Optimal Design of Experiments: A Case Study Approach

Vif: intercept, x2, x3 and x2 x3p. 41

Book 1.25Data 1.12

Table 2.11's runs give 12 × 3/32 = 1.125: the runs at the four combinations of x2 and x3 number 4, 4, 2 and 2, and the intercept, x2, x3 and x2 x3, orthogonal to the other four columns, each have the variance (1/16)(1/4 + 1/4 + 1/2 + 1/2) = 3/32 in units of σ². A coordinate exchange from 3,000 random starts finds no 12-run design with a larger |X'X| for this model than Table 2.11's 339,738,624, and that best design's VIFs are 1.125 too

Vif: Interceptp. 42

Book 1.25Data 1.12

Table 2.11's runs give 12 × 3/32 = 1.125: the runs at the four combinations of x2 and x3 number 4, 4, 2 and 2, and the intercept, x2, x3 and x2 x3, orthogonal to the other four columns, each have the variance (1/16)(1/4 + 1/4 + 1/2 + 1/2) = 3/32 in units of σ². A coordinate exchange from 3,000 random starts finds no 12-run design with a larger |X'X| for this model than Table 2.11's 339,738,624, and that best design's VIFs are 1.125 too

Vif: Ethanolp. 42

Book 1.25Data 1.12

Table 2.11's runs give 12 × 3/32 = 1.125: the runs at the four combinations of x2 and x3 number 4, 4, 2 and 2, and the intercept, x2, x3 and x2 x3, orthogonal to the other four columns, each have the variance (1/16)(1/4 + 1/4 + 1/2 + 1/2) = 3/32 in units of σ². A coordinate exchange from 3,000 random starts finds no 12-run design with a larger |X'X| for this model than Table 2.11's 339,738,624, and that best design's VIFs are 1.125 too

Vif: Propanolp. 42

Book 1.25Data 1.12

Table 2.11's runs give 12 × 3/32 = 1.125: the runs at the four combinations of x2 and x3 number 4, 4, 2 and 2, and the intercept, x2, x3 and x2 x3, orthogonal to the other four columns, each have the variance (1/16)(1/4 + 1/4 + 1/2 + 1/2) = 3/32 in units of σ². A coordinate exchange from 3,000 random starts finds no 12-run design with a larger |X'X| for this model than Table 2.11's 339,738,624, and that best design's VIFs are 1.125 too

Vif: Ethanol × Propanolp. 42

Book 1.25Data 1.12

Table 2.11's runs give 12 × 3/32 = 1.125: the runs at the four combinations of x2 and x3 number 4, 4, 2 and 2, and the intercept, x2, x3 and x2 x3, orthogonal to the other four columns, each have the variance (1/16)(1/4 + 1/4 + 1/2 + 1/2) = 3/32 in units of σ². A coordinate exchange from 3,000 random starts finds no 12-run design with a larger |X'X| for this model than Table 2.11's 339,738,624, and that best design's VIFs are 1.125 too

Rmse: the reduced modelp. 58

Book >1.5Data 1.06662

The reduced model of Table 3.7, fitted to the 20 runs, has an RMSE of 1.07; the RMSE above one and a half is that of the model fitted to the 12 original runs (Table 3.6), 1.56

Xtx: the augmented design, Equation 3.5, the block's rowp. 64

The block's row of X'X is the products of the block column with each column of Equation 3.13, which that equation prints: 4 2 2 -2 2 2 0 0 2 0 2 -2 2 20; the first entry is the 12 original runs at +1 less the 8 follow-up runs at -1. The printed row and column differ from these in every entry but the diagonal, by 2

Determinant: the augmented design, Equation 3.5p. 64

Book 6.5541e17Data 6.92428e+17

|X'X| for the runs of Equation 3.13 is 6.9243e17, and Table 3.10's variances are its inverse's diagonal; the matrix printed in Equation 3.14, whose block row is wrong, has the determinant 6.5246e17, so 6.5541e17 is the determinant of neither

Determinant: I-optimal designp. 77

Book 1.5121e20Data 2.08428e+18

The designs of Tables 4.2 and 4.3, whose relative variances Table 4.4 prints, have determinants in the ratio 0.4113 under any coding of the supplier (D-optimal 5.0672e18 and I-optimal 2.0843e18 with the supplier coded +1, 0 and -1 as here; 4.1045e20 and 1.6883e20 with JMP's orthonormal coding), so the relative D-efficiency is 0.952; the printed determinants are in the ratio 0.2828

Relative d efficiency: I-optimal design versus D-optimal designp. 77

Book 0.93Data 0.95

The designs of Tables 4.2 and 4.3, whose relative variances Table 4.4 prints, have determinants in the ratio 0.4113 under any coding of the supplier (D-optimal 5.0672e18 and I-optimal 2.0843e18 with the supplier coded +1, 0 and -1 as here; 4.1045e20 and 1.6883e20 with JMP's orthonormal coding), so the relative D-efficiency is 0.952; the printed determinants are in the ratio 0.2828

Relative d efficiency: I-optimal design versus D-optimal designp. 77

Book 0.93Data 0.95

The designs of Tables 4.2 and 4.3, whose relative variances Table 4.4 prints, have determinants in the ratio 0.4113 under any coding of the supplier (D-optimal 5.0672e18 and I-optimal 2.0843e18 with the supplier coded +1, 0 and -1 as here; 4.1045e20 and 1.6883e20 with JMP's orthonormal coding), so the relative D-efficiency is 0.952; the printed determinants are in the ratio 0.2828

