* air.sas, air velocity experiment, (D. Wilkie (1962), Applied Statistics, vol. 11, pp. 184-195); options ls=76 nodate; ; data air; input y A B; A2=A*A; B2=B*B; B3=B**3; B4=B**4; B5=B**5; AB=A*B; AB2=A*B2; AB3=A*B3; AB4=A*B4; A2B=A2*B; A2B2=A2*B2; A2B3=A2*B3; lines; -24 1 1 -23 1 2 1 1 3 8 1 4 29 1 5 23 1 6 33 2 1 28 2 2 45 2 3 57 2 4 74 2 5 80 2 6 37 3 1 79 3 2 79 3 3 95 3 4 101 3 5 111 3 6 ; proc print; var y A B; title 'Air Velocity Experiment (Wilkie, 1962)'; title2 'Raw data'; ; * standard ANOVA full model; proc glm; class A B; model y = A B A*B; title2 'Standard ANOVA full model'; ; * standard ANOVA main-effects model; proc glm; class A B; model y = A B; means A B / Tukey; title2 'Standard ANOVA main-effects model'; ; * plot the data; proc plot; plot y*B=A / vpos=27; title2 'Plot of raw data'; ; * regression model including some but not all interaction df; proc glm; model y = A A2 B B2 B3 B4 B5 AB AB2 AB3 AB4 A2B A2B2 A2B3; title2 'Regression model including all main effects and some interactions'; ; * same as regression model but using orthogonal polynomials; data air2; input Y A B Aln Aqd Bln Bqd Bcb Bqr Bqn; AlnBln = Aln*Bln; AlnBqd = Aln*Bqd; AlnBcb = Aln*Bcb; AlnBqr = Aln*Bqr; AqdBln = Aqd*Bln; AqdBqd = Aqd*Bqd; AqdBcb = Aqd*Bcb; lines; -24 1 1 -1 1 -5 5 -5 1 -1 -23 1 2 -1 1 -3 -1 7 -3 5 1 1 3 -1 1 -1 -4 4 2 -10 8 1 4 -1 1 1 -4 -4 2 10 29 1 5 -1 1 3 -1 -7 -3 -5 23 1 6 -1 1 5 5 5 1 1 33 2 1 0 -2 -5 5 -5 1 -1 28 2 2 0 -2 -3 -1 7 -3 5 45 2 3 0 -2 -1 -4 4 2 -10 57 2 4 0 -2 1 -4 -4 2 10 74 2 5 0 -2 3 -1 -7 -3 -5 80 2 6 0 -2 5 5 5 1 1 37 3 1 1 1 -5 5 -5 1 -1 79 3 2 1 1 -3 -1 7 -3 5 79 3 3 1 1 -1 -4 4 2 -10 95 3 4 1 1 1 -4 -4 2 10 101 3 5 1 1 3 -1 -7 -3 -5 111 3 6 1 1 5 5 5 1 1 ; proc print; title2 'Data and orthogonal polynomials for main effects'; ; proc glm; * omit the class statement; model Y = Aln Aqd Bln Bqd Bcb Bqr Bqn AlnBln AlnBqd AlnBcb AlnBqr AqdBln AqdBqd AqdBcb; title2 'Analysis using orthogonal polynomials'; ; proc glm; model Y = A B; title2 'Model suggested by EDA'; Air Velocity Experiment (Wilkie, 1962) 1 Raw data Obs y A B 1 -24 1 1 2 -23 1 2 3 1 1 3 4 8 1 4 5 29 1 5 6 23 1 6 7 33 2 1 8 28 2 2 9 45 2 3 10 57 2 4 11 74 2 5 12 80 2 6 13 37 3 1 14 79 3 2 15 79 3 3 16 95 3 4 17 101 3 5 18 111 3 6 Air Velocity Experiment (Wilkie, 1962) 2 Standard ANOVA full model The GLM Procedure Class Level Information Class Levels Values A 3 1 2 3 B 6 1 2 3 4 5 6 Number of Observations Read 18 Number of Observations Used 18 Air Velocity Experiment (Wilkie, 1962) 3 Standard ANOVA full model The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 17 28411.61111 1671.27124 . . Error 0 0.00000 . Corrected Total 17 28411.61111 R-Square Coeff Var Root MSE y Mean 1.000000 . . 