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