Experimentos Fatoriais do tipo 2k
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• Analyze residuals (graphical)• Interpret results
The 23 Factorial Design
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
An Example of a 23 Factorial Design
A = gap, B = Flow, C = Power, y = Etch Rate
Table of – and + Signs for the 23 Factorial Design (pg. 218)
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
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)
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
Estimation of Factor Effects
ANOVA Summary – Full Model
Model Coefficients – Full Model
Refine Model – Remove Nonsignificant Factors
Model Coefficients – Reduced Model
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
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
Model Interpretation
Cube plots are often useful visual displays of experimental results
Cube Plot of Ranges
What do the large ranges
when gap and power are at the high level tell
you?
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
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