实验设计分析第六版(美 道格拉斯-蒙哥马利)课后作业4

Home Work #4 (Class note 105, 106) 1. Montgomery Problem 6-18

2. Montgomery Problem 6-26

3. Montgomery Problem 8-4

英文版题目6-18

（a）Factorial Fit: Etch rate versus A极距, B压强, C气流, D功率

Estimated Effects and Coefficients for Etch rate (coded units) Term Effect Coef Constant 776.06 A极距 -101.62 -50.81 B压强 -1.62 -0.81 C气流 7.37 3.69 D功率 306.13 153.06 A极距*B压强 -7.88 -3.94 A极距*C气流 -24.88 -12.44 A极距*D功率 -153.63 -76.81 B压强*C气流 -43.87 -21.94 B压强*D功率 -0.63 -0.31 C气流*D功率 -2.13 -1.06 A极距*B压强*C气流 -15.63 -7.81 A极距*B压强*D功率 4.12 2.06 A极距*C气流*D功率 5.62 2.81 B压强*C气流*D功率 -25.38 -12.69 A极距*B压强*C气流*D功率 -40.13 -20.06

Effects Plot for Etch rate

According to Figure , Alpha=0.05 ,the factors of A, AD and D seem significant. Main Effects Plot (data means) for Etch rate

Interaction Plot (data means) for Etch rate

结论：从主因子效应图看出A,D因子的效应比较显著；从因子交互作用看出，A*D的交互作用比其他的任何项的交互效应都大得多。

（b）General Linear Model: Etch rate versus A极距, D功率

Factor Type Levels Values A极距 fixed 2 -1, 1 D功率 fixed 2 -1, 1

Analysis of Variance for Etch rate, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P A极距 1 41311 41311 41311 23.77 0.000 D功率 1 374850 374850 374850 215.66 0.000 A极距*D功率 1 94403 94403 94403 .31 0.000 Error 12 20858 20858 1738 Total 15 531421

S = 41.6911 R-Sq = 96.08% R-Sq(adj) = 95.09%

(c)Factorial Fit: Etch rate versus A极距, D功率

Estimated Effects and Coefficients for Etch rate (coded units) Term Effect Coef SE Coef T P Constant 776.06 10.42 74.46 0.000 A极距 -101.62 -50.81 10.42 -4.88 0.000 D功率 306.12 153.06 10.42 14.69 0.000 A极距*D功率 -153.62 -76.81 10.42 -7.37 0.000 S = 41.6911 R-Sq = 96.08% R-Sq(adj) = 95.09% Analysis of Variance for Etch rate (coded units)

Source DF Seq SS Adj SS Adj MS F P Main Effects 2 416161 416161 208080 119.71 0.000 2-Way Interactions 1 94403 94403 94403 .31 0.000 Residual Error 12 20858 20858 1738 Pure Error 12 20858 20858 1738 Total 15 531421

So we get Regression: Y = 776.06 - 50.81A + 153.06D - 76.81A*D

(d)Residual Plots for Etch rate

残差分析：从正态概率图中，残差紧密的分布在直线的两侧，呈线性分布，很好的吻合了正态性分布假定；残差关于拟合值也是上下基本大致对称的分布

的，证明回归模型的方程结论是非常符合这个实验数据统计理论的。

（e）根据前面的分析可知，A,D的影响效应是B,C的好几倍，B,C的影响基本可以忽略了。剩余的两个因子A,D是非常重要的，变成重复2^2=4次.但要注意实验次序应重新随机化排列。 A极距 D功率 Etch rate -1 -1 550 1 -1 669 -1 -1 604 1 -1 650 -1 -1 633 1 -1 2 -1 -1 601 1 -1 635 -1 1 1037 1 1 749 -1 1 1052 1 1 868 -1 1 1075 1 1 860 -1 1 1063 1 1 729 General Linear Model: Etch rate versus A极距, D功率

Factor Type Levels Values A极距 fixed 2 -1, 1 D功率 fixed 2 -1, 1

Analysis of Variance for Etch rate, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P A极距 1 41311 41311 41311 23.77 0.000 D功率 1 374850 374850 374850 215.66 0.000 A极距*D功率 1 94403 94403 94403 .31 0.000 Error 12 20858 20858 1738 Total 15 531421

S = 41.6911 R-Sq = 96.08% R-Sq(adj) = 95.09%

（f） Interaction Plot (data means) for Etch rate

表明当A低水平，D高水平时，二者的交互作用非常明显。

(g)Residuals vs Order for Etch rate

我发现残差有峰有谷近似沿着某条正弦曲线分布的趋势。 题目6-26

（a） According to the following Figure , Alpha=0.05 ,the factors of A, A*B and C seem significant for M -W only.