46.27778 Source DF Type I SS Mean Square F Value Pr > F A 2 20232.11111 10116.05556 . . B 5 7386.94444 1477.38889 . . A*B 10 792.55556 79.25556 . . Source DF Type III SS Mean Square F Value Pr > F A 2 20232.11111 10116.05556 . . B 5 7386.94444 1477.38889 . . A*B 10 792.55556 79.25556 . . Air Velocity Experiment (Wilkie, 1962) 4 Standard ANOVA main-effects model The GLM Procedure Class Level Information Class Levels Values A 3 1 2 3 B 6 1 2 3 4 5 6 Number of Observations Read 18 Number of Observations Used 18 Air Velocity Experiment (Wilkie, 1962) 5 Standard ANOVA main-effects model The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 7 27619.05556 3945.57937 49.78 <.0001 Error 10 792.55556 79.25556 Corrected Total 17 28411.61111 R-Square Coeff Var Root MSE y Mean 0.972105 19.23722 8.902559 46.27778 Source DF Type I SS Mean Square F Value Pr > F A 2 20232.11111 10116.05556 127.64 <.0001 B 5 7386.94444 1477.38889 18.64 <.0001 Source DF Type III SS Mean Square F Value Pr > F A 2 20232.11111 10116.05556 127.64 <.0001 B 5 7386.94444 1477.38889 18.64 <.0001 Air Velocity Experiment (Wilkie, 1962) 6 Standard ANOVA main-effects model The GLM Procedure Tukey's Studentized Range (HSD) Test for y NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.05 Error Degrees of Freedom 10 Error Mean Square 79.25556 Critical Value of Studentized Range 3.87678 Minimum Significant Difference 14.09 Means with the same letter are not significantly different. Tukey Grouping Mean N A A 83.667 6 3 B 52.833 6 2 C 2.333 6 1 Air Velocity Experiment (Wilkie, 1962) 7 Standard ANOVA main-effects model The GLM Procedure Tukey's Studentized Range (HSD) Test for y NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.05 Error Degrees of Freedom 10 Error Mean Square 79.25556 Critical Value of Studentized Range 4.91202 Minimum Significant Difference 25.247 Means with the same letter are not significantly different. Tukey Grouping Mean N B A 71.333 3 6 A A 68.000 3 5 A B A 53.333 3 4 B B C 41.667 3 3 C D C 28.000 3 2 D D 15.333 3 1 Air Velocity Experiment (Wilkie, 1962) 8 Plot of raw data Plot of y*B. Symbol is value of A. y | 150 + | | | | | 3 100 + 3 | 3 | 2 | 3 3 2 | | 2 50 + | 2 |2 | 2 1 1 | | 1 0 + 1 | | |1 1 | | -50 + | -+-------------+-------------+-------------+-------------+-------------+ 1 2 3 4 5 6 B NOTE: 1 obs hidden. Air Velocity Experiment (Wilkie, 1962) 9 Regression model including all main effects and some interactions The GLM Procedure Number of Observations Read 18 Number of Observations Used 18 Air Velocity Experiment (Wilkie, 1962) 10 Regression model including all main effects and some interactions The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 14 28235.87235 2016.84803 34.43 0.0070 Error 3 175.73876 58.57959 Corrected Total 17 28411.61111 R-Square Coeff Var Root MSE y Mean 0.993815 16.53867 7.653730 46.27778 Source DF Type I SS Mean Square F Value Pr > F A 1 19845.33333 19845.33333 338.78 0.0003 A2 1 386.77778 386.77778 6.60 0.0825 B 1 7262.97619 7262.97619 123.98 0.0016 B2 1 65.01587 65.01587 1.11 0.3695 B3 1 36.29630 36.29630 0.62 0.4887 B4 1 13.76190 13.76190 0.23 0.6611 B5 1 8.89418 8.89418 0.15 0.7228 AB 1 20.82857 20.82857 0.36 0.5930 AB2 1 47.14881 47.14881 0.80 