李亦秦DOE實(shí)驗(yàn)計(jì)劃

實(shí)驗(yàn)設(shè)計(jì)

(DesignofExperiments)李亦秦

老師主講3/17/041DOE課程大綱實(shí)驗(yàn)設(shè)計(jì)之簡介一因子隨機(jī)化實(shí)驗(yàn)變異數(shù)分析例子說明變異數(shù)分析模型適當(dāng)性檢驗(yàn)實(shí)務(wù)結(jié)果解釋統(tǒng)計(jì)軟體應(yīng)用區(qū)集化隨機(jī)設(shè)計(jì)因子設(shè)計(jì)原理2k因子設(shè)計(jì)2k全因子設(shè)計(jì)(22,23)2k-p部份因子設(shè)計(jì)統(tǒng)計(jì)軟體應(yīng)用反應(yīng)曲面簡介統(tǒng)計(jì)軟體應(yīng)用3/17/042DOE實(shí)驗(yàn)設(shè)計(jì)-簡介實(shí)驗(yàn)是一個(gè)試驗(yàn)或是一序列的試驗(yàn)實(shí)驗(yàn)設(shè)計(jì)之目的：發(fā)展一穩(wěn)健製程(即受外來變異原因影響極小化的製程)實(shí)驗(yàn)設(shè)計(jì)之應(yīng)用製程特徵化及最佳化(冶金工程：油化或鹽化)評(píng)估材料特性產(chǎn)品設(shè)計(jì)及研發(fā)決定系統(tǒng)及零件公差3/17/043DOE實(shí)驗(yàn)設(shè)計(jì)(簡介)-例子冶金工程：油焠或鹽水焠鋁合金何者的硬度將最大？本例子還可以有其它考慮？？只有這兩種溶液？還有那些因子可能影響硬度？是否在實(shí)驗(yàn)中加入控制？每一種溶液的樣本數(shù)應(yīng)該抽多少？實(shí)驗(yàn)的順序?yàn)楹危吭撚媚切┓治龇椒?？所觀測的平均硬度值的差異，要大到多少才算是真正有差異？3/17/044DOE實(shí)驗(yàn)設(shè)計(jì)的四段歷史農(nóng)業(yè)時(shí)代，1918–1940s英人R.A.Fisher(漁夫？F檢定)影響農(nóng)業(yè)新品種改良因子設(shè)計(jì)(FactorialDesign)，變異數(shù)分析(ANOVA)工業(yè)時(shí)期一,1951–late1970sBox&Wilson,反應(yīng)曲面(ResponseSurfaceMethod)應(yīng)用在化工製程工業(yè)時(shí)期二,late1970s–1990品質(zhì)改善田口穩(wěn)健設(shè)計(jì)(TaguchiMethod)、參數(shù)設(shè)計(jì)近代,1990年以後3/17/045DOE製程或系統(tǒng)模型實(shí)驗(yàn)設(shè)計(jì)之目的：製程良率提昇縮短製程開發(fā)時(shí)程改善現(xiàn)有製程績效改善產(chǎn)品可靠度降低產(chǎn)品與製程變異決定材料元件設(shè)計(jì)之公差3/17/046DOE實(shí)驗(yàn)設(shè)計(jì)的基本原理隨機(jī)化(randomization)按隨機(jī)順序進(jìn)行實(shí)驗(yàn)平衡式的設(shè)計(jì)(每組資料量一樣多)重複性(replication)樣本大小重複實(shí)驗(yàn)v.s重複量測區(qū)集化(blocking)處理干擾因子3/17/047DOE實(shí)驗(yàn)設(shè)計(jì)的策略一次改變一因子(未同時(shí)考慮交互作用)相關(guān)科學(xué)(理論式)及工程方法(經(jīng)驗(yàn)法則)可決定反應(yīng)變數(shù)及可控因子過度強(qiáng)調(diào)交互作用方法不有效率以統(tǒng)計(jì)方法設(shè)計(jì)實(shí)驗(yàn)根據(jù)Fisher’s的因子設(shè)計(jì)觀念3/17/048DOE因子設(shè)計(jì)(FactorialDesigns)在因子設(shè)計(jì)實(shí)驗(yàn)中，所有可能的組合被試驗(yàn)打高爾夫球的實(shí)驗(yàn):球座的種類球的種類走路或者是坐電動(dòng)車飲料的型式(水或beer)上午或下午打球氣候高爾夫球鞋的型式等等…3/17/049DOE因子設(shè)計(jì)高爾夫球的例子(二因子)3/17/0410DOE高爾夫球的例子(三、四因子)3/17/0411DOE部份因子設(shè)計(jì)高爾夫球的例子(四因子)3/17/0412DOE實(shí)驗(yàn)的規(guī)劃進(jìn)行及分析問題的描述因子的選擇，水準(zhǔn)的選擇和水準(zhǔn)的範(fàn)圍反應(yīng)變數(shù)的選擇因子設(shè)計(jì)的選擇實(shí)驗(yàn)的進(jìn)行統(tǒng)計(jì)分析結(jié)論及建議3/17/0413DOE實(shí)驗(yàn)的規(guī)劃進(jìn)行及分析-續(xù)統(tǒng)計(jì)的想法(DOE,ANOVA,…)非統(tǒng)計(jì)的知識(shí)有關(guān)實(shí)驗(yàn)的成敗(DomainKnowledge)實(shí)驗(yàn)前的規(guī)劃實(shí)驗(yàn)的順序3/17/0414DOE實(shí)驗(yàn)設(shè)計(jì)的基本統(tǒng)計(jì)觀念兩組樣本平均數(shù)是否相等的--t檢定變異數(shù)分析(多組平均數(shù)是否有差異？)ANOVA統(tǒng)計(jì)檢定及分析檢驗(yàn)假設(shè)及模型正確性ANOVA分析後之平均數(shù)檢定樣本大小的決定3/17/0415DOEPortlandCementFormulation(Table2-1,pp.22)Observation(sample),jModifiedMortar(Formulation1)UnmodifiedMortar(Formulation2)116.8517.50216.4017.63317.2118.25416.3518.00516.5217.86617.0417.75716.9618.22817.1517.90916.5917.961016.5718.153/17/0416DOEGraphicalViewoftheData

