实验设计分析第六版美 道格拉斯蒙哥马利课后作业4.docx
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实验设计分析第六版美道格拉斯蒙哥马利课后作业4
实验设计分析作业4机械电子工程
HomeWork#4(Classnote105,106)
1.MontgomeryProblem6-18
2.MontgomeryProblem6-26
3.MontgomeryProblem8-4
英文版题目6-18
(a)FactorialFit:
EtchrateversusA极距,B压强,C气流,D功率
EstimatedEffectsandCoefficientsforEtchrate(codedunits)
TermEffectCoef
Constant776.06
A极距-101.62-50.81
B压强-1.62-0.81
C气流7.373.69
D功率306.13153.06
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4.122.06
A极距*C气流*D功率5.622.81
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
EffectsPlotforEtchrate
AccordingtoFigure,Alpha=0.05,thefactorsofA,ADandDseemsignificant.
MainEffectsPlot(datameans)forEtchrate
InteractionPlot(datameans)forEtchrate
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)GeneralLinearModel:
EtchrateversusA极距,D功率
FactorTypeLevelsValues
A极距fixed2-1,1
D功率fixed2-1,1
AnalysisofVarianceforEtchrate,usingAdjustedSSforTests
SourceDFSeqSSAdjSSAdjMSFP
A极距141311413114131123.770.000
D功率1374850374850374850215.660.000
A极距*D功率194403944039440354.310.000
Error1220858208581738
Total15531421
S=41.6911R-Sq=96.08%R-Sq(adj)=95.09%
(c)FactorialFit:
EtchrateversusA极距,D功率
EstimatedEffectsandCoefficientsforEtchrate(codedunits)
TermEffectCoefSECoefTP
Constant776.0610.4274.460.000
A极距-101.62-50.8110.42-4.880.000
D功率306.12153.0610.4214.690.000
A极距*D功率-153.62-76.8110.42-7.370.000
S=41.6911R-Sq=96.08%R-Sq(adj)=95.09%
AnalysisofVarianceforEtchrate(codedunits)
SourceDFSeqSSAdjSSAdjMSFP
MainEffects2416161416161208080119.710.000
2-WayInteractions194403944039440354.310.000
ResidualError1220858208581738
PureError1220858208581738
Total15531421
SowegetRegression:
Y=776.06-50.81A+153.06D-76.81A*D
(d)ResidualPlotsforEtchrate
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
Etchrate
-1
-1
550
1
-1
669
-1
-1
604
1
-1
650
-1
-1
633
1
-1
642
-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
GeneralLinearModel:
EtchrateversusA极距,D功率
FactorTypeLevelsValues
A极距fixed2-1,1
D功率fixed2-1,1
AnalysisofVarianceforEtchrate,usingAdjustedSSforTests
SourceDFSeqSSAdjSSAdjMSFP
A极距141311413114131123.770.000
D功率1374850374850374850215.660.000
A极距*D功率194403944039440354.310.000
Error1220858208581738
Total15531421
S=41.6911R-Sq=96.08%R-Sq(adj)=95.09%
(f)InteractionPlot(datameans)forEtchrate
表明当A低水平,D高水平时,二者的交互作用非常明显。
(g)ResidualsvsOrderforEtchrate
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)AccordingtothefollowingFigure,Alpha=0.05,thefactorsofA,A*BandCseemsignificantforM-Wonly.
(b)FactorialFit:
MolecularWeightversusA,B,C,D
EstimatedEffectsandCoefficientsforMolecularWeight(codedunits)
TermEffectCoefSECoefTP
Constant2506.2511.597216.120.000
A123.6561.8311.5965.330.006
B-11.28-5.6411.596-0.490.652
C201.21100.6011.5968.680.001
D6.173.0911.5960.270.803
A*B120.2960.1411.5945.190.007
A*C20.4810.2411.5900.880.427
A*D-16.64-8.3211.575-0.720.512
B*C-22.37-11.1811.596-0.960.389
B*D7.733.8711.5950.330.755
C*D12.896.4511.5920.560.608
A*B*C14.827.4111.5360.640.556
A*B*D-13.82-6.9111.398-0.610.577
A*C*D-23.04-11.5211.037-1.040.356
B*C*D2.581.2911.5560.110.916
A*B*C*D-9.63-4.814.524-1.060.347
S=46.3860R-Sq=97.17%R-Sq(adj)=86.54%
AnalysisofVarianceforMolecularWeight(codedunits)
SourceDFSeqSSAdjSSAdjMSFP
MainEffects413804322376855942.026.000.004
2-WayInteractions61523486358910598.24.930.072
3-WayInteractions4231140841021.00.470.756
4-WayInteractions1243724372436.81.130.347
ResidualError4860786072151.7
LackofFit1782782781.70.300.622
PureError3782578252608.3
Total19303745
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MolecularWeight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)FactorialFit:
MolecularWeightversusA,B,C
EstimatedEffectsandCoefficientsforMolecularWeight(codedunits)
