experimentos fatoriais do tipo 2 k
DESCRIPTION
Experimentos Fatoriais do tipo 2 k. Capítulo 6. Analysis Procedure for a Factorial Design. Estimate factor effects Formulate model With replication, use full model With an unreplicated design, use normal probability plots Statistical testing (ANOVA) Refine the model - PowerPoint PPT PresentationTRANSCRIPT
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Experimentos Fatoriais do tipo 2k
Capítulo 6
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Analysis Procedure for a Factorial Design
• Estimate factor effects• Formulate model
– With replication, use full model– With an unreplicated design, use normal
probability plots
• Statistical testing (ANOVA)• Refine the model• Analyze residuals (graphical)• Interpret results
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The 23 Factorial Design
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Effects in The 23 Factorial Design
etc, etc, ...
A A
B B
C C
A y y
B y y
C y y
Analysis done via computer
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An Example of a 23 Factorial Design
A = gap, B = Flow, C = Power, y = Etch Rate
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Table of – and + Signs for the 23 Factorial Design (pg. 218)
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Properties of the Table
• Except for column I, every column has an equal number of + and – signs
• The sum of the product of signs in any two columns is zero• Multiplying any column by I leaves that column unchanged
(identity element)• The product of any two columns yields a column in the table:
• Orthogonal design• Orthogonality is an important property shared by all factorial
designs
2
A B AB
AB BC AB C AC
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Ajuste do Modelo usando o R
• dados=read.table("e:\\dox\\pfat2cubo.txt",header=T)
• A=as.factor(dados$A)• B=as.factor(dados$B)• C=as.factor(dados$C)• modeloC=dados$y~A+B+C+A:B+A:C+B:C+A:B:C• fitC=aov(modeloC)• summary(fitC)
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Resultados
Df Sum Sq Mean Sq F value Pr(>F) A 1 41311 41311 18.3394 0.0026786 ** B 1 218 218 0.0966 0.7639107 C 1 374850 374850 166.4105 1.233e-06 ***A:B 1 2475 2475 1.0988 0.3251679 A:C 1 94403 94403 41.9090 0.0001934 ***B:C 1 18 18 0.0080 0.9308486 A:B:C 1 127 127 0.0562 0.8185861 Residuals 8 18020 2253 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
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Estimation of Factor Effects
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ANOVA Summary – Full Model
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Model Coefficients – Full Model
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Refine Model – Remove Nonsignificant Factors
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Model Coefficients – Reduced Model
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Ajuste pelo R
modeloP=dados$y~A+C+A:C fitP=aov(modeloP) summary(fitP)
Df Sum Sq Mean Sq F value Pr(>F) A 1 41311 41311 23.767 0.0003816 ***C 1 374850 374850 215.661 4.951e-09 ***A:C 1 94403 94403 54.312 8.621e-06 ***Residuals 12 20858 1738 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
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Model Summary Statistics for Reduced Model
• R2 and adjusted R2
• R2 for prediction (based on PRESS)
52
5
25
5.106 100.9608
5.314 10
/ 20857.75 /121 1 0.9509
/ 5.314 10 /15
Model
T
E EAdj
T T
SSR
SS
SS dfR
SS df
2Pred 5
37080.441 1 0.9302
5.314 10T
PRESSR
SS
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Model Interpretation
Cube plots are often useful visual displays of experimental results
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Cube Plot of Ranges
What do the large ranges
when gap and power are at the high level tell
you?
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The 2k Factorial Design• Special case of the general factorial design; k
factors, all at two levels• The two levels are usually called low and high
(they could be either quantitative or qualitative)• Very widely used in industrial experimentation• Form a basic “building block” for other very
useful experimental designs (DNA)• Special (short-cut) methods for analysis
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The General 2k Factorial Design
• Section 6-4, pg. 227, Table 6-9, pg. 228
• There will be k main effects, and
two-factor interactions2
three-factor interactions3
1 factor interaction
k
k
k