MTB > read 'a:\351\brogkut.dat' c1-c4 Entering data from file: a:\351\brogkut.dat 42 rows read. MTB > name c1 'method' MTB > name c2 'prepost' MTB > name c3 'outcome' MTB > name c4 'subject' MTB > info Column Name Count C1 method 42 C2 prepost 42 C3 outcome 42 C4 subject 42 REPEATED MEASURES ANALYSIS MTB > glm outcome = subject(method) + method|prepost; SUBC> random subject; SUBC> ems; SUBC> means method|prepost. General Linear Model Factor Type Levels Values subject(method) random 21 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 method fixed 2 1 2 prepost fixed 2 1 2 Analysis of Variance for outcome, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P subject(method) 19 4440.00 4440.00 233.68 6.52 0.000 method 1 847.48 847.48 847.48 3.63 0.072 prepost 1 542.88 317.69 317.69 8.86 0.008 method*prepost 1 407.41 407.41 407.41 11.36 0.003 Error 19 681.21 681.21 35.85 Total 41 6918.98 Unusual Observations for outcome Obs outcome Fit StDev Fit Residual St Resid 3 35.0000 44.6250 4.4908 -9.6250 -2.43R 4 55.0000 45.3750 4.4908 9.6250 2.43R R denotes an observation with a large standardized residual. Expected Mean Squares, using Adjusted SS Source Expected Mean Square for Each Term 1 subject(method) (5) + 2.0000(1) 2 method (5) + 2.0000(1) + Q[2, 4] 3 prepost (5) + Q[3, 4] 4 method*prepost (5) + Q[4] 5 Error (5) Error Terms for Tests, using Adjusted SS Source Error DF Error MS Synthesis of Error MS 1 subject(method) 19.00 35.85 (5) 2 method 19.00 233.68 (1) 3 prepost 19.00 35.85 (5) 4 method*prepost 19.00 35.85 (5) Variance Components, using Adjusted SS Source Estimated Value subject(method) 98.92 Error 35.85 Least Squares Means for outcome method Mean StDev 1 46.75 3.822 2 37.50 2.998 prepost 1 44.96 1.345 2 39.29 1.345 method*prepost 1 1 46.38 2.117 1 2 47.13 2.117 2 1 43.54 1.661 2 2 31.46 1.661 BROGAN-KUTNER Section 5 equivalences MTB > Unstack 'outcome' c60-c61; SUBC> Subscripts 'prepost'. MTB > print c60-c61 Data Display Row C60 C61 1 51 48 2 35 55 3 66 60 4 40 35 5 39 36 6 46 43 7 52 46 8 42 54 9 34 16 10 40 36 11 34 16 12 36 18 13 38 32 14 32 14 15 44 20 16 50 43 17 60 45 18 63 67 19 50 36 20 42 34 21 43 32 MTB > Set c62 DATA> 1( 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 )1 DATA> End. MTB > print c60-c62 Data Display Row C60 C61 C62 1 51 48 1 2 35 55 1 3 66 60 1 4 40 35 1 5 39 36 1 6 46 43 1 7 52 46 1 8 42 54 1 9 34 16 2 10 40 36 2 11 34 16 2 12 36 18 2 13 38 32 2 14 32 14 2 15 44 20 2 16 50 43 2 17 60 45 2 18 63 67 2 19 50 36 2 20 42 34 2 21 43 32 2 ===Item #1 MTB > let c63=c60+c61 MTB > twot c63 c62 Two Sample T-Test and Confidence Interval Two sample T for C63 C62 N Mean StDev SE Mean 1 8 93.5 16.2 5.7 2 13 75.0 24.2 6.7 95% CI for mu (1) - mu (2): ( -0.0, 37.0) T-Test mu (1) = mu (2) (vs not =): T = 2.10 P = 0.050 DF = 18 MTB > twot c63 c62; SUBC> pooled. Two Sample T-Test and Confidence Interval Two sample T for C63 C62 N Mean StDev SE Mean 1 8 93.5 16.2 5.7 2 13 75.0 24.2 6.7 95% CI for mu (1) - mu (2): ( -1.8, 38.8) T-Test mu (1) = mu (2) (vs not =): T = 1.90 P = 0.072 