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Program 14 TITLE '2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk'; OPTIONS LS=75 NODATE PAGENO=1; DATA ANCOVA; INPUT a $ b $ y x; LABEL a='degree of shock' b='race of conferderate' x='covariate-TASA score' Y='dependent score-change from pre-post'; datalines; 1 1 14 21 1 1 10 18 1 1 2 17 1 1 -2 18 1 1 6 17 1 2 0 16 1 2 -8 16 1 2 -4 15 1 2 8 19 1 2 4 17 2 1 -6 15 2 1 -2 17 2 1 -10 14 2 1 -18 13 2 1 -14 15 2 2 9 19 2 2 -7 14 2 2 5 18 2 2 1 15 2 2 -3 15 ; PROC REG; MODEL y=x; PLOT y*x; BY a b; TITLE2 'Regression of y on x'; RUN; PROC GLM; CLASS a b; MODEL y=x a b a*b/solution; MEANS a b a*b; LSMEANS a b a*b/E STDERR PDIFF; TITLE2 'ANCOVA analysis through TYPE III SS'; RUN; PROC GLM; CLASS a b; MODEL y=x a b a*b x*a*b; TITLE2 'Test of equal slopes: CRFAC-22'; RUN; 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 1 ---------------- degree of shock=1 race of conferderate=1 ----------------- Model: MODEL1 Dependent Variable: Y dependent score-change from pre-post Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 1 72.59259 72.59259 2.492 0.2126 Error 3 87.40741 29.13580 C Total 4 160.00000 Root MSE 5.39776 R-square 0.4537 Dep Mean 6.00000 Adj R-sq 0.2716 C.V. 89.96265 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -41.185185 29.99054873 -1.373 0.2633 X 1 2.592593 1.64248576 1.578 0.2126 Variable Variable DF Label INTERCEP 1 Intercept X 1 covariate-TASA score 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 2 Regression of y on x ---------------- degree of shock=1 race of conferderate=1 ----------------- |---|----|----|----|----|----|----|----|----|----|----|----|----|----| Y | | 14 + 1 + | | d | | e | | p | | e 12 + + n | | d | | e | | n | | t 10 + 1 + | | s | | c | | o | | r 8 + + e | | - | | c | | h | | a 6 + 1 + n | | g | | e | | | | f 4 + + r | | o | | m | | | | p 2 + 1 + r | | e | | - | | p | | o 0 + + s | | t | | | | | | -2 + 1 + |---|----|----|----|----|----|----|----|----|----|----|----|----|----| 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 covariate-TASA score X 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 3 ---------------- degree of shock=1 race of conferderate=2 ----------------- Model: MODEL1 Dependent Variable: Y dependent score-change from pre-post Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 1 111.30435 111.30435 6.857 0.0791 Error 3 48.69565 16.23188 C Total 4 160.00000 Root MSE 4.02888 R-square 0.6957 Dep Mean 0.00000 Adj R-sq 0.5942 C.V. . Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -57.739130 22.12298784 -2.610 0.0797 X 1 3.478261 1.32828281 2.619 0.0791 Variable Variable DF Label INTERCEP 1 Intercept X 1 covariate-TASA score 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 4 Regression of y on x ---------------- degree of shock=1 race of conferderate=2 ----------------- |---|----|----|----|----|----|----|----|----|----|----|----|----|----| Y | | 8 + 1 + | | d | | e | | p | | e 6 + + n | | d | | e | | n | | t 4 + 1 + | | s | | c | | o | | r 2 + + e | | - | | c | | h | | a 0 + 1 + n | | g | | e | | | | f -2 + + r | | o | | m | | | | p -4 + 1 + r | | e | | - | | p | | o -6 + + s | | t | | | | | | -8 + 1 + |---|----|----|----|----|----|----|----|----|----|----|----|----|----| 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 covariate-TASA score X 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 5 ---------------- degree of shock=2 race of conferderate=1 ----------------- Model: MODEL1 Dependent Variable: Y dependent score-change from pre-post Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 1 116.36364 116.36364 8.000 0.0663 Error 3 43.63636 14.54545 C Total 4 160.00000 Root MSE 3.81385 R-square 0.7273 Dep Mean -10.00000 Adj R-sq 0.6364 C.V. -38.13850 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -63.818182 19.10389169 -3.341 0.0444 X 1 3.636364 