DotDiagram,Fig.2-1,pp.223/17/0417DOEBoxPlots,Fig.2-3,pp.243/17/0418DOE兩因子以上該如何檢定？

t-test無法使用實(shí)務(wù)上常有超過兩因子及兩水準(zhǔn)的情況發(fā)生ANOVA適合分析這種型態(tài)的實(shí)驗(yàn)ANOVA是在1920年由Fisher發(fā)明主要應(yīng)用在農(nóng)業(yè)的實(shí)驗(yàn)上目前已大量用在工業(yè)的實(shí)驗(yàn)上3/17/0419DOE一因子實(shí)驗(yàn)變異數(shù)分析(例子說明)探討人造纖維含棉量對(duì)伸張力之影響？？反應(yīng)變數(shù)是伸張力實(shí)驗(yàn)者想要決定最佳的含綿量含綿量範(fàn)圍在10%~40%實(shí)驗(yàn)者選擇五種不同的含綿量:15%，20%，25%，30%，35%每組實(shí)驗(yàn)進(jìn)行五次重複，以隨機(jī)順序進(jìn)行3/17/0420DOE含棉量v.s.伸張力改變含綿量會(huì)改變平均伸張力的強(qiáng)度嗎?是否有最佳含綿量的水準(zhǔn)?3/17/0421DOE一因子變異數(shù)分析完全隨機(jī)試驗(yàn)completelyrandomizeddesign

(CRD)N=所有實(shí)驗(yàn)次數(shù)目的是檢定有關(guān)處理(Treatment)平均數(shù)間是否有差異存在3/17/0422DOE一因子變異數(shù)分析變異數(shù)分析來自反應(yīng)變數(shù)的總變異量和模型的變異單因子ANOVA模型3/17/0423DOE一因子變異數(shù)分析總變異為sumofsquares:ANOVA的組成為:3/17/0424DOETheAnalysisofVariance大的處理平方和表示處理平均數(shù)間有大的差異小的處理平方和表示處理平均數(shù)間沒有差異SSE代表實(shí)驗(yàn)誤差或是實(shí)驗(yàn)中無法解釋部份ANOVA的統(tǒng)計(jì)假設(shè)為:3/17/0425DOE一因子變異數(shù)分析均方為平方和除以自由度:如果處理平均數(shù)間有差異則處理均方將顯著地大於誤差均方如果處理平均數(shù)沒有差異則處理均方將不顯著地大於誤差均方3/17/0426DOE一因子變異數(shù)分析請參考F分配forF0istheFa-1,a(n-1)distribution拒絕虛無假設(shè)3/17/0427DOEANOVAComputerOutput

(Design-Expert)

Response:

Strength

ANOVAforSelectedFactorialModel

Analysisofvariancetable[Partialsumofsquares]

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Model 475.76 4 118.94 14.76 <0.0001

A

475.76

4

118.94

14.76

<0.0001

PureError 161.20 20 8.06

CorTotal 636.96 24

Std.Dev. 2.84 R-Squared 0.7469

Mean 15.04 AdjR-Squared 0.6963

C.V. 18.88 PredR-Squared 0.6046 PRESS 251.88 AdeqPrecision 9.2943/17/0428DOETheReferenceF-Distribution:3/17/0429DOEGraphicalViewoftheResultsDESIGN-EXPERTPlotStrength

X=A:CottonWeight%DesignPointsA:CottonWeight%StrengthOneFactorPlot1520253035711.51620.525222222222222223/17/0430DOEANOVA模型適當(dāng)性檢驗(yàn)檢驗(yàn)ANOVA的假設(shè)是非常重要的一、常態(tài)性(normality)二、變異數(shù)為常數(shù)(equalvariance)三、獨(dú)立性檢驗(yàn)(independent)這是一個(gè)合適的模型嗎?3/17/0431DOEANOVA模型適當(dāng)性檢驗(yàn)殘差的檢驗(yàn)Design-Expert產(chǎn)生殘差圖殘差圖檢驗(yàn)常態(tài)性DESIGN-EXPERTPlotStrengthResidualNormal%probabilityNormalplotofresiduals-3.8-1.550.72.955.2151020305070809095993/17/0432DOE其他重要的殘差圖Strength22222222222222PredictedResidualsResidualsvs.Predicted-3.8-1.550.72.955.29.8012.7515.7018.6521.60StrengthRunNumberResidualsResidualsvs.Run-3.8-1.550.72.955.21471013161922253/17/0433DOE處理平均數(shù)間的比較ANOVA檢定處理平均數(shù)是否均相等？假設(shè)滿足殘差分析拒絕虛無假設(shè)表示處理平均數(shù)有差異多重平均數(shù)差異的比較有許多方法可做多重平均數(shù)差異的比較例如Fisher’sLSD、Tukey方法3/17/0434DOEDesign-ExpertOutput

TreatmentMeans(Adjusted,IfNecessary)

