A 2_{III}^{7-4} design with 8 observations - YOUR NAME 1 We can extend the design to get more information by a second replication with `foldover' in column D Times for riding a bicycle up a hill with High/Low settings of A=Seat B=Dynamo C=Handlebars D=Gear E=Raincoat F=Breakfast G=Tires 22:38 Wednesday, February 20, 2008 See Table 6.4 p245 and Table 6.8-6.9 p250-251 THE DATA AS SAS SEES IT D E F G are functions of A B C The only change in the second replication is D=-D Row Rep A B C D E F G Yield 1 1 -1 -1 -1 1 1 1 -1 69 2 1 1 -1 -1 -1 -1 1 1 52 3 1 -1 1 -1 -1 1 -1 1 60 4 1 1 1 -1 1 -1 -1 -1 83 5 1 -1 -1 1 1 -1 -1 1 71 6 1 1 -1 1 -1 1 -1 -1 50 7 1 -1 1 1 -1 -1 1 -1 59 8 1 1 1 1 1 1 1 1 88 9 2 -1 -1 -1 -1 1 1 -1 47 10 2 1 -1 -1 1 -1 1 1 74 11 2 -1 1 -1 1 1 -1 1 84 12 2 1 1 -1 -1 -1 -1 -1 62 13 2 -1 -1 1 -1 -1 -1 1 53 14 2 1 -1 1 1 1 -1 -1 78 15 2 -1 1 1 1 -1 1 -1 87 16 2 1 1 1 -1 1 1 1 60 A 2_{III}^{7-4} design with 8 observations - YOUR NAME 2 ANALYSIS OF THE FIRST REPLICATION (A 2_{III}^{7-4} DESIGN): Only B and D are large in the first replication. This suggests that only B and D are active However, the 2_{III}^{7-3} design has confounding relations A=A+BD+CE+FG B=B+AD+CF+EG C=C+AE+BF+DG, D=D+AB+CG+EF E=E+AC+BG+DF F=F+AG+BC+DE G=G+AF+BE+CD So, for example, B could be small but AD large The second (foldover) replication will resolve these issues USING PROC REG TO ANALYZE BOTH REPLICATIONS INDIVIDUALLY: 22:38 Wednesday, February 20, 2008 Rep=1 The REG Procedure Model: MODEL1 Dependent Variable: Yield Number of Observations Read 8 Number of Observations Used 8 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 1342.00000 191.71429 . . Error 0 0 . Corrected Total 7 1342.00000 Root MSE . R-Square 1.0000 Dependent Mean 66.50000 Adj R-Sq . Coeff Var . Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 66.50000 . . . A Seat 1 1.75000 . . . B Dynamo 1 6.00000 . . . C Handlebars 1 0.50000 . . . D Gear 1 11.25000 . . . E Raincoat 1 0.25000 . . . F Breakfast 1 0.50000 . . . G Tires 1 1.25000 . . . A 2_{III}^{7-4} design with 8 observations - YOUR NAME 3 ANALYSIS OF THE FIRST REPLICATION (A 2_{III}^{7-4} DESIGN): Only B and D are large in the first replication. This suggests that only B and D are active However, the 2_{III}^{7-3} design has confounding relations A=A+BD+CE+FG B=B+AD+CF+EG C=C+AE+BF+DG, D=D+AB+CG+EF E=E+AC+BG+DF F=F+AG+BC+DE G=G+AF+BE+CD So, for example, B could be small but AD large The second (foldover) replication will resolve these issues USING PROC REG TO ANALYZE BOTH REPLICATIONS INDIVIDUALLY: 22:38 Wednesday, February 20, 2008 Rep=2 The REG Procedure Model: MODEL1 Dependent Variable: Yield Number of Observations Read 8 Number of Observations Used 8 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 1518.87500 216.98214 . . Error 0 0 . Corrected Total 7 1518.87500 Root MSE . R-Square 1.0000 Dependent Mean 68.12500 Adj R-Sq . Coeff Var . Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 68.12500 . . . A Seat 1 0.37500 . . . B Dynamo 1 5.12500 . . . C Handlebars 1 1.37500 . . . D Gear 1 12.62500 . . . E Raincoat 1 -0.87500 . . . F Breakfast 1 -1.12500 . . . G Tires 1 -0.37500 . . . A 2_{III}^{7-4} design with 8 observations - YOUR NAME 4 2^2 DESIGN FOR B AND D USING THE FIRST REPLICATION Rep1: B and D are significant, but BD is not 22:38 Wednesday, February 20, 2008 The GLM Procedure Class Level Information Class Levels Values B 2 -1 1 D 2 -1 1 Number of Observations Read 8 Number of Observations Used 8 A 2_{III}^{7-4} design with 8 observations - YOUR NAME 5 2^2 DESIGN FOR B AND D USING THE FIRST REPLICATION Rep1: B and D are significant, but BD is not 22:38 Wednesday, February 20, 2008 The GLM Procedure Dependent Variable: Yield Sum of Source DF Squares Mean Square F Value Pr > F Model 3 1325.000000 441.666667 103.92 0.0003 Error 4 17.000000 4.250000 Corrected Total 7 1342.000000 R-Square Coeff Var Root MSE Yield Mean 0.987332 3.100079 2.061553 66.50000 Source DF Type I SS Mean Square F Value Pr > F B 1 288.000000 288.000000 67.76 0.0012 D 1 1012.500000 1012.500000 238.24 0.0001 B*D 1 24.500000 24.500000 5.76 0.0743 Source DF Type III SS Mean Square F Value Pr > F B 1 288.000000 288.000000 67.76 0.0012 D 1 1012.500000 1012.500000 238.24 0.0001 B*D 1 24.500000 24.500000 5.76 0.0743 A 2_{III}^{7-4} design with 8 observations - YOUR NAME 6 PROC REG FOR 16X16 DESIGN MATRIX WITH 22:38 Wednesday, February 20, 2008 2-WAY ITERACTIONS WITH REPLICATION NUMBER NOTE THAT REP*A IS ESSENTIALLY B*D The REG Procedure Model: MODEL1 Dependent Variable: Yield Number of Observations Read 16 Number of Observations Used 16 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 15 2871.43750 191.42917 . . Error 0 0 . Corrected Total 15 2871.43750 Root MSE . R-Square 1.0000 Dependent Mean 67.31250 Adj R-Sq . Coeff Var . Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 67.31250 . . . A Seat 1 1.06250 . . . B Dynamo 1 5.56250 . . . C Handlebars 1 0.93750 . . . D Gear 1 11.93750 . . . E Raincoat 1 -0.31250 . . . F Breakfast 1 -0.31250 . . . G Tires 1 0.43750 . . . SRep 1 0.81250 . . . SRepA B*D 1 -0.68750 . . . SRepB A*D 1 -0.43750 . . . SRepC D*G 1 0.43750 . . . SRepD A*B,E*F,C*G 1 0.68750 . . . SRepE D*F 1 -0.56250 . . . SRepF D*E 1 -0.81250 . . . SRepG C*D 1 -0.81250 . . . A 2_{III}^{7-4} design with 8 observations - YOUR NAME 7 SORTED LIST OF PARAMETER ESIMATES for 16x16 model (See Table 6.5 p245 and Table 6.8b p250) 22:38 Wednesday, February 20, 2008 Abs Obs Effect _LABEL_ PARMS Bookparms Parms 1 D Gear 11.9375 23.875 11.9375 2 B Dynamo 5.5625 11.125 5.5625 3 A Seat 1.0625 2.125 1.0625 4 C Handlebars 0.9375 1.875 0.9375 5 SRep 0.8125 1.625 0.8125 6 SRepF D*E -0.8125 -1.625 0.8125 7 SRepG C*D -0.8125 -1.625 0.8125 8 SRepA B*D -0.6875 -1.375 0.6875 9 SRepD A*B,E*F,C*G 0.6875 1.375 0.6875 10 SRepE D*F -0.5625 -1.125 0.5625 11 G Tires 0.4375 0.875 0.4375 12 SRepB A*D -0.4375 -0.875 0.4375 13 SRepC D*G 0.4375 0.875 0.4375 14 E Raincoat -0.3125 -0.625 0.3125 15 F Breakfast -0.3125 -0.625 0.3125 A 2_{III}^{7-4} design with 8 observations - YOUR NAME 8 CALCULATE AVERAGES AND SEMI-DIFFERENCES DIRECTLY ESTIMATES FROM PROC REG FOR TWO REPLICATIONS Two sets of estimated values in two long rows 22:38 Wednesday, February 20, 2008 Obs Rep _MODEL_ _TYPE_ _DEPVAR_ _RMSE_ Intercept A 1 1 MODEL1 PARMS Yield . 