DF = 19 Both use Pooled StDev = 21.6 ==Item #2 MTB > let c64 = c61 - c60 MTB > twot c64 c62; SUBC> pooled. Two Sample T-Test and Confidence Interval Two sample T for C64 C62 N Mean StDev SE Mean 1 8 0.75 9.74 3.4 2 13 -12.08 7.63 2.1 95% CI for mu (1) - mu (2): ( 4.9, 20.8) T-Test mu (1) = mu (2) (vs not =): T = 3.37 P = 0.0032 DF = 19 Both use Pooled StDev = 8.47 ===Item #3===== MTB > ttest c64 T-Test of the Mean Test of mu = 0.00 vs mu not = 0.00 Variable N Mean StDev SE Mean T P C64 21 -7.19 10.43 2.28 -3.16 0.0049 MTB > REPEATED MEASURES ANOVA the long way ============let's start an analysis here============= ==First part gives us MTB > glm c3 = c2|c1 Factor Levels Values prepost 2 1 2 method 2 1 2 Analysis of Variance for outcome Source DF Seq SS Adj SS Adj MS F P prepost 1 542.9 317.7 317.7 2.36 0.133 = repeated measure (Within subj part) method 1 847.5 847.5 847.5 6.29 0.017 = Group (Factor A) (Betwen subjects part) prepost*method 1 407.4 407.4 407.4 3.02 0.090 = GroupxRepeated measure Interaction (Within subjects part) Error 38 5121.2 5121.2 134.8 Total 41 6919.0 Unusual Observations for outcome Obs. outcome Fit Stdev.Fit Residual St.Resid 36 67.0000 31.4615 3.2198 35.5385 3.19R R denotes an obs. with a large st. resid. ================== ====Now create the within-groups anova to obtain the 2 error terms MTB > copy c2-c4 c5-c7; SUBC> use c1=1. MTB > twoway c6 c5 c7 ANALYSIS OF VARIANCE C6 SOURCE DF SS MS C5 1 2.2 2.2 C7 7 915.0 130.7 = piece of subjects within groups Between sujects error term ERROR 7 331.7 47.4 = piece of subjectsxrepeated measure within group interaction Within subjects error term TOTAL 15 1249.0 MTB > copy c2-c4 c8-c10; SUBC> use c1=2. MTB > twoway c9 c8 c10 ANALYSIS OF VARIANCE C9 SOURCE DF SS MS C8 1 948.0 948.0 C10 12 3525.0 293.7 = piece of subjects within groups Between sujects error term ERROR 12 349.5 29.1 = piece of subjectsxrepeated measure within group interaction Within subjects error term TOTAL 25 4822.5 915 + 3525 = 4440 (and 7 + 12 = 19df) Between sujects error term 331.7 + 349.5 = 681.2 (and 7 + 12 = 19df) Within subjects error term ================== SAS implementation data brogk; input m1 m2 grp; datalines; 51 48 1 35 55 1 66 60 1 40 35 1 39 36 1 46 43 1 52 46 1 42 54 1 34 16 2 40 36 2 34 16 2 36 18 2 38 32 2 32 14 2 44 20 2 50 43 2 60 45 2 63 67 2 50 36 2 42 34 2 43 32 2 ; proc glm data=brogk; class grp; model m1--m2 = grp /nouni; repeated Time 2 (1 2) / summary printe; run; OUTPUT (selected) The SAS System 16:13 Tuesday, May 16, 2000 35 The GLM Procedure Repeated Measures Analysis of Variance Tests of Hypotheses for Between Subjects Effects Source DF Type III SS Mean Square F Value Pr > F grp 1 847.476190 847.476190 3.63 0.0721 Error 19 4440.000000 233.684211 The GLM Procedure Repeated Measures Analysis of Variance Univariate Tests of Hypotheses for Within Subject Effects Source DF Type III SS Mean Square F Value Pr > F Time 1 317.6932234 317.6932234 8.86 0.0078 Time*grp 1 407.4075092 407.4075092 11.36 0.0032 Error(Time) 19 681.2115385 35.8532389