Estimated

Standard

Mean

Error

1-15 9.80 1.27

2-20 15.40 1.27

3-25 17.60 1.27

4-30 21.60 1.27

5-35 10.80 1.27

Mean

Standard

tforH0

Treatment

Difference

DF

Error

Coeff=0

Prob>|t|

1vs2 -5.60 1 1.80 -3.12 0.0054

1vs3 -7.80 1 1.80 -4.34 0.0003

1vs4 -11.80 1 1.80 -6.57 <0.0001

1vs5 -1.00 1 1.80 -0.56 0.5838

2vs3 -2.20 1 1.80 -1.23 0.2347

2vs4 -6.20 1 1.80 -3.45 0.0025

2vs5 4.60 1 1.80 2.56 0.0186

3vs4 -4.00 1 1.80 -2.23 0.0375

3vs5 6.80 1 1.80 3.79 0.0012

4vs5 10.80 1 1.80 6.01 <0.0001

3/17/0435DOEGraphicalComparisonofMeans

Text,pg.893/17/0436DOEFortheCaseofQuantitativeFactors,aRegressionModelisoftenUseful

Response:

Strength

ANOVAforResponseSurfaceCubicModel

Analysisofvariancetable[Partialsumofsquares]

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Model 441.81 3 147.27 15.85 <0.0001

A

90.84

1

90.84

9.78

0.0051

A2

343.21

1

343.21

36.93

<0.0001

A3

64.98

1

64.98

6.99

0.0152

Residual 195.15 21 9.29

LackofFit

33.95

1

33.95

4.21

0.0535

PureError

161.20

20

8.06

CorTotal 636.96 24

Coefficient

Standard

95%CI

95%CI

Factor

Estimate

DF

Error

Low

High

VIF Intercept 19.47 1 0.95 17.49 21.44 A-Cotton% 8.10 1 2.59 2.71 13.49 9.03

A2 -8.86 1 1.46 -11.89 -5.83 1.00

A3 -7.60 1 2.87 -13.58 -1.62 9.033/17/0437DOETheRegressionModel最後的方程式:

Strength=+62.61143

-9.01143*CottonWeight%+0.48143*CottonWeight%^2-7.60000E-003*CottonWeight%^3StrengthX=A:CottonWeight%DesignPoints15.0020.0025.0030.0035.00711.51620.525A:CottonWeight%StrengthOneFactorPlot222222222222223/17/0438DOE樣本大小的決定樣本大小的決定與型I誤差與型II誤差有關(guān)TypeIerror–當(dāng)虛無假設(shè)為真時(shí)，拒絕虛無假設(shè)TypeIIerror–當(dāng)虛無假設(shè)為偽時(shí)，無法拒絕虛無假設(shè)檢定力Power=1-特性曲線Operatingcharacteristiccurvesplotagainstaparameterwhere3/17/0439DOE樣本大小的決定使用OC曲線利用OC

以決定樣本大小時(shí)必須查實(shí)驗(yàn)設(shè)計(jì)課本後附錄OC曲線圖使用OC曲線圖需先定義兩平均數(shù)之差

D，然後求的極小值，公式如下：基本上，需先計(jì)算出

比率，而後嘗試不同的n值，直到所求的檢定力出現(xiàn)為止。3/17/0440DOE區(qū)集設(shè)計(jì)的主要原理區(qū)集化與干擾因子區(qū)集化設(shè)計(jì)(randomizedcompleteblockdesignorRCBD)將ANOVA延伸至RCBD的設(shè)計(jì)3/17/0441DOE區(qū)集設(shè)計(jì)的主要原理區(qū)集化的設(shè)計(jì)是一種用來處理干擾因子的方法干擾因子(nuisancefactor)是一種因子對(duì)反應(yīng)變數(shù)有某種效應(yīng)但對(duì)實(shí)驗(yàn)沒有益處，在實(shí)驗(yàn)過程中，我們希望此項(xiàng)效應(yīng)對(duì)反應(yīng)變數(shù)可以降至最低。典型的干擾因子如不同批量的材料、不同的作業(yè)員、不同的測試機(jī)臺(tái)、不同班別、不同日子的作業(yè)等等。許多工業(yè)上的實(shí)驗(yàn)包括區(qū)集化設(shè)計(jì)。無法將區(qū)集化設(shè)計(jì)在實(shí)驗(yàn)中將是一項(xiàng)缺陷？3/17/0442DOE區(qū)集設(shè)計(jì)的主要原理如果干擾因子是可知而且可控制，我們即可使用區(qū)集化設(shè)計(jì)。如果干擾因子是可知，但是是不可控制的，有時(shí)候我們可以利用共變異數(shù)處理。如果干擾因子是不可知，而且是不可控制的，我們希望可以利用隨機(jī)化的過程處理干擾因子的效果。3/17/0443DOERCBD--硬度測試的例子決定四種不同尖銳物打在金屬物品上在Rockwell硬度測試機(jī)測是否產(chǎn)生不同的讀數(shù)量測系統(tǒng)常常使用實(shí)驗(yàn)設(shè)計(jì)法以決定機(jī)器的能力capability隨機(jī)指派尖銳物給16個(gè)實(shí)驗(yàn)單位即金屬物品。如果使用完全隨機(jī)實(shí)驗(yàn)(CRD)具有潛在危險(xiǎn)：因?yàn)榭赡苁褂玫慕饘傥锲繁旧砑从凶儺惔嬖?。所以，我們將此例子以區(qū)集化設(shè)計(jì)即可免除金屬物品可能存在的干擾因子。3/17/0444DOERCBD--硬度測試的例子RCBD的作法：指派所有四種不同的尖銳物給每一金屬試片每一試片稱為區(qū)集，即使用同一種材料的試片。區(qū)集間的變異是大的；區(qū)集內(nèi)的變異是相對(duì)的小。一般而言，區(qū)集是一特定的干擾因子的水準(zhǔn)每一區(qū)集內(nèi)執(zhí)行重複次數(shù)的實(shí)驗(yàn)。在每一區(qū)集內(nèi)的實(shí)驗(yàn)必須是隨機(jī)的。3/17/0445DOERCBD--硬度測試的例子假設(shè)有4區(qū)集:Noticethetwo-waystructureoftheexperiment我們所興趣的仍是檢驗(yàn)各種處理平均數(shù)是否相等，但在計(jì)算之前我們必須先移出區(qū)集中的干擾因子所造成的變異。3/17/0446DOERCBD是ANOVA的延伸Supposethattherearea