66.500 1.750 2 2 MODEL1 PARMS Yield . 68.125 0.375 Obs B C D E F G Yield 1 6.000 0.500 11.250 0.250 0.500 1.250 -1 2 5.125 1.375 12.625 -0.875 -1.125 -0.375 -1 A 2_{III}^{7-4} design with 8 observations - YOUR NAME 9 CALCULATE AVERAGES AND SEMI-DIFFERENCES DIRECTLY PARAMETER AND BOOK PARAMETER VALUES IN TWO COLUMNS (Same as Table 6.5 p245 and Table 6.8b p250) 22:38 Wednesday, February 20, 2008 Obs Effect _LABEL_ Parms1 Parms2 Book1 Book2 1 A Seat 1.75 0.375 3.5 0.75 2 B Dynamo 6.00 5.125 12.0 10.25 3 C Handlebars 0.50 1.375 1.0 2.75 4 D Gear 11.25 12.625 22.5 25.25 5 E Raincoat 0.25 -0.875 0.5 -1.75 6 F Breakfast 0.50 -1.125 1.0 -2.25 7 G Tires 1.25 -0.375 2.5 -0.75 A 2_{III}^{7-4} design with 8 observations - YOUR NAME 10 Averages (except for D) are unconfounded with 2-way interactions with D. 22:38 Wednesday, February 20, 2008 Semi-Df are interactions with D that are unconfounded with all other main and 2-way effects. These suggest that interactions with D are unimportant In particular, factors B and D appear to be active Note two apparent typos in Tables 6.8b and 6.9 in text Effect _LABEL_ Book1 Book2 Average SemiDf A Seat 3.5 0.75 2.125 1.375 B Dynamo 12.0 10.25 11.125 0.875 C Handlebars 1.0 2.75 1.875 -0.875 D Gear 22.5 25.25 23.875 -1.375 E Raincoat 0.5 -1.75 -0.625 1.125 F Breakfast 1.0 -2.25 -0.625 1.625 G Tires 2.5 -0.75 0.875 1.625 A 2_{III}^{7-4} design with 8 observations - YOUR NAME 11 FULL ANALYSIS OF 2^2 DESIGN (4 OBSERVATIONS/CELL) FOR B AND D B and D are significant, but BD is not 22:38 Wednesday, February 20, 2008 The REG Procedure Model: MODEL1 Dependent Variable: Yield Number of Observations Read 16 Number of Observations Used 16 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 2782.68750 927.56250 125.42 <.0001 Error 12 88.75000 7.39583 Corrected Total 15 2871.43750 Root MSE 2.71953 R-Square 0.9691 Dependent Mean 67.31250 Adj R-Sq 0.9614 Coeff Var 4.04015 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 67.31250 0.67988 99.01 <.0001 B Dynamo 1 5.56250 0.67988 8.18 <.0001 D Gear 1 11.93750 0.67988 17.56 <.0001 BD 1 0.68750 0.67988 1.01 0.3319 A 2_{III}^{7-4} design with 8 observations - YOUR NAME 12 ANALYSIS OF 2^3 DESIGN (2 OBSERVATIONS/CELL) FOR A B D Again, only B and D are significant 22:38 Wednesday, February 20, 2008 The REG Procedure Model: MODEL1 Dependent Variable: Yield Number of Observations Read 16 Number of Observations Used 16 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 7 2821.93750 403.13393 65.15 <.0001 Error 8 49.50000 6.18750 Corrected Total 15 2871.43750 Root MSE 2.48747 R-Square 0.9828 Dependent Mean 67.31250 Adj R-Sq 0.9677 Coeff Var 3.69540 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 67.31250 0.62187 108.24 <.0001 A Seat 1 1.06250 0.62187 1.71 0.1259 B Dynamo 1 5.56250 0.62187 8.94 <.0001 D Gear 1 11.93750 0.62187 19.20 <.0001 AB 1 -0.68750 0.62187 -1.11 0.3011 AD 1 0.43750 0.62187 0.70 0.5017 BD 1 0.68750 0.62187 1.11 0.3011 ABD 1 -0.81250 0.62187 -1.31 0.2277