(b)Factorial Fit: Molecular Weight versus A, B, C, D

Estimated Effects and Coefficients for Molecular Weight (coded units) Term Effect Coef SE Coef T P Constant 2506.25 11.597 216.12 0.000 A 123.65 61.83 11.596 5.33 0.006 B -11.28 -5. 11.596 -0.49 0.652 C 201.21 100.60 11.596 8.68 0.001 D 6.17 3.09 11.596 0.27 0.803 A*B 120.29 60.14 11.594 5.19 0.007 A*C 20.48 10.24 11.590 0.88 0.427 A*D -16. -8.32 11.575 -0.72 0.512 B*C -22.37 -11.18 11.596 -0.96 0.3 B*D 7.73 3.87 11.595 0.33 0.755 C*D 12. 6.45 11.592 0.56 0.608 A*B*C 14.82 7.41 11.536 0. 0.556 A*B*D -13.82 -6.91 11.398 -0.61 0.577 A*C*D -23.04 -11.52 11.037 -1.04 0.356 B*C*D 2.58 1.29 11.556 0.11 0.916 A*B*C*D -9.63 -4.81 4.524 -1.06 0.347

S = 46.3860 R-Sq = 97.17% R-Sq(adj) = 86.%

Analysis of Variance for Molecular Weight (coded units)

Source DF Seq SS Adj SS Adj MS F P Main Effects 4 138043 223768 55942.0 26.00 0.004 2-Way Interactions 6 152348 635 10598.2 4.93 0.072

3-Way Interactions 4 2311 4084 1021.0 0.47 0.756 4-Way Interactions 1 2437 2437 2436.8 1.13 0.347 Residual Error 4 8607 8607 2151.7

Lack of Fit 1 782 782 781.7 0.30 0.622 Pure Error 3 7825 7825 2608.3 Total 19 303745

结论：从上面的分析中看出A、C、A*B的P值均小于0.05，故A、C、A*B对

Molecular Weight的影响最大。因为有A,B的交互作用明显，所以暗示了回归模型的方程不是线性的，是曲线形状的。

(c)Factorial Fit: Molecular Weight versus A, B, C

Estimated Effects and Coefficients for Molecular Weight (coded units) Term Effect Coef SE Coef T P Constant 2505.57 10.781 232.40 0.000 A 138.67 69.33 10.250 6.76 0.000 B -7.18 -3.59 10.747 -0.33 0.743 C 208.03 104.02 10.677 9.74 0.000 A*B 75.25 37.62 3.860 9.75 0.000 S = 43.1422 R-Sq = 90.81% R-Sq(adj) = 88.36%

Analysis of Variance for Molecular Weight (coded units)

Source DF Seq SS Adj SS Adj MS F P Main Effects 3 99017 2738 91296 49.05 0.000 2-Way Interactions 1 176810 176810 176810 95.00 0.000 Residual Error 15 27919 27919 1861

Lack of Fit 4 13869 13869 3467 2.71 0.085 Pure Error 11 14050 14050 1277 Total 19 303745

Regression Equation: Y = 2505.57+69.33A-3.59B+104.02C+37.62A*B

(d)Residual Plots for Molecular Weight

分析回归方程模型的残差正态概率图，看到残差点近似的分布呈一直线，正负拟合的情况也比较好，说明模型的假设是非常合适的，验证了我得到的统计结果是很有效的。当然图中显示出现了两个异常点，对于模型的适用来说出现这个情况是允许的，不影响广泛性。

Residuals from Molecular Weight vs Molecular Weight

（e） Factorial Fit: Viscosity versus A, B, C, D

Estimated Effects and Coefficients for Viscosity (coded units) Term Effect Coef SE Coef T P Constant 1500.62 7.524 199.46 0.000 A 96.38 48.19 7.523 6.41 0.003 B 91.29 45. 7.524 6.07 0.004 C 7.56 3.78 7.524 0.50 0.2 D -17.39 -8.70 7.523 -1.16 0.312 A*B -14.14 -7.07 7.522 -0.94 0.400 A*C 11.85 5.92 7.519 0.79 0.475 A*D -23.67 -11.84 7.509 -1.58 0.190 B*C 9.82 4.91 7.523 0.65 0.9

B*D -25.32 -12.66 7.522 -1.68 0.168 C*D 13.22 6.61 7.521 0.88 0.429 A*B*C 16.95 8.48 7.484 1.13 0.321 A*B*D -6.49 -3.24 7.395 -0.44 0.683 A*C*D 4.60 2.30 7.160 0.32 0.7 B*C*D 20.35 10.17 7.497 1.36 0.246 A*B*C*D 6.20 3.10 2.935 1.06 0.351

S = 30.0942 R-Sq = 95.79% R-Sq(adj) = 80.01% Analysis of Variance for Viscosity (coded units)