0.4358 AB3 1 265.22500 265.22500 4.53 0.1233 AB4 1 33.01786 33.01786 0.56 0.5073 A2B 1 15.23810 15.23810 0.26 0.6452 A2B2 1 170.33532 170.33532 2.91 0.1867 A2B3 1 65.02315 65.02315 1.11 0.3694 Source DF Type III SS Mean Square F Value Pr > F A 1 163.0890675 163.0890675 2.78 0.1938 A2 1 245.3347578 245.3347578 4.19 0.1332 B 1 4.3115592 4.3115592 0.07 0.8038 B2 1 13.2567698 13.2567698 0.23 0.6668 B3 1 14.4770098 14.4770098 0.25 0.6532 B4 1 11.8930963 11.8930963 0.20 0.6829 B5 1 8.8941799 8.8941799 0.15 0.7228 AB 1 31.3990268 31.3990268 0.54 0.5171 AB2 1 3.8264618 3.8264618 0.07 0.8148 AB3 1 4.9751694 4.9751694 0.08 0.7897 AB4 1 33.0178571 33.0178571 0.56 0.5073 A2B 1 134.4741323 134.4741323 2.30 0.2270 A2B2 1 92.6559525 92.6559525 1.58 0.2975 A2B3 1 65.0231481 65.0231481 1.11 0.3694 Air Velocity Experiment (Wilkie, 1962) 11 Regression model including all main effects and some interactions The GLM Procedure Dependent Variable: y Standard Parameter Estimate Error t Value Pr > |t| Intercept -113.0555557 199.0460407 -0.57 0.6098 A 238.9166667 143.1881802 1.67 0.1938 A2 -69.1666667 33.7979573 -2.05 0.1332 B -93.3709874 344.1659675 -0.27 0.8038 B2 106.1524469 223.1435302 0.48 0.6668 B3 -34.5077160 69.4144565 -0.50 0.6532 B4 4.6458333 10.3107274 0.45 0.6829 B5 -0.2277778 0.5845633 -0.39 0.7228 AB -126.1620370 172.3231982 -0.73 0.5171 AB2 16.2941468 63.7538011 0.26 0.8148 AB3 2.7939815 9.5872209 0.29 0.7897 AB4 -0.4479167 0.5966174 -0.75 0.5073 A2B 58.3465608 38.5096024 1.52 0.2270 A2B2 -15.4980159 12.3228904 -1.26 0.2975 A2B3 1.2268519 1.1644780 1.05 0.3694 Air Velocity Experiment (Wilkie, 1962) 12 Data and orthogonal polynomials for main effects Aln Aln Aln Aln Aqd Aqd Aqd Obs Y A B Aln Aqd Bln Bqd Bcb Bqr Bqn Bln Bqd Bcb Bqr Bln Bqd Bcb 1 -24 1 1 -1 1 -5 5 -5 1 -1 5 -5 5 -1 -5 5 -5 2 -23 1 2 -1 1 -3 -1 7 -3 5 3 1 -7 3 -3 -1 7 3 1 1 3 -1 1 -1 -4 4 2 -10 1 4 -4 -2 -1 -4 4 4 8 1 4 -1 1 1 -4 -4 2 10 -1 4 4 -2 1 -4 -4 5 29 1 5 -1 1 3 -1 -7 -3 -5 -3 1 7 3 3 -1 -7 6 23 1 6 -1 1 5 5 5 1 1 -5 -5 -5 -1 5 5 5 7 33 2 1 0 -2 -5 5 -5 1 -1 0 0 0 0 10 -10 10 8 28 2 2 0 -2 -3 -1 7 -3 5 0 0 0 0 6 2 -14 9 45 2 3 0 -2 -1 -4 4 2 -10 0 0 0 0 2 8 -8 10 57 2 4 0 -2 1 -4 -4 2 10 0 0 0 0 -2 8 8 11 74 2 5 0 -2 3 -1 -7 -3 -5 0 0 0 0 -6 2 14 12 80 2 6 0 -2 5 5 5 1 1 0 0 0 0 -10 -10 -10 13 37 3 1 1 1 -5 5 -5 1 -1 -5 5 -5 1 -5 5 -5 14 79 3 2 1 1 -3 -1 7 -3 5 -3 -1 7 -3 -3 -1 7 15 79 3 3 1 1 -1 -4 4 2 -10 -1 -4 4 2 -1 -4 4 16 95 3 4 1 1 1 -4 -4 2 10 1 -4 -4 2 1 -4 -4 17 101 3 5 1 1 3 -1 -7 -3 -5 3 -1 -7 -3 3 -1 -7 18 111 3 6 1 1 5 5 5 1 1 5 5 5 1 5 5 5 Air Velocity Experiment (Wilkie, 1962) 13 Analysis using orthogonal polynomials The GLM Procedure Number of Observations Read 18 Number of Observations Used 18 Air Velocity Experiment (Wilkie, 1962) 14 Analysis using orthogonal polynomials The GLM Procedure Dependent Variable: Y Sum of Source DF Squares Mean Square F Value Pr > F Model 14 28235.87235 2016.84803 34.43 0.0070 Error 3 175.73876 58.57959 Corrected