treatments(factorlevels)andb

blocksAstatisticalmodel(effectsmodel)fortheRCBDisTherelevant(fixedeffects)hypothesesare3/17/0447DOERCBD是ANOVA的延伸ANOVApartitioningoftotalvariability:3/17/0448DOE

Thedegreesoffreedomforthesumsofsquaresinareasfollows:Therefore,ratiosofsumsofsquarestotheirdegreesoffreedomresultinmeansquaresandtheratioofthemeansquarefortreatmentstotheerrormeansquareisanFstatisticthatcanbeusedtotestthehypothesisofequaltreatmentmeansExtensionoftheANOVAtotheRCBD3/17/0449DOEANOVADisplayfortheRCBDManualcomputing(ugh!)…seeEquations(4-9)–(4-12),page131Design-ExpertanalyzestheRCBD3/17/0450DOEHardnessTestingRevisited

Design-ExpertOutput

Response:

Hardness

ANOVAforSelectedFactorialModel

Analysisofvariancetable[Partialsumofsquares]

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Block 82.50 3 27.50

Model 38.50 3 12.83 14.44 0.0009

A

38.50

3

12.83

14.44

0.0009

Residual 8.00 9 0.89

CorTotal 129.00 15

Std.Dev. 0.94 R-Squared 0.8280

Mean 1.25 AdjR-Squared 0.7706

C.V. 75.42 PredR-Squared 0.4563 PRESS 25.28 AdeqPrecision 15.6353/17/0451DOEResidualAnalysisfortheHardness

TestingExperimentHardnessResidualNormal%probabilityNormalplotofresiduals-1-0.3750.250.8751.5151020305070809095993/17/0452DOEResidualAnalysisforthe

HardnessTestingExperimentHardness2222PredictedResidualsResidualsvs.Predicted-1-0.3750.250.8751.5-2.75-0.312.134.567.00HardnessRunNumberResidualsResidualsvs.Run-1-0.3750.250.8751.51471013163/17/0453DOEResidualAnalysisforthe

HardnessTestingExperiment殘差圖(residualplots)顯示常態(tài)性假設(shè)、變異數(shù)相等假設(shè)是滿足的。沒有相關(guān)隨機(jī)性的問題(randomization)同時(shí)畫出(殘差v.s.尖銳物種類)圖；(殘差v.s.區(qū)集因子)圖，將可從中得到更多相關(guān)資訊。3/17/0454DOEMultipleComparisonsfortheHardnessTestingExperiment–WhichTipsareDifferent?

TreatmentMeans(Adjusted,IfNecessary)

Estimated

Standard

Mean

Error

1-T1 0.75 0.47

2-T2 1.00 0.47

3-T3 -0.50 0.47

4-T4 3.75 0.47

Mean

StandardtforH0

Treatment

Difference

DF

Error

Coeff=0Prob>|t|

1vs2 -0.25 1 0.67 -0.38 0.7163

1vs3 1.25 1 0.67 1.87 0.0935

1vs4 -3.00 1 0.67 -4.50 0.0015

2vs3 1.50 1 0.67 2.25 0.0510

2vs4 -2.75 1 0.67 -4.13 0.0026

3vs4 -4.25 1 0.67 -6.38 0.0001

AlsoseeFigure4-3,Pg.1353/17/0455DOEOtherAspectsoftheRCBD

SeeText,Section4-1.3,pg.136TheRCBDutilizesan(加法模式)additivemodel–無交互作用存在於因子與區(qū)集之間。Treatmentsand/orblocksasrandomeffectsMissingvaluesSamplesizingintheRCBD?TheOCcurveapproachcanbeusedtodeterminethenumberofblockstorun..seepage1393/17/0456DOE拉丁方格

(TheLatinSquareDesign)此設(shè)計(jì)用於同時(shí)處理兩項(xiàng)干擾因子所造成的變異來源。(twosourcesofnuisancevariability)重要的假設(shè)是三因子(處理及兩干擾因子)沒有交互作用。如果違反以上的假設(shè)則拉丁方格將無法算出正確的結(jié)果。工業(yè)上拉丁方格不如RCBD常用。3/17/0457DOETheRocketPropellantProblem–ALatinSquareDesignThisisaPage145showssomeotherLatinsquaresTable4-13(page148)containspropertiesofLatinsquaresStatisticalanalysis?3/17/0458DOE統(tǒng)計(jì)軟體應(yīng)用一因子變異數(shù)分析含棉量與伸張力模式適當(dāng)性檢驗(yàn)處理平均數(shù)間的比較RCBD區(qū)集化設(shè)計(jì)變異數(shù)分析RCBD模式適當(dāng)性檢驗(yàn)3/17/0459DOE實(shí)驗(yàn)設(shè)計(jì)-因子設(shè)計(jì)簡介主要的原理二因子設(shè)計(jì)因子設(shè)計(jì)的ANOVA

