Experimentos Fatoriais do tipo 2 k Capítulo 6. Analysis Procedure for a Factorial Design Estimate factor effects Formulate model –With replication, use.

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Presentation transcript:

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 Analyze residuals (graphical) Interpret results

The 2 3 Factorial Design

Effects in The 2 3 Factorial Design Analysis done via computer

An Example of a 2 3 Factorial Design A = gap, B = Flow, C = Power, y = Etch Rate

Table of – and + Signs for the 2 3 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

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 ** B C e-06 *** A:B A:C *** B:C A:B:C Residuals Signif. codes: 0 ‘***’ ‘**’ 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 *** C e-09 *** A:C e-06 *** Residuals Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Model Summary Statistics for Reduced Model R 2 and adjusted R 2 R 2 for prediction (based on PRESS)

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 2 k 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 2 k Factorial Design Section 6-4, pg. 227, Table 6-9, pg. 228 There will be k main effects, and