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Experimentos Fatoriais do tipo 2 k

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

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Experimentos Fatoriais do tipo 2 k

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  1. Experimentos Fatoriais do tipo 2k Capítulo 6

  2. 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

  3. The 23 Factorial Design

  4. Effects in The 23 Factorial Design Analysis done via computer

  5. An Example of a 23 Factorial Design A = gap, B = Flow, C = Power, y = Etch Rate

  6. Table of – and + Signs for the 23 Factorial Design (pg. 218)

  7. 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

  8. 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)

  9. 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

  10. Estimation of Factor Effects

  11. ANOVA Summary – Full Model

  12. Model Coefficients – Full Model

  13. Refine Model – Remove Nonsignificant Factors

  14. Model Coefficients – Reduced Model

  15. 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

  16. Model Summary Statistics for Reduced Model • R2 and adjusted R2 • R2 for prediction (based on PRESS)

  17. Model Interpretation Cubeplots are often useful visual displays of experimental results

  18. Cube Plot of Ranges What do the large ranges when gap and power are at the high level tell you?

  19. 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

  20. The General 2kFactorial Design • Section 6-4, pg. 227, Table 6-9, pg. 228 • There will be k main effects, and

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