分析兩個(gè)以上因子的設(shè)計(jì)計(jì)量值與計(jì)數(shù)值因子的設(shè)計(jì)-反應(yīng)曲線跟反應(yīng)曲面3/17/0460DOE二因子全因子設(shè)計(jì)因子效果:當(dāng)因子的設(shè)定由低水準(zhǔn)到高水準(zhǔn)所改變的平均反應(yīng)值3/17/0461DOE二因子設(shè)計(jì)交互作用3/17/0462DOE迴歸模式和

相關(guān)的反應(yīng)曲面3/17/0463DOE反應(yīng)曲面上的

交互作用效果將交互作用項(xiàng)目加入模式中:交互作用實(shí)際上是一個(gè)曲線3/17/0464DOE二因子23設(shè)計(jì)範(fàn)例電池壽命的實(shí)驗(yàn)A=Materialtype;B=Temperature(A是計(jì)量變數(shù)(quantitative))材料的型式跟溫度對(duì)電池壽命有何影響?2.是否可選擇材料的型式以增加電池壽命而不管溫度的效應(yīng)?3/17/0465DOE二因子實(shí)驗(yàn)設(shè)計(jì)一般型式alevelsoffactorA;blevelsoffactorB;nreplicates這是完全隨機(jī)試驗(yàn)CRD(completelyrandomizeddesign)3/17/0466DOEStatistical(effects)model:3/17/0467DOE二因子設(shè)計(jì)的ANOVA分析3/17/0468DOE二因子設(shè)計(jì)之ANOVATable3/17/0469DOEDesign-ExpertOutput–Example5-1

Response:

Life

ANOVAforSelectedFactorialModel

Analysisofvariancetable[Partialsumofsquares]

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Model 59416.22 8 7427.03 11.00<0.0001

A

10683.72

2

5341.86

7.91

0.0020

B

39118.72

2

19559.36

28.97

<0.0001

AB

9613.78

4

2403.44

3.56

0.0186

PureE18230.7527 675.21

CTotal 77646.9735

Std.Dev. 25.98 R-Squared 0.7652

Mean 105.53 AdjR-Squared 0.6956

C.V. 24.62 PredR-Squared 0.5826 PRESS 32410.22 AdeqPrecision 8.1783/17/0470DOE殘差圖分析LifeResidualNormal%probabilityNormalplotofresiduals-60.75-34.25-7.7518.7545.2515102030507080909599LifePredictedResidualsResidualsvs.Predicted-60.75-34.25-7.7518.7545.2549.5076.06102.62129.19155.753/17/0471DOE殘差圖分析DESIGN-EXPERTPlotLifeRunNumberResidualsResidualsvs.Run-60.75-34.25-7.7518.7545.25161116212631363/17/0472DOE殘差圖分析DESIGN-EXPERTPlotLifeMaterialResidualsResidualsvs.Material-60.75-34.25-7.7518.7545.25123DESIGN-EXPERTPlotLifeTemperatureResidualsResidualsvs.Temperature-60.75-34.25-7.7518.7545.251233/17/0473DOE交互作用圖DESIGN-EXPERTPlotLifeX=B:TemperatureY=A:MaterialA1A1A2A2A3A3A:MaterialInteractionGraphLifeB:Temperature157012520621041461882222223/17/0474DOE計(jì)量值與計(jì)數(shù)值的因子基本的ANOVA分析步驟可以處理計(jì)數(shù)值資料有時(shí)候?qū)嶒?yàn)包含計(jì)數(shù)值與計(jì)量值的因子如此資料型態(tài)可以用迴歸模式在計(jì)量值的因子上表示計(jì)數(shù)值的水準(zhǔn)這些反應(yīng)曲線和反應(yīng)曲面通常在實(shí)務(wù)上是可以解釋的3/17/0475DOE計(jì)量值與計(jì)數(shù)值因子

Response:

Life

***WARNING:TheCubicModelisAliased!***

SequentialModelSumofSquares

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F Mean4.009E+0051 4009E+005

Linear

49726.39

3

16575.46

19.00

<0.0001

Suggested

2FI 2315.08 2 1157.54 1.36 0.2730

Quadratic 76.06 1 76.06 0.086 0.7709

Cubic 7298.69 2 3649.35 5.40 0.0106 Aliased

Residual 18230.75 27 675.21

Total4.785E+00536 13292.97

"SequentialModelSumofSquares":Selectthehighestorderpolynomialwherethe

additionaltermsaresignificant.3/17/0476DOE計(jì)量值與計(jì)數(shù)值因子CandidatemodeltermsfromDesign-Expert: Intercept

A

B

B2

AB

B3

AB2A=MaterialtypeB=LineareffectofTemperatureB2=Quadraticeffectof TemperatureAB=Materialtype–TempLinearAB2=Materialtype-TempQuadB3=Cubiceffectof Temperature(Aliased)3/17/0477DOE計(jì)量值與計(jì)數(shù)值因子

LackofFitTests

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Linear

9689.83

5

1937.97

2.87

0.0333

Suggested 2FI 7374.75 3 2458.25 3.64 0.0252

Quadratic 7298.69 2 3649.35 5.40 0.0106

Cubic 0.00 0 Aliased

PureError18230.7527 675.21

"LackofFitTests":Wanttheselectedmodeltohaveinsignificantlack-of-fit.3/17/0478DOE計(jì)量值與計(jì)數(shù)值因子

ModelSummaryStatistics

Std.