Determinant: D-optimal designp. 77

Book 5.3467e20Data 5.06723e+18

The designs of Tables 4.2 and 4.3, whose relative variances Table 4.4 prints, have determinants in the ratio 0.4113 under any coding of the supplier (D-optimal 5.0672e18 and I-optimal 2.0843e18 with the supplier coded +1, 0 and -1 as here; 4.1045e20 and 1.6883e20 with JMP's orthonormal coding), so the relative D-efficiency is 0.952; the printed determinants are in the ratio 0.2828

Relative d efficiency: Table 8.7 versus Table 8.6p. 172

Book 1.03Data 1.00

The design printed as Table 8.7 holds the same 64 runs as Table 8.6, in another order: its blocks 1 to 8 confound x1 x2, x3 x4 and x5 x6 and its blocks 9 to 16 x1 x6, x2 x3 and x4 x5, as Peter's construction does, so the two have the same |X'X| and a relative D-efficiency of 1

Relative d efficiency: the I-optimal design's gainp. 228

Book 12.1Data 9.4

The designs of Figure 10.2 and Table 10.2 give a relative D-efficiency of 91.4%, 9.4% more information for the I-optimal design; like Table 10.3's I-optimal column, the 89.2% belongs to an I-optimal design other than the one printed

Relative d efficiency: against the I-optimal designp. 228

Book 89.2%Data 91.378

The designs of Figure 10.2 and Table 10.2 give a relative D-efficiency of 91.4%, 9.4% more information for the I-optimal design; like Table 10.3's I-optimal column, the 89.2% belongs to an I-optimal design other than the one printed

Variance: yaw angle and grille coveragep. 229

Book 0.032Data 0.034

The I-optimal design printed in Table 10.2 gives 0.0340: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: yaw angle and grille coveragep. 229

Book 0.032Data 0.033

The I-optimal design printed in Table 10.2 gives 0.0333: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: the other interaction effectsp. 229

Book 0.046Data 0.048

The I-optimal design printed in Table 10.2 gives 0.0485: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: the other interaction effectsp. 229

Book 0.046Data 0.047

The I-optimal design printed in Table 10.2 gives 0.0466: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: the other interaction effectsp. 229

Book 0.046Data 0.049

The I-optimal design printed in Table 10.2 gives 0.0487: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: the other interaction effectsp. 229

Book 0.046Data 0.047

The I-optimal design printed in Table 10.2 gives 0.0468: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: Interceptp. 230

Book 0.454Data 0.447

The I-optimal design printed in Table 10.2 gives 0.4471: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: Yaw anglep. 230

Book 0.032Data 0.034

The I-optimal design printed in Table 10.2 gives 0.0340: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: Grille coveragep. 230

Book 0.032Data 0.033

The I-optimal design printed in Table 10.2 gives 0.0333: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: FRH × Yaw anglep. 230

Book 0.046Data 0.048

The I-optimal design printed in Table 10.2 gives 0.0485: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: FRH × Grille coveragep. 230

Book 0.046Data 0.047

The I-optimal design printed in Table 10.2 gives 0.0466: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: RRH × Yaw anglep. 230

Book 0.046Data 0.049

The I-optimal design printed in Table 10.2 gives 0.0487: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: RRH × Grille coveragep. 230

Book 0.046Data 0.047

The I-optimal design printed in Table 10.2 gives 0.0468: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: Yaw angle × Grille coveragep. 230

Book 0.042Data 0.049

The I-optimal design printed in Table 10.2 gives 0.0487: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: FRH × FRHp. 230

Book 0.523Data 0.524

The I-optimal design printed in Table 10.2 gives 0.5242: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: RRH × RRHp. 230

Book 0.523Data 0.524

The I-optimal design printed in Table 10.2 gives 0.5235: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: Yaw angle × Yaw anglep. 230

Book 0.102Data 0.100

The I-optimal design printed in Table 10.2 gives 0.1001: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Variance: Grille coverage × Grille coveragep. 230

Book 0.102Data 0.105

The I-optimal design printed in Table 10.2 gives 0.1050: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Average variance: excluding the interceptp. 230

Book 0.160Data 0.161

The I-optimal design printed in Table 10.2 gives 0.1614: Table 10.3's I-optimal column, symmetric in the two ride heights and in the two easy-to-change factors, is not that of the printed design, which reproduces its own average prediction variance 0.512 (p. 231); a coordinate-exchange search finds many I-optimal split-plot designs with this master design and an average prediction variance of 0.5125 to 0.513, whose variances differ in the third decimal, and no single changed run of Table 10.2 gives the column

Df: β0, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β1, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β2, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β3, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β4, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β12, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β13, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β14, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β23, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β24, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β34, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β11, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β22, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β33, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Df: β44, ordinary least squaresp. 237

Book 45Data 35

The full quadratic model has 15 parameters, so ordinary least squares on the 50 runs leaves 35 residual degrees of freedom; the table's p values are those of 35 (the t ratio 0.93 of β12 has p 0.3592 on 35 and 0.3575 on 45)

Squares: (Yaw angle)² and (Grille coverage)²p. 247

The first run of whole plot 6 has yaw angle 1, whose square is 1; the table prints 0