An antihistamine is frequently studied using a model to examine its effectiveness (compared to a placebo) in inhibiting a positive skin reaction to a known allergen. Consider the following situation. Individuals are screened to find 10 subjects who demonstrate sensitivity to the allergen to be used in the study. The 10 subjects are then randomly assigned to one of two treatment groups---level 1 is the known antihistamine and level 2 is an identical-appearing placebo---with five subjects per group. At the start of the study, a baseline (pre-drug) sensitivity reading is obtained, and then each patient begins taking the assigned medication for three days. Skin sensitivity readings are taken at 1, 2, 3, 4 and 8 hours following the first dose. The percentage inhibition of skin sensitivity reaction (reduction in swelling area where the allergen is applied, compared to baseline) is the response variable. (A negative value means there was an increase in swelling compared to baseline.) This experiment involves subjects and is a repeated measures design to study the factors "treatment" and "time". On what factors are there repeated measures? For the three factors "patient", "treatment" and "time", which factors are crossed and which are nested? Discuss the plots provided. Conduct the analysis of variance for these data, testing each testable source of variation appropriately. For each test, state the null hypothesis, give the observed significance level, and state your conclusion. Obtain and interpret relevant confidence intervals to investigate the effectiveness of the treatment. Try first to do this using prog glm output, then if possible verify the results with proc mixed output. * repeat05.sas; * Repeated measures design; options ls=78 nodate; ; data pit; input trtmt patient y time; lines; 1 1 -10.2 1 1 2 -12.9 1 1 3 27.1 1 1 4 13.0 1 1 5 28.9 1 1 1 27.2 2 1 2 22.1 2 1 3 26.5 2 1 4 19.7 2 1 5 26.1 2 : : : : 2 2 15.8 8 2 3 23.2 8 2 4 26.1 8 2 5 10.6 8 ; proc plot; plot y*time=trtmt; plot y*time=patient; title 'Repeated Measures Design'; ; proc glm; class patient trtmt time; model y = trtmt time trtmt*time patient(trtmt); random patient(trtmt); ; proc mixed CL CovTest; class patient trtmt time; model y = trtmt | time; random patient(trtmt); lsmeans trtmt time / adjust=Tukey CL; Repeated Measures Design 1 Plot of y*time. Symbol is value of trtmt. 50 + | | | | | 1 | 40 + 1 | 1 | | 1 | 1 2 1 | 2 | 30 + 1 2 | 1 2 1 | 1 1 y | 1 2 | | 1 2 | 1 2 2 20 + 1 | 1 1 | 1 | 2 2 2 | | 1 2 2 | 1 10 + 2 2 | | | 2 | | | 0 + 2 | | | 2 | | | -10 + 1 | | 1 | | | 2 | -20 + 2 --+---------+---------+---------+---------+---------+---------+---------+- 1 2 3 4 5 6 7 8 time NOTE: 8 obs hidden. Repeated Measures Design 2 Plot of y*time. Symbol is value of patient. 50 + | | | | | 2 | 40 + 4 | 1 | | 1 | 2 2 2 | 2 | 30 + 4 1 | 5 2 3 | 3 1 y | 5 4 | | 4 3 | 2 4 5 20 + 4 | 5 1 | 3 | 3 1 2 | | 4 5 1 | 5 10 + 3 5 | | | 1 | | | 0 + 5 | | | 1 | | | -10 + 1 | | 2 | | | 2 | -20 + 4 --+---------+---------+---------+---------+---------+---------+---------+- 1 2 3 4 5 6 7 8 time NOTE: 8 obs hidden. Repeated Measures Design 3 The GLM Procedure Class Level Information Class Levels Values patient 5 1 2 3 4 5 trtmt 2 1 2 time 5 1 2 3 4 8 Number of observations 50 Repeated Measures Design 4 The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 17 7053.70960 414.92409 3.25 0.0020 Error 32 4086.02320 127.68823 Corrected Total 49 11139.73280 R-Square Coeff Var Root MSE y Mean 0.633203 59.19909 11.29992 19.08800 Source DF Type I SS Mean Square F Value Pr > F trtmt 1 1085.780000 1085.780000 8.50 0.0064 time 4 4847.458800 1211.864700 9.49 <.0001 trtmt*time 4 318.214000 79.553500 0.62 0.6494 patient(trtmt) 8 802.256800 100.282100 0.79 0.6189 Source DF Type III SS