Adjusted

Predicted

Source

Dev.

R-Squared

R-Squared

R-Squared

PRESS

Linear

29.54

0.6404

0.6067

0.5432

35470.60

Suggested 2FI 29.22 0.6702 0.6153 0.5187 37371.08

Quadratic 29.67 0.6712 0.6032 0.4900 39600.97

Cubic 25.98 0.7652 0.6956 0.5826 32410.22 Aliased

"ModelSummaryStatistics":Focusonthemodelmaximizingthe"AdjustedR-Squared"

andthe"PredictedR-Squared".

3/17/0479DOE計(jì)量值與計(jì)數(shù)值因子

Response:

Life

ANOVAforResponseSurfaceReducedCubicModel

Analysisofvariancetable[Partialsumofsquares]

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Model 59416.22 8 7427.03 11.00 <0.0001

A

10683.72

2

5341.86

7.91

0.0020

B

39042.67

1

39042.67

57.82

<0.0001

B2

76.06

1

76.06

0.11

0.7398

AB

2315.08

2

1157.54

1.71

0.1991

AB2

7298.69

2

3649.35

5.40

0.0106

PureE18230.75 27 675.21

CTotal77646.9735 Std.Dev. 25.98 R-Squared 0.7652

Mean 105.53 AdjR-Squared 0.6956

C.V. 24.62 PredR-Squared 0.5826 PRESS 32410.22 AdeqPrecision 8.1783/17/0480DOE迴歸模式的結(jié)果

FinalEquationinTermsofActualFactors:

Material A1

Life =

+169.38017

-2.50145 *Temperature

+0.012851 *Temperature2

Material A2

Life =

+159.62397

-0.17335 *Temperature

-5.66116E-003*Temperature2

Material A3

Life =

+132.76240

+0.90289 *Temperature

-0.010248 *Temperature2

3/17/0481DOE迴歸模式的結(jié)果DESIGN-EXPERTPlotLifeX=B:TemperatureY=A:MaterialA1A1A2A2A3A3A:MaterialInteractionGraphLifeB:Temperature15.0042.5070.0097.50125.0020621041461882222223/17/0482DOE兩因子以上的因子設(shè)計(jì)基本步驟與兩因子設(shè)計(jì)類似;所有的實(shí)驗(yàn)組合以隨機(jī)的方式進(jìn)行ANOVA的分析也是相同的3/17/0483DOE2k

的因子設(shè)計(jì)因子設(shè)計(jì)的特例k

個(gè)因子均具有兩水準(zhǔn)兩水準(zhǔn)通常稱一個(gè)為低水準(zhǔn)一個(gè)為高水準(zhǔn)(他們也可以是計(jì)量值或計(jì)數(shù)值)大量使用於工業(yè)上的實(shí)驗(yàn)有特殊的簡易的解法可以用Design-Expert軟體計(jì)算3/17/0484DOETheSimplestCase:The22“-”and“+”代表低和高的水準(zhǔn)低和高的水準(zhǔn)是自訂的幾何上而言，四次的實(shí)驗(yàn)形成方形的四個(gè)角因子可以是計(jì)量值或計(jì)數(shù)值，雖然他們最終的處理結(jié)果不同3/17/0485DOE化工製程的範(fàn)例A=反應(yīng)的內(nèi)容,B=觸媒的量,y=recovery

3/17/0486DOE因子設(shè)計(jì)的分析步驟估計(jì)因子的效果模型的形成使用全因子模型重複實(shí)驗(yàn)使用常態(tài)機(jī)率圖不重複實(shí)驗(yàn)統(tǒng)計(jì)檢定ANOVA分析重新調(diào)整模型分析殘差(殘差圖分析)解釋分析的結(jié)果3/17/0487DOE因子效果的估計(jì)效果估計(jì)是:A=8.33,B=-5.00,AB=1.67實(shí)務(wù)上的解釋?Design-Expert輸出的分析3/17/0488DOE因子效果的估計(jì)

Term Effect SumSqr %Contribution

Model Intercept

Model A 8.33333208.333 64.4995

Model B -5 75 23.2198

Model AB 1.666678.333332.57998

Error LackOfFit0 0

Error PError31.33339.70072

Lenth'sME 6.15809

Lenth'sSME 7.95671 3/17/0489DOE統(tǒng)計(jì)檢定ANOVA分析

Response:

Conversion

ANOVAforSelectedFactorialModel

Analysisofvariancetable[Partialsumofsquares]

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Model 291.67 3 97.22 24.82 0.0002

A

208.33

1

208.33

53.19

<0.0001

B

75.00

1

75.00

19.15

0.0024

AB

8.33

1

8.33

2.13

0.1828

PureError 31.33 8 3.92

CorTotal 323.00 11

Std.Dev. 1.98 R-Squared 0.9030

Mean 27.50 AdjR-Squared 0.8666

C.V. 7.20 PredR-Squared 0.7817 PRESS 70.50 AdeqPrecision 11.669

TheF-testforthe“model”sourceistestingthesignificanceoftheoverallmodel;thatis,iseitherA,B,orABorsomecombinationoftheseeffectsimportant?3/17/0490DOE統(tǒng)計(jì)檢定ANOVA分析

Coefficient

Standard

95%CI

95%CI

Factor

Estimate

DF

Error

Low

High

VIF

Intercept 27.501 0.57 26.18 28.82

A-Concent4.171 0.57 2.85 5.48 1.00

B-Catalyst-2.501 0.57-3.82-1.18 1.00

AB 0.831 0.57-0.48 2.15 1.00

Generalformulasforthestandarderrorsofthemodelcoefficientsandtheconfidenceintervalsareavailable.Theywillbegivenlater.3/17/0491DOE調(diào)整模型

