.30 45.51 33.07 Standard Deviation of Pain Score 2.46 1.83 10.16 Sample Size 20 35 55 Mean Pain Score 40.33 59.80 52.72 Standard Deviation of Pain Score 2.16 2.49 9.74 a. Is there a statistically significant difference in mean pain scores between patients assigned to the new medication as compared to the standard medication? Run the appropriate test at =0.05. (Ignore age in this analysis.) Step 1. Set up hypotheses and determine level of significance H0: = 2 H1: 2 =0.05 Step 2. Select the appropriate test statistic. The appropriate test statistic is X1 - X 2 Z Sp . 1 1 n1 n 2 Check whether the assumption of equality of population variances is reasonable: s12/s22 = 7.522/7.442 = 1.02. The ratio of the sample variances is between 0.5 and 2 suggesting that the assumption of equality of population variances is reasonable. Step 3. Set up decision rule. This is a two-tailed test, using a Z statistic and a 5% level of significance. The appropriate critical values can be found in Table 1C and the decision rule is as follows: Reject H0 if Z < -1.960 or is Z > 1.960. Step 4. Compute the test statistic. We first compute Sp, the pooled estimate of the common standard deviation. Sp 2 (n 1 1)s 1 (n 2 1)s 2 2 n1 n 2 2 =7.48. 2 Sp (60 1)7.52 (60 1)7.44 2 60 60 2 Now the test statistic, Z 30.31 53.85 7.48 Step 5. 1 1 60 60 23.54 17.18. 1.37 Conclusion. We reject H0 because -17.18 < -1.960. We have statistically significant evidence at =0.05 to show that there is a difference in mean pain scores between treatments. The p-value can be found in Table 1C and is equal to p < 0.0001. Question 8 Use the data in Problem 7 to determine whether age a confounding variable. Run the tests of hypothesis to determine whether the age is related to treatment assignment and whether there is a difference in mean pain scores by age group. Is age a confounder? Justify your conclusion. Age Group and Treatment Age < 65 Age 65+ Step 1. New Medication 40 20 Set up hypotheses and determine level of significance. Standard Medication 25 35 H0: Age and treatment are independent H1: H0 is false. Step 2. =0.05 Select the appropriate test statistic. The formula for the test statistic is in Table 7.8 and is given below. (O - E) 2 2 E The condition for appropriate use of the above test statistic is that each expected frequency is 5 or more. In Step 4 we will compute the expected frequencies and ensure that the condition is met. Step 3. Set up decision rule. df=(2-1)(2-1)=1. Reject H0 if 2 > 3.84. Step 4. Compute the test statistic. Age Group and Treatment Age < 65 Age 65+ New Medication 40 (32.5) 20 (27.5) Standard Medication 25 (32.5) 35 (27.5) The test statistic is computed as follows: (40 32.5) 2 ( 25 32.5) 2 ( 20 27.5) 2 (35 27.5) 2 2 32.5 32.5 27.5 27.5 2 = 1.73 + 1.73 + 2.05 + 2.05 = 7.56 Step 5. Conclusion. We reject H0 because 7.56 > 3.84. We have statistically significant evidence at =0.05 to show that H0 is false; age and treatment are not independent (i.e., they are related). Using Table 3, the p-value is p < 0.005. Age Group and Mean Pain Scores Step 1. Set up hypotheses and determine level of significance H0: = 2 H1: 2 =0.05 Step 2. Select the appropriate test statistic. The appropriate test statistic is . X1 - X 2 Z Sp 1 1 n1 n 2 Check whether the assumption of equality of population variances is reasonable: s12/s22 = 10.162/9.742 = 1.16. The ratio of the sample variances is between 0.5 and 2 suggesting that the assumption of equality of population variances is reasonable. Step 3. Set up decision rule. This is a two-tailed test, using a Z statistic and a 5% level of significance. The appropriate critical values can be found in Table 1C and the decision rule is as follows: Reject H0 if Z < -1.960 or is Z > 1.960. Step 4. Compute the test statistic. We first compute Sp, the pooled estimate of the common standard deviation. 2 (n 1 1)s 1 (n 2 1)s 2 2 Sp n1 n 2 2 = 9.97. 2 Sp (65 1)10.16 (55 1)9.74 2 65 55 2 Now the test statistic, Z 33.07 52.72 9.97 Step 5. 1 1 65 55 19.65 10.74. 1.83 Conclusion. We reject H0 because -10.74 < -1.960. We have statistically significant evidence at =0.05 to show that there is a difference in mean pain scores between age groups. The p-value can be found in Table 1C and is equal to p < 0.0001. Age is confounder as it is related to both to treatment and to pain scores. Question 9 The data presented in Problem 7 are analyzed using multiple linear regression analysis and the models are shown below. In the models below, the data are coded as follows: treatment is coded as 1=new medication and 0=standard medication and age 65+ is coded as 1=yes and 0=no. Y Y Y = 53.85 - 23.54 Medication = 45.31 - 19.88 Medication + 14.64 Age 65+ = 45.51 - 20.21 Medication + 14.29 Age > 65 + 0.75 Medication *Age 65+ Patients < 65: Patients > 65: Y Y = 45.51 - 20.21 Medication = 59.80 - 19.47 Medication Does it appear that there is effect modification by age? Justify your response using the models above. It does not appear that there is effect modification by age because the regression coefficient associated with the treatment by age interaction is b3 = 0.75 which is close to zero (and likely non-significant in a statistical test). In addition, the magnitude of the treatment effect is similar in patients < 65 years of age as compared to patients 65 years and older. In both groups, mean pain scores are lower in patients assigned to the new medication. The differences in means are 20.21 and 19.47 units, respectively. Question 10 Based on your answers to Problem 8 and Problem 9, how should the effect of the treatment be summarized? Should results be reported separately by age group or combined? Should the effect of treatment be adjusted for age? Justify your response using the models presented in Problem 9