Mean Square F Value Pr > F trtmt 1 1085.780000 1085.780000 8.50 0.0064 time 4 4847.458800 1211.864700 9.49 <.0001 trtmt*time 4 318.214000 79.553500 0.62 0.6494 patient(trtmt) 8 802.256800 100.282100 0.79 0.6189 Repeated Measures Design 5 The GLM Procedure Source Type III Expected Mean Square trtmt Var(Error) + 5 Var(patient(trtmt)) + Q(trtmt,trtmt*time) time Var(Error) + Q(time,trtmt*time) trtmt*time Var(Error) + Q(trtmt*time) patient(trtmt) Var(Error) + 5 Var(patient(trtmt)) Repeated Measures Design 6 The Mixed Procedure Model Information Data Set WORK.PIT Dependent Variable y Covariance Structure Variance Components Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Containment Class Level Information Class Levels Values patient 5 1 2 3 4 5 trtmt 2 1 2 time 5 1 2 3 4 8 Dimensions Covariance Parameters 2 Columns in X 18 Columns in Z 10 Subjects 1 Max Obs Per Subject 50 Observations Used 50 Observations Not Used 0 Total Observations 50 Iteration History Iteration Evaluations -2 Res Log Like Criterion 0 1 321.83811491 1 1 321.83811491 0.00000000 Convergence criteria met. Covariance Parameter Estimates Standard Z Cov Parm Estimate Error Value Pr Z Alpha Lower patient(trtmt) 0 . . . . . Residual 122.21 27.3263 4.47 <.0001 0.05 82.3751 Repeated Measures Design 7 The Mixed Procedure Covariance Parameter Estimates Cov Parm Upper patient(trtmt) . Residual 200.07 Fit Statistics -2 Res Log Likelihood 321.8 AIC (smaller is better) 323.8 AICC (smaller is better) 323.9 BIC (smaller is better) 324.1 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trtmt 1 8 8.88 0.0176 time 4 32 9.92 <.0001 trtmt*time 4 32 0.65 0.6304 Least Squares Means Standard Effect trtmt time Estimate Error DF t Value Pr > |t| Alpha trtmt 1 23.7480 2.2109 8 10.74 <.0001 0.05 trtmt 2 14.4280 2.2109 8 6.53 0.0002 0.05 time 1 0.2000 3.4958 32 0.06 0.9547 0.05 time 2 19.9500 3.4958 32 5.71 <.0001 0.05 time 3 23.5500 3.4958 32 6.74 <.0001 0.05 time 4 28.6400 3.4958 32 8.19 <.0001 0.05 time 8 23.1000 3.4958 32 6.61 <.0001 0.05 Least Squares Means Effect trtmt time Lower Upper trtmt 1 18.6496 28.8464 trtmt 2 9.3296 19.5264 time 1 -6.9207 7.3207 time 2 12.8293 27.0707 time 3 16.4293 30.6707 time 4 21.5193 35.7607 time 8 15.9793 30.2207 Repeated Measures Design 8 The Mixed Procedure Differences of Least Squares Means Standard Effect trtmt time _trtmt _time Estimate Error DF t Value trtmt 1 2 9.3200 3.1267 8 2.98 time 1 2 -19.7500 4.9438 32 -3.99 time 1 3 -23.3500 4.9438 32 -4.72 time 1 4 -28.4400 4.9438 32 -5.75 time 1 8 -22.9000 4.9438 32 -4.63 time 2 3 -3.6000 4.9438 32 -0.73 time 2 4 -8.6900 4.9438 32 -1.76 time 2 8 -3.1500 4.9438 32 -0.64 time 3 4 -5.0900 4.9438 32 -1.03 time 3 8 0.4500 4.9438 32 0.09 time 4 8 5.5400 4.9438 32 1.12 Differences of Least Squares Means Effect trtmt time _trtmt _time Pr > |t| Adjustment Adj P Alpha trtmt 1 2 0.0176 Tukey 0.0176 0.05 time 1 2 0.0004 Tukey 0.0031 0.05 time 1 3 <.0001 Tukey 0.0004 0.05 time 1 4 <.0001 Tukey <.0001 0.05 time 1 8 <.0001 Tukey 0.0005 0.05 time 2 3 0.4718 Tukey 0.9483 0.05 time 2 4 0.0884 Tukey 0.4148 0.05 time 2 8 0.5286 Tukey 0.9678 0.05 time 3 4 0.3109 Tukey 0.8398 0.05 time 3 8 0.9280 Tukey 1.0000 0.05 time 4 8 0.2708 Tukey 0.7945 0.05 Differences of Least Squares Means Adj Adj Effect trtmt time _trtmt _time Lower Upper Lower Upper trtmt 1 2 2.1097 16.5303 2.1098 16.5302 time 1 2 -29.8202 -9.6798 -34.0347 -5.4653 time 1 3 -33.4202 -13.2798 -37.6347 -9.0653 time 1 4 -38.5102 -18.3698 -42.7247 -14.1553 time 1 8 -32.9702 -12.8298 -37.1847 -8.6153 time 2 3 -13.6702 6.4702 -17.8847 10.6847 time 2 4 -18.7602 1.3802 -22.9747 5.5947 time 2 8 -13.2202 6.9202 -17.4347 11.1347 time 3 4 -15.1602 4.9802 -19.3747 9.1947 time 3 8 -9.6202 10.5202 -13.8347 14.7347 time 4 8 -4.5302 15.6102 -8.7447 19.8247