Response:

Conversion

ANOVAforSelectedFactorialModel

Analysisofvariancetable[Partialsumofsquares]

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Model 283.33 2 141.67 32.14 <0.0001

A

208.33

1

208.33

47.27

<0.0001

B

75.00

1

75.00

17.02

0.0026

Residual 39.67 9 4.41

LackofFit

8.33

1

8.33

2.13

0.1828

PureError

31.33

8

3.92

CorTotal 323.00 11

Std.Dev. 2.10 R-Squared 0.8772

Mean 27.50 AdjR-Squared 0.8499

C.V. 7.63 PredR-Squared 0.7817 PRESS 70.52 AdeqPrecision 12.702

Thereisnowaresidualsumofsquares,partitionedintoa“l(fā)ackoffit”component(theABinteraction)anda“pureerror”component3/17/0492DOERegressionModelfortheProcess3/17/0493DOE殘差與診斷檢查DESIGN-EXPERTPlotConversionResidualNormal%probabilityNormalplotofresiduals-2.83333-1.58333-0.3333330.9166672.1666715102030507080909599DESIGN-EXPERTPlotConversion22PredictedResidualsResidualsvs.Predicted-2.83333-1.58333-0.3333330.9166672.1666720.8324.1727.5030.8334.173/17/0494DOE反應(yīng)曲面DESIGN-EXPERTPlotConversionX=A:ConcentrationY=B:CatalystDesignPointsConversionA:ConcentrationB:Catalyst15.0017.5020.0022.5025.001.001.251.501.752.0023.055625.277827.529.722231.94443333DESIGN-EXPERTPlotConversionX=A:ConcentrationY=B:Catalyst20.833324.166727.530.833334.1667

Conversion15.0017.5020.0022.5025.001.001.251.501.752.00

A:Concentration

B:Catalyst3/17/0495DOE23

因子設(shè)計(jì)3/17/0496DOE23因子設(shè)計(jì)的效果Analysisdoneviacomputer3/17/0497DOE23因子設(shè)計(jì)的範(fàn)例A=carbonation,B=pressure,C=speed,y=filldeviation3/17/0498DOEFactorialEffectTreatmentCombinationIABABCACBCABC(1)=-4+--+-++-a=1++----++b=-1+-+--+-+ab=5++++----c=-1+--++--+ac=3++--++--bc=2+-+-+-+-abc=11++++++++Contrast

2418614244Effect

3.002.250.751.750.250.500.50

Tableof–and+Signsforthe23FactorialDesign3/17/0499DOE表格內(nèi)的性質(zhì)每一欄有一樣數(shù)目的“+”跟“–”

每一欄的總和為零每一欄乘上I使得每一欄的值不改變?nèi)蝺尚械某朔e產(chǎn)生表格中的另一行直交設(shè)計(jì)因子設(shè)計(jì)中直交性是非常重要的特性3/17/04100DOE因子效果的估計(jì)

Term Effect SumSqr %Contribution

Model Intercept

Error A 3 36 46.1538

Error B 2.25 20.25 25.9615

Error C 1.75 12.25 15.7051

Error AB 0.75 2.25 2.88462

Error AC 0.25 0.25 0.320513

Error BC 0.5 1 1.28205

Error ABC 0.5 1 1.28205

Error LOF 0

Error PError 5 6.41026

Lenth'sME 1.25382

Lenth'sSME 1.88156 3/17/04101DOE全因子模型的ANOVA分析

Response:

Fill-deviation

ANOVAforSelectedFactorialModel

Analysisofvariancetable[Partialsumofsquares]

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Model 73.00 7 10.43 16.69 0.0003

A

36.00

1

36.00

57.60

<0.0001

B

20.25

1

20.25

32.40

0.0005

C

12.25

1

12.25

19.60

0.0022

AB

2.25

1

2.25

3.60

0.0943

AC

0.25

1

0.25

0.40

0.5447

BC

1.00

1

1.00

1.60

0.2415

ABC

1.00

1

1.00

1.60

0.2415

PureError 5.00 8 0.63

CorTotal 78.00 15

Std.Dev. 0.79 R-Squared 0.9359

Mean 1.00 AdjR-Squared 0.8798

C.V. 79.06 PredR-Squared 0.7436 PRESS 20.00 AdeqPrecision 13.4163/17/04102DOE全因子模型的

係數(shù)(Coefficients)

Coefficient

Standard

95%CI95%CI

Factor

Estimate

DF

Error

Low

High

VIF Intercept 1.00 1 0.20 0.54 1.46 A-Carbonation1.50 1 0.20 1.04 1.961.00

B-Pressure1.13 1 0.20 0.67 1.581.00

C-Speed 0.88 1 0.20 0.42 1.331.00

AB 0.38 1 0.20-0.081 0.831.00

AC 0.13 1 0.20-0.33 0.581.00

BC 0.25 1 0.20 -0.21 0.711.00

ABC 0.25 1 0.20 -0.21 0.711.00

3/17/04103DOE調(diào)整模型-移除不顯著的因子

Response:

Fill-deviation

ANOVAforSelectedFactorialModel

Analysisofvariancetable[Partialsumofsquares]