1.28564869 2.828 0.0663 Variable Variable DF Label INTERCEP 1 Intercept X 1 covariate-TASA score 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 6 Regression of y on x ---------------- degree of shock=2 race of conferderate=1 ----------------- |---|----|----|----|----|----|----|----|----|----|----|----|----|----| Y | | -2 + 1 + | | d | | e | | p | | e -4 + + n | | d | | e | | n | | t -6 + 1 + | | s | | c | | o | | r -8 + + e | | - | | c | | h | | a -10 + 1 + n | | g | | e | | | | f -12 + + r | | o | | m | | | | p -14 + 1 + r | | e | | - | | p | | o -16 + + s | | t | | | | | | -18 + 1 + |---|----|----|----|----|----|----|----|----|----|----|----|----|----| 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 covariate-TASA score X 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 7 ---------------- degree of shock=2 race of conferderate=2 ----------------- Model: MODEL1 Dependent Variable: Y dependent score-change from pre-post Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 1 143.82979 143.82979 26.684 0.0141 Error 3 16.17021 5.39007 C Total 4 160.00000 Root MSE 2.32165 R-square 0.8989 Dep Mean 1.00000 Adj R-sq 0.8652 C.V. 232.16526 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -43.808511 8.73619543 -5.015 0.0153 X 1 2.765957 0.53544925 5.166 0.0141 Variable Variable DF Label INTERCEP 1 Intercept X 1 covariate-TASA score 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 8 Regression of y on x ---------------- degree of shock=2 race of conferderate=2 ----------------- |---|----|----|----|----|----|----|----|----|----|----|----|----|----| Y | | | | | | d 10 + + e | | p | 1 | e | | n 8 + + d | | e | | n | | t 6 + + | | s | 1 | c | | o 4 + + r | | e | | - | | c 2 + + h | | a | 1 | n | | g 0 + + e | | | | f | | r -2 + + o | | m | 1 | | | p -4 + + r | | e | | - | | p -6 + + o | | s | 1 | t | | -8 + + | | | | |---|----|----|----|----|----|----|----|----|----|----|----|----|----| 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 covariate-TASA score X 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 9 ANCOVA analysis through TYPE III SS General Linear Models Procedure Class Level Information Class Levels Values A 2 1 2 B 2 1 2 Number of observations in data set = 20 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 10 ANCOVA analysis through TYPE III SS General Linear Models Procedure Dependent Variable: Y dependent score-change from pre-post Sum of Mean Source DF Squares Square F Value Pr > F Model 4 1109.38025 277.34506 20.36 0.0001 Error 15 204.36975 13.62465 Corrected Total 19 1313.75000 R-Square C.V. Root MSE Y Mean 0.844438 -492.1544 3.69116 -0.75000 Source DF Type I SS Mean Square F Value Pr > F X 1 1002.53493 1002.53493 73.58 0.0001 A 1 2.69215 2.69215 0.20 0.6630 B 1 40.66396 40.66396 2.98 0.1046 A*B 1 63.48922 63.48922 4.66 0.0475 Source DF Type III SS Mean Square F Value Pr > F X 1 435.630252 435.630252 31.97 0.0001 A 1 11.129110 11.129110 0.82 0.3804 B 1 39.229413 39.229413 2.88 0.1104 A*B 1 63.489216 63.489216 4.66 0.0475 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT -48.00840336 B -5.44 0.0001 8.82290727 X 3.02521008 5.65 0.0001 0.53500667 A 1 -2.21008403 B -0.94 0.3607 2.34428178 2 0.00000000 B . . . B 1 -6.76470588 B -2.76 0.0146 2.45170858 2 0.00000000 B . . . A*B 1 1 7.92436975 B 2.16 0.0475 3.67094118 1 2 0.00000000 B . . . 2 1 0.00000000 B . . . 2 2 0.00000000 B . . . NOTE: The X'X matrix has been found to be singular and a generalized inverse was used to solve the normal equations. Estimates followed by the letter 'B' are biased, and are not unique estimators of the parameters. 