Source DF Seq SS Adj SS Adj MS F P Main Effects 4 47824 71930 17982.4 19.86 0.007 2-Way Interactions 6 288 7258 1209.7 1.34 0.407 3-Way Interactions 4 4755 3121 780.2 0.86 0.556 4-Way Interactions 1 1009 1009 1008.9 1.11 0.351 Residual Error 4 3623 3623 905.7

Lack of Fit 1 14 14 1453.9 2.01 0.251 Pure Error 3 2169 2169 722.9 Total 19 86074

从下面因子效应图中可以看出，A,B对viscosity的影响是最为显著的，需要着重分析。

 Factorial Fit: Viscosity versus A, B

Estimated Effects and Coefficients for Viscosity (coded units) Term Effect Coef SE Coef T P Constant 1505.53 15.703 95.88 0.000 A -11.61 -5.81 5.076 -1.14 0.269 B 61.83 30.92 15.227 2.03 0.058

S = 63.0455 R-Sq = 21.50% R-Sq(adj) = 12.26% Analysis of Variance for Viscosity (coded units)

Source DF Seq SS Adj SS Adj MS F P Main Effects 2 18503 18503 9251.7 2.33 0.128 Residual Error 17 67570 67570 3974.7

Lack of Fit 2 52677 52677 26338.3 26.53 0.000 Pure Error 15 144 144 992.9 Total 19 86074

结论：因为Viscosity 显著影响只和A,B有关，所以回归方程肯定是一条直线，是关于A和B的线性方程，所以不会是曲线的情形。

回归方程：U = 1505.53 -5.81A + 30.92B ④残差分析与模型合适性检验

在正态概率图看到残差很紧密的分布，近似呈现一条直线，表明残差很好的服从正态分布；残差关于零线上下分布，拟合的较好，出现的序列也是杂乱随机的，验证了这个回归模型对实验是非常适用的。

题目8-4

（a）Fractional Factorial Design

Factors: 5 Base Design: 5, 8 Resolution: III Runs: 8 Replicates: 1 Fraction: 1/4 Blocks: 1 Center pts (total): 0

Design Generators: D = AB, E = AC Alias Structure

I + ABD + ACE + BCDE A + BD + CE + ABCDE B + AD + CDE + ABCE C + AE + BDE + ABCD D + AB + BCE + ACDE

E + AC + BCD + ABDE BC + DE + ABE + ACD BE + CD + ABC + ADE

From above all, we can get Design Generators: D = AB, E = AC

Run 1 2 3 4 5 6 7 8 Basic Design Generated Design A -1 1 -1 1 -1 1 -1 1 B -1 -1 1 1 -1 -1 1 1 C -1 -1 -1 -1 1 1 1 1 D=AB 1 -1 -1 1 1 -1 -1 1 E=AC 1 -1 1 -1 -1 1 -1 1 根据6-21题意可以得到下表 Run 1 2 3 4 5 6 7 8 (a)Factorial Fit: yield versus A, B, C, D, E

Estimated Effects and Coefficients for yield (coded units) Term Effect Coef Constant 30.3750 A 11.2500 5.6250 B 33.2500 16.6250 C 10.7500 5.3750 D 7.7500 3.8750 E 2.2500 1.1250 B*C -1.7500 -0.8750 B*E 1.7500 0.8750

A -1 1 -1 1 -1 1 -1 1 B -1 -1 1 1 -1 -1 1 1 C -1 -1 -1 -1 1 1 1 1 D 1 -1 -1 1 1 -1 -1 1 E 1 -1 1 -1 -1 1 -1 1 Yield 6 9 35 50 18 22 40 63 Lables de a be abd cd ace bc abcde Effects Plot for yield

从各个因子效应的正态概率图中得出，当显著性水平α=0.10时，只有主因素B是非常显著的;但是当α=0.28时，因子A,B,C都显得比较重要了（由于试验次数很少，这里采用较大的α水平来估计）。

（b）对于α=0.28时，General Linear Model: yield versus A, B, C Factor Type Levels Values

A fixed 2 -1, 1 B fixed 2 -1, 1 C fixed 2 -1, 1

Analysis of Variance for yield, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P A 1 253.12 253.13 253.13 7.11 0.056 B 1 2211.12 2211.12 2211.12 62.07 0.001

C 1 231.12 231.12 231.12 6.49 0.0 Error 4 142.50 142.50 35.63 Total 7 2837.87

S = 5.96867 R-Sq = 94.98% R-Sq(adj) = 91.21%

（c）regression equation: Y = 30.375 + 5.625A + 16.625B + 5.375C

(d)Residual Plots for yield

根据上面的残差概率图和残差分布图，残差是随机的出现，并符合正态性分布，假设的模型是非常合适的，结果是非常适用有效的。

（e）Residuals from yield vs FITS1

（f）

A*D的交互作用最为明显了。

（h）