Total 17 28411.61111 R-Square Coeff Var Root MSE Y Mean 0.993815 16.53867 7.653730 46.27778 Source DF Type I SS Mean Square F Value Pr > F Aln 1 19845.33333 19845.33333 338.78 0.0003 Aqd 1 386.77778 386.77778 6.60 0.0825 Bln 1 7262.97619 7262.97619 123.98 0.0016 Bqd 1 65.01587 65.01587 1.11 0.3695 Bcb 1 36.29630 36.29630 0.62 0.4887 Bqr 1 13.76190 13.76190 0.23 0.6611 Bqn 1 8.89418 8.89418 0.15 0.7228 AlnBln 1 20.82857 20.82857 0.36 0.5930 AlnBqd 1 47.14881 47.14881 0.80 0.4358 AlnBcb 1 265.22500 265.22500 4.53 0.1233 AlnBqr 1 33.01786 33.01786 0.56 0.5073 AqdBln 1 15.23810 15.23810 0.26 0.6452 AqdBqd 1 170.33532 170.33532 2.91 0.1867 AqdBcb 1 65.02315 65.02315 1.11 0.3694 Source DF Type III SS Mean Square F Value Pr > F Aln 1 19845.33333 19845.33333 338.78 0.0003 Aqd 1 386.77778 386.77778 6.60 0.0825 Bln 1 7262.97619 7262.97619 123.98 0.0016 Bqd 1 65.01587 65.01587 1.11 0.3695 Bcb 1 36.29630 36.29630 0.62 0.4887 Bqr 1 13.76190 13.76190 0.23 0.6611 Bqn 1 8.89418 8.89418 0.15 0.7228 AlnBln 1 20.82857 20.82857 0.36 0.5930 AlnBqd 1 47.14881 47.14881 0.80 0.4358 AlnBcb 1 265.22500 265.22500 4.53 0.1233 AlnBqr 1 33.01786 33.01786 0.56 0.5073 AqdBln 1 15.23810 15.23810 0.26 0.6452 AqdBqd 1 170.33532 170.33532 2.91 0.1867 AqdBcb 1 65.02315 65.02315 1.11 0.3694 Air Velocity Experiment (Wilkie, 1962) 15 Analysis using orthogonal polynomials The GLM Procedure Dependent Variable: Y Standard Parameter Estimate Error t Value Pr > |t| Intercept 46.27777778 1.80400150 25.65 0.0001 Aln 40.66666667 2.20944159 18.41 0.0003 Aqd -3.27777778 1.27562170 -2.57 0.0825 Bln 5.88095238 0.52815756 11.13 0.0016 Bqd -0.50793651 0.48213968 -1.05 0.3695 Bcb -0.25925926 0.32936411 -0.79 0.4887 Bqr -0.40476190 0.83509043 -0.48 0.6611 Bqn -0.10846561 0.27836348 -0.39 0.7228 AlnBln 0.38571429 0.64685826 0.60 0.5930 AlnBqd -0.52976190 0.59049810 -0.90 0.4358 AlnBcb 0.85833333 0.40338700 2.13 0.1233 AlnBqr -0.76785714 1.02277272 -0.75 0.5073 AqdBln 0.19047619 0.37346379 0.51 0.6452 AqdBqd -0.58134921 0.34092424 -1.71 0.1867 AqdBcb 0.24537037 0.23289559 1.05 0.3694 Air Velocity Experiment (Wilkie, 1962) 16 Model suggested by EDA The GLM Procedure Number of Observations Read 18 Number of Observations Used 18 Air Velocity Experiment (Wilkie, 1962) 17 Model suggested by EDA The GLM Procedure Dependent Variable: Y Sum of Source DF Squares Mean Square F Value Pr > F Model 2 27108.30952 13554.15476 156.00 <.0001 Error 15 1303.30159 86.88677 Corrected Total 17 28411.61111 R-Square Coeff Var Root MSE Y Mean 0.954128 20.14208 9.321307 46.27778 Source DF Type I SS Mean Square F Value Pr > F A 1 19845.33333 19845.33333 228.40 <.0001 B 1 7262.97619 7262.97619 83.59 <.0001 Source DF Type III SS Mean Square F Value Pr > F A 1 19845.33333 19845.33333 228.40 <.0001 B 1 7262.97619 7262.97619 83.59 <.0001 Standard Parameter Estimate Error t Value Pr > |t| Intercept -76.22222222 7.35274647 -10.37 <.0001 A 40.66666667 2.69082968 15.11 <.0001 B 11.76190476 1.28646265 9.14 <.0001