Sumof

Mean

F

Source

Squares

DF

Square

Value

Prob>F

Model 70.75 4 17.69 26.84 <0.0001

A

36.00

1

36.00

54.62

<0.0001

B

20.25

1

20.25

30.72

0.0002

C

12.25

1

12.25

18.59

0.0012

AB

2.25

1

2.25

3.41

0.0917

Residual 7.25 11 0.66

LOF

2.25

3

0.75

1.20

0.3700

PureE

5.00

8

0.63

CTotal 78.00 15

Std.Dev. 0.81 R-Squared 0.9071

Mean 1.00 AdjR-Squared 0.8733

C.V. 81.18 PredR-Squared 0.8033 PRESS 15.34 AdeqPrecision 15.4243/17/04104DOE模型係數(shù)–縮減模型

Coefficient

Standard

95%CI95%CI

Factor

Estimate

DF

Error

Low

High

Intercept 1.00 1 0.20 0.55 1.45

A-Carbonation 1.50 1 0.20 1.05 1.95

B-Pressure 1.13 1 0.20 0.68 1.57

C-Speed 0.88 1 0.20 0.43 1.32

AB 0.38 1 0.20 -0.072 0.82

3/17/04105DOE模型的統(tǒng)計(jì)分析表R2andadjustedR2

R2forprediction(basedonPRESS)3/17/04106DOE迴歸模型

FinalEquationinTermsofCodedFactors:

Fill-deviation =

+1.00

+1.50 *A

+1.13 *B

+0.88 *C

+0.38 *A*B

FinalEquationinTermsofActualFactors:

Fill-deviation =

+9.62500

-2.62500 *Carbonation

-1.20000 *Pressure

+0.035000 *Speed

+0.15000 *Carbonation*Pressure

3/17/04107DOE殘差圖是滿足的3/17/04108DOE模型的解釋DESIGN-EXPERTPlotFill-deviationX=A:CarbonationY=B:PressureB-25.000B+30.000ActualFactorC:Speed=225.00B:PressureInteractionGraphFill-deviationA:Carbonation10.0010.5011.0011.5012.00-3-0.751.53.756有溫和的交互作用於碳與壓力之間3/17/04109DOE模型的解釋DESIGN-EXPERTPlotFill-deviationX=A:CarbonationY=B:PressureZ=C:SpeedCubeGraphFill-deviationA:CarbonationB:PressureC:SpeedA-A+B-B+C-C+-2.13-0.37-0.631.130.121.883.134.88立體圖通常可以幫助實(shí)驗(yàn)結(jié)果的視覺化3/17/04110DOE反應(yīng)曲線曲面圖–

在高水準(zhǔn)的速度DESIGN-EXPERTPlotFill-deviationX=A:CarbonationY=B:PressureDesignPointsActualFactorC:Speed=250.00Fill-deviationA:CarbonationB:Pressure10.0010.5011.0011.5012.0025.0026.2527.5028.7530.000.51.3752.253.12522223/17/04111DOE一般2k

的因子設(shè)計(jì)k

個(gè)主要因子效果3/17/04112DOE不重複實(shí)驗(yàn)的2k

因子設(shè)計(jì)2k因子設(shè)計(jì)每一實(shí)驗(yàn)組合只有一個(gè)觀測值不重複實(shí)驗(yàn)的2k因子設(shè)計(jì)也稱為一次的2k實(shí)驗(yàn)這種設(shè)計(jì)常常被使用風(fēng)險(xiǎn)…如果只有一個(gè)觀測值在角落上會(huì)有一些異常的值影響到整個(gè)結(jié)果這是模型的“雜訊”嗎?3/17/04113DOE不重複實(shí)驗(yàn)2k

因子設(shè)計(jì)的水準(zhǔn)間距如果因子水準(zhǔn)設(shè)的太接近，將增加雜訊的機(jī)會(huì)影響資料，加大水準(zhǔn)的間隔將使結(jié)果較佳。3/17/04114DOE缺乏重複實(shí)驗(yàn)將造成統(tǒng)計(jì)檢定的問題-重複實(shí)驗(yàn)可以估計(jì)內(nèi)部的誤差-沒有重複實(shí)驗(yàn)將造成零自由度-可以解決的辦法如-選擇高次項(xiàng)交互作用估計(jì)誤差-常態(tài)機(jī)率圖的效果不重複實(shí)驗(yàn)2k

因子設(shè)計(jì)3/17/04115DOE不重複實(shí)驗(yàn)因子設(shè)計(jì)的2k範(fàn)例一個(gè)

24

的因子設(shè)計(jì)被用來探討一個(gè)過濾器的過濾效果A=溫度,B=壓力,C=分子量,D=攪拌率3/17/04116DOETheResinPlantExperiment3/17/04117DOETheResinPlantExperiment3/17/04118DOE因子效果的估計(jì)

Term Effect SumSqr %Contribution

Model Intercept

Error A 21.625 1870.56 32.6397

Error B 3.125 39.0625 0.681608

Error C 9.875 390.062 6.80626

Error D 14.625 855.563 14.9288

Error AB 0.125 0.0625 0.00109057

Error AC -18.125 1314.06 22.9293

Error AD 16.625 1105.56 19.2911

Error BC 2.375 22.5625 0.393696

Error BD -0.375 0.5625 0.00981515

Error CD -1.125 5.0625 0.0883363

Error ABC 1.875 14.0625 0.245379

Error ABD 4.125 68.0625 1.18763

Error ACD -1.625 10.5625 0.184307

Error BCD -2.625 27.5625 0.480942

Error ABCD 1.375 7.5625 0.131959

Lenth'sME 6.74778

Lenth'sSME 13.699 3/17/04119DOE常態(tài)機(jī)率圖的效果DESIGN-EXPERTPlotFiltrationRateA:TemperatureB:PressureC:ConcentrationD:StirringRateNo