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Experimentos Fatoriais do tipo 2 k Capítulo 6

Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

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Page 1: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Experimentos Fatoriais do tipo 2k

Capítulo 6

Page 2: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

6.5 Unreplicated 2k Factorial Designs

• These are 2k factorial designs with one observation at each corner of the “cube”

• An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k

• These designs are very widely used• Risks…if there is only one observation at

each corner, is there a chance of unusual response observations spoiling the results?

• Modeling “noise”?

Page 3: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Spacing of Factor Levels in the Unreplicated 2k Factorial

Designs

If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data

More aggressive spacing is usually best

Page 4: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Unreplicated 2k Factorial Designs

• Lack of replication causes potential problems in statistical testing– Replication admits an estimate of “pure error” (a

better phrase is an internal estimate of error)– With no replication, fitting the full model results in

zero degrees of freedom for error

• Potential solutions to this problem– Pooling high-order interactions to estimate error– Normal probability plotting of effects (Daniels,

1959)– Other methods…see text

Page 5: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Example of an Unreplicated 2k Design

• A 24 factorial was used to investigate the effects of four factors on the filtration rate of a resin

• The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate

• Experiment was performed in a pilot plant

Page 6: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

The Resin Plant Experiment

Page 7: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

The Resin Plant Experiment

Page 8: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner
Page 9: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

dados=read.table("e:\\dox\\pfat2a4sr.txt",header=T) A=as.factor(dados$A) B=as.factor(dados$B) C=as.factor(dados$C) D=as.factor(dados$D)modeloC=dados$y~A+B+C+D+A:B+A:C+A:D+B:C+B:D+C:D+A:B:C+A:B:D+A:C:D+B:C:D+A:B:C:D

fitC=aov(modeloC)summary(fitC)

Df Sum Sq Mean SqA 1 1870.56 1870.56B 1 39.06 39.06C 1 390.06 390.06D 1 855.56 855.56A:B 1 0.06 0.06A:C 1 1314.06 1314.06A:D 1 1105.56 1105.56B:C 1 22.56 22.56B:D 1 0.56 0.56C:D 1 5.06 5.06A:B:C 1 14.06 14.06A:B:D 1 68.06 68.06A:C:D 1 10.56 10.56B:C:D 1 27.56 27.56A:B:C:D 1 7.56 7.56

Page 10: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Estimates of the Effects

Page 11: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

The Half-Normal Probability Plot of Effects

Page 12: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Design Projection: ANOVA Summary for the Model as a 23 in Factors A, C, and D

Page 13: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

modeloP=dados$y~A+C+D+A:C+A:D+C:D+A:C:D fitP=aov(modeloP)summary(fitP)

Df Sum Sq Mean Sq F value Pr(>F) A 1 1870.56 1870.56 83.3677 1.667e-05 ***C 1 390.06 390.06 17.3844 0.0031244 ** D 1 855.56 855.56 38.1309 0.0002666 ***A:C 1 1314.06 1314.06 58.5655 6.001e-05 ***A:D 1 1105.56 1105.56 49.2730 0.0001105 ***C:D 1 5.06 5.06 0.2256 0.6474830 A:C:D 1 10.56 10.56 0.4708 0.5120321 Residuals 8 179.50 22.44 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Page 14: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

The Regression Model

Page 15: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

regP=lm(dados$y~dados$A+dados$C+dados$D+dados$A*dados$C+dados$A*dados$D)summary(regP)

Residuals: Min 1Q Median 3Q Max -6.3750 -1.5000 0.0625 2.9063 5.7500

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 70.062 1.104 63.444 2.30e-14 ***dados$A 10.812 1.104 9.791 1.93e-06 ***dados$C 4.938 1.104 4.471 0.00120 ** dados$D 7.313 1.104 6.622 5.92e-05 ***dados$A:dados$C -9.062 1.104 -8.206 9.41e-06 ***dados$A:dados$D 8.312 1.104 7.527 2.00e-05 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.417 on 10 degrees of freedomMultiple R-squared: 0.966, Adjusted R-squared: 0.9489 F-statistic: 56.74 on 5 and 10 DF, p-value: 5.14e-07

Page 16: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Model Residuals are Satisfactory

Page 17: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Model Interpretation – Main Effects and Interactions

Page 18: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Outliers: suppose that cd = 375 (instead of 75)

Page 19: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Dealing with Outliers

• Replace with an estimate

• Make the highest-order interaction zero

• In this case, estimate cd such that ABCD = 0

• Analyze only the data you have

• Now the design isn’t orthogonal

• Consequences?