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 11 ANCOVA analysis through TYPE III SS General Linear Models Procedure Level of -------------Y------------- -------------X------------- A N Mean SD Mean SD 1 10 3.00000000 6.74948558 17.4000000 1.71269768 2 10 -4.50000000 8.31664997 15.5000000 1.90029238 Level of -------------Y------------- -------------X------------- B N Mean SD Mean SD 1 10 -2.00000000 10.3279556 16.5000000 2.32139805 2 10 0.50000000 5.9860950 16.4000000 1.77638835 Level of Level of ------------Y------------ ------------X------------ A B N Mean SD Mean SD 1 1 5 6.00000000 6.32455532 18.2000000 1.64316767 1 2 5 0.00000000 6.32455532 16.6000000 1.51657509 2 1 5 -10.00000000 6.32455532 14.8000000 1.48323970 2 2 5 1.00000000 6.32455532 16.2000000 2.16794834 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 12 ANCOVA analysis through TYPE III SS General Linear Models Procedure Least Squares Means Coefficients for A Least Square Means A 1 2 Effect Coefficients INTERCEPT 1 1 X 16.45 16.45 A 1 1 0 2 0 1 B 1 0.5 0.5 2 0.5 0.5 A*B 1 1 0.5 0 1 2 0.5 0 2 1 0 0.5 2 2 0 0.5 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 13 ANCOVA analysis through TYPE III SS General Linear Models Procedure Least Squares Means A Y Std Err Pr > |T| Pr > |T| H0: LSMEAN LSMEAN H0:LSMEAN=0 LSMEAN1=LSMEAN2 1 0.12605042 1.27310231 0.9224 0.3804 2 -1.62605042 1.27310231 0.2209 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 14 ANCOVA analysis through TYPE III SS General Linear Models Procedure Least Squares Means Coefficients for B Least Square Means B 1 2 Effect Coefficients INTERCEPT 1 1 X 16.45 16.45 A 1 0.5 0.5 2 0.5 0.5 B 1 1 0 2 0 1 A*B 1 1 0.5 0 1 2 0 0.5 2 1 0.5 0 2 2 0 0.5 B Y Std Err Pr > |T| Pr > |T| H0: LSMEAN LSMEAN H0:LSMEAN=0 LSMEAN1=LSMEAN2 1 -2.15126050 1.16755324 0.0853 0.1104 2 0.65126050 1.16755324 0.5852 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 15 ANCOVA analysis through TYPE III SS General Linear Models Procedure Least Squares Means Coefficients for A*B Least Square Means A 1 1 2 B 1 2 1 Effect Coefficients INTERCEPT 1 1 1 X 16.45 16.45 16.45 A 1 1 1 0 2 0 0 1 B 1 1 0 1 2 0 1 0 A*B 1 1 1 0 0 1 2 0 1 0 2 1 0 0 1 2 2 0 0 0 A 2 B 2 Effect Coefficients INTERCEPT 1 X 16.45 A 1 0 2 1 B 1 0 2 1 A*B 1 1 0 1 2 0 2 1 0 2 2 1 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 16 ANCOVA analysis through TYPE III SS General Linear Models Procedure Least Squares Means A B Y Std Err Pr > |T| LSMEAN LSMEAN LSMEAN H0:LSMEAN=0 Number 1 1 0.70588235 1.89776603 0.7151 1 1 2 -0.45378151 1.65268575 0.7874 2 2 1 -5.00840336 1.87195005 0.0173 3 2 2 1.75630252 1.65614597 0.3057 4 Pr > |T| H0: LSMEAN(i)=LSMEAN(j) i/j 1 2 3 4 1 . 0.6476 0.0726 0.6883 2 0.6476 . 0.0914 0.3607 3 0.0726 0.0914 . 0.0146 4 0.6883 0.3607 0.0146 . NOTE: To ensure overall protection level, only probabilities associated with pre-planned comparisons should be used. 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 17 Test of equal slopes: CRFAC-22 General Linear Models Procedure Class Level Information Class Levels Values A 2 1 2 B 2 1 2 Number of observations in data set = 20 2-WAY ANCOVA, Q.12 (p.748), Chapter 15 of Kirk 18 Test of equal slopes: CRFAC-22 General Linear Models Procedure Dependent Variable: Y dependent score-change from pre-post Sum of Mean Source DF Squares Square F Value Pr > F Model 7 1117.84036 159.69148 9.78 0.0004 Error 12 195.90964 16.32580 Corrected Total 19 1313.75000 R-Square C.V. Root MSE Y Mean 0.850878 -538.7360 4.04052 -0.75000 Source DF Type I SS Mean Square F Value Pr > F X 1 1002.53493 1002.53493 61.41 0.0001 A 1 2.69215 2.69215 0.16 0.6918 B 1 40.66396 40.66396 2.49 0.1405 A*B 1 63.48922 63.48922 3.89 0.0721 X*A*B 3 8.46011 2.82004 0.17 0.9128 Source DF Type III SS Mean Square F Value Pr > F X 1 422.638630 422.638630 25.89 0.0003 A 1 0.755252 0.755252 0.05 0.8333 B 1 0.119095 0.119095 0.01 0.9333 A*B 1 13.332579 13.332579 0.82 0.3839 X*A*B 3 8.460112 2.820037 0.17 0.9128 LINKS... Schedule Lectures Discussion Resources Comments: peng@indiana.edu Dr. Peng's Home Page: Dr. Chao-Ying Joanne Peng Copyright 1999, The Trustees of Indiana University