TermEffectCoefSECoefTP
Constant2505.5710.781232.400.000
A138.6769.3310.2506.760.000
B-7.18-3.5910.747-0.330.743
C208.03104.0210.6779.740.000
A*B75.2537.623.8609.750.000
S=43.1422R-Sq=90.81%R-Sq(adj)=88.36%
AnalysisofVarianceforMolecularWeight(codedunits)
SourceDFSeqSSAdjSSAdjMSFP
MainEffects3990172738899129649.050.000
2-WayInteractions117681017681017681095.000.000
ResidualError1527919279191861
LackofFit4138691386934672.710.085
PureError1114050140501277
Total19303745
RegressionEquation:
Y=2505.57+69.33A-3.59B+104.02C+37.62A*B
(d)ResidualPlotsforMolecularWeight
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。
ResidualsfromMolecularWeightvsMolecularWeight
(e)FactorialFit:
ViscosityversusA,B,C,D
EstimatedEffectsandCoefficientsforViscosity(codedunits)
TermEffectCoefSECoefTP
Constant1500.627.524199.460.000
A96.3848.197.5236.410.003
B91.2945.647.5246.070.004
C7.563.787.5240.500.642
D-17.39-8.707.523-1.160.312
A*B-14.14-7.077.522-0.940.400
A*C11.855.927.5190.790.475
A*D-23.67-11.847.509-1.580.190
B*C9.824.917.5230.650.549
B*D-25.32-12.667.522-1.680.168
C*D13.226.617.5210.880.429
A*B*C16.958.487.4841.130.321
A*B*D-6.49-3.247.395-0.440.683
A*C*D4.602.307.1600.320.764
B*C*D20.3510.177.4971.360.246
A*B*C*D6.203.102.9351.060.351
S=30.0942R-Sq=95.79%R-Sq(adj)=80.01%
AnalysisofVarianceforViscosity(codedunits)
SourceDFSeqSSAdjSSAdjMSFP
MainEffects4478247193017982.419.860.007
2-WayInteractions62886472581209.71.340.407
3-WayInteractions447553121780.20.860.556
4-WayInteractions1100910091008.91.110.351
ResidualError436233623905.7
LackofFit1145414541453.92.010.251
PureError321692169722.9
Total1986074
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。
FactorialFit:
ViscosityversusA,B
EstimatedEffectsandCoefficientsforViscosity(codedunits)
TermEffectCoefSECoefTP
Constant1505.5315.70395.880.000
A-11.61-5.815.076-1.140.269
B61.8330.9215.2272.030.058
S=63.0455R-Sq=21.50%R-Sq(adj)=12.26%
AnalysisofVarianceforViscosity(codedunits)
SourceDFSeqSSAdjSSAdjMSFP
MainEffects218503185039251.72.330.128
ResidualError1767570675703974.7
LackofFit2526775267726338.326.530.000
PureError151489414894992.9
Total1986074
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)FractionalFactorialDesign
Factors:
5BaseDesign:
5,8Resolution:
III
Runs:
8Replicates:
1Fraction:
1/4
Blocks:
1Centerpts(total):
0
DesignGenerators:
D=AB,E=AC
AliasStructure
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
Fromaboveall,wecangetDesignGenerators:
D=AB,E=AC
BasicDesign
GeneratedDesign
Run
A
B
C
D=AB
E=AC
1
-1
-1
-1
1
1
2
1
-1
-1
-1
-1
3
-1
1
-1
-1
1
4
1
1
-1
1
-1
5
-1
-1
1
1
-1
6
1
-1
1
-1
1
7
-1
1
1
-1
-1
8
1
1
1
1
1
根据6-21题意可以得到下表
Run
A
B
C
D
E
Yield
Lables
1
-1
-1
-1
1
1
6
de
2
1
-1
-1
-1
-1
9
a
3
-1
1
-1
-1
1
35
be
4
1
1
-1
1
-1
50
abd
5
-1
-1
1
1
-1
18
cd
6
1
-1
1
-1
1
22
ace
7
-1
1
1
-1
-1
40
bc
8
1
1
1
1
1
63
abcde
(a)FactorialFit:
yieldversusA,B,C,D,E
EstimatedEffectsandCoefficientsforyield(codedunits)
TermEffectCoef
Constant30.3750
A11.25005.6250
B33.250016.6250
C10.75005.3750
D7.75003.8750
E2.25001.1250
B*C-1.7500-0.8750
B*E1.75000.8750
EffectsPlotforyield
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,GeneralLinearModel:
yieldversusA,B,C
FactorTypeLevelsValues
Afixed2-1,1
Bfixed2-1,1
Cfixed2-1,1
AnalysisofVarianceforyield,usingAdjustedSSforTests
SourceDFSeqSSAdjSSAdjMSFP
A1253.12253.13253.137.110.056
B12211.122211.122211.1262.070.001
C1231.12231.12231.126.490.064
Error4142.50142.5035.63
Total72837.87
S=5.96867R-Sq=94.98%R-Sq(adj)=91.21%
(c)regressionequation:
Y=30.375+5.625A+16.625B+5.375C
(d)ResidualPlotsforyield
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)ResidualsfromyieldvsFITS1
(f)
A*D的交互作用最为明显了。
(h)