Page 20: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

The Drilling Experiment Example 6.3

A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill

Page 21: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Normal Probability Plot of Effects –The Drilling Experiment

Page 22: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Residual Plots

DESIGN-EXPERT Plotadv._rate

Predicted

Re

sid

ua

ls

Residuals vs. Predicted

-1.96375

-0.82625

0.31125

1.44875

2.58625

1.69 4.70 7.70 10.71 13.71

Page 23: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

• The residual plots indicate that there are problems with the equality of variance assumption

• The usual approach to this problem is to employ a transformation on the response

• Power family transformations are widely used

Transformations are typically performed to – Stabilize variance– Induce at least approximate normality– Simplify the model

Residual Plots

*y y

Page 24: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Selecting a Transformation

• Empirical selection of lambda• Prior (theoretical) knowledge or experience can

often suggest the form of a transformation• Analytical selection of lambda…the Box-Cox

(1964) method (simultaneously estimates the model parameters and the transformation parameter lambda)

• Box-Cox method implemented in Design-Expert

Page 25: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

(15.1)

Page 26: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

The Box-Cox MethodDESIGN-EXPERT Plotadv._rate

LambdaCurrent = 1Best = -0.23Low C.I. = -0.79High C.I. = 0.32

Recommend transform:Log (Lambda = 0)

Lambda

Ln

(Re

sid

ua

lSS

)

Box-Cox Plot for Power Transforms

1.05

2.50

3.95

5.40

6.85

-3 -2 -1 0 1 2 3

A log transformation is recommended

The procedure provides a confidence interval on the transformation parameter lambda

If unity is included in the confidence interval, no transformation would be needed

Page 27: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Effect Estimates Following the Log Transformation

Three main effects are large

No indication of large interaction effects

What happened to the interactions?

Page 28: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

ANOVA Following the Log Transformation

Page 29: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Following the Log Transformation

Page 30: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

The Log Advance Rate Model

• Is the log model “better”?

• We would generally prefer a simpler model in a transformed scale to a more complicated model in the original metric

• What happened to the interactions?

• Sometimes transformations provide insight into the underlying mechanism

Page 31: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Other Examples of Unreplicated 2k Designs

• The sidewall panel experiment (Example 6.4, pg. 245)– Two factors affect the mean number of defects– A third factor affects variability– Residual plots were useful in identifying the dispersion

effect

• The oxidation furnace experiment (Example 6.5, pg. 245)– Replicates versus repeat (or duplicate) observations?– Modeling within-run variability

Page 32: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

Por que trabalha-se com as variáveis de planejamento codificadas?

I R A B V

4 1 -1 -1 3.802

4 1 -1 -1 4.013

6 1 1 -1 6.065

6 1 1 -1 5.992

4 2 -1 1 7.934

4 2 -1 1 8.159

6 2 1 1 11.865

6 2 1 1 12.138

EXEMPLO

Page 33: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

• Na análise com as variáveis codificadas, as magnitudes dos coeficientes do modelo são diretamente comparáveis, isto é, elas são adimensionais, e medem os efeitos da variação de cada fator de planejamento sobre um intervalo unitário.

• Além disso, são todas estimadas com a mesma precisão.

• Variáveis codificadas são muito efetivas para determinar o tamanho relativo dos efeitos dos fatores.

Page 34: Experimentos Fatoriais do tipo 2 k Capítulo 6. 6.5 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner

• Em geral, os coeficientes obtidos usando-se as unidades originais não são diretamente comparáveis, mas eles podem ter significado físico.

• Em quase todas as situações, a análise codificada é preferível.


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