Unreplicated 2 k Factorial Designs
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1 Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner of the cube An unreplicated 2 k factorial design is also sometimes called a single replicate of the 2 k More Statistics tutorial at 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?
2 Spacing of Factor Levels in the Unreplicated 2 k 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
3 Unreplicated 2 k 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, pp. 234
4 Example of an Unreplicated 2 k Design A 2 4 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
5 The Resin Plant Experiment
6 The Resin Plant Experiment
7 Main Effect A B C D Two-way Interaction AB AC AD BC BD CD Three-way interaction ABC ABD ACD BCD Four-way Interaction ABCD 1.375
8 Analysis of Variance Table Response: FRate Df Sum Sq Mean Sq F value Pr(>F) A B C D AB A:B A:C B:C A:D B:D C:D A:B:C ABD A:B:D A:C:D B:C:D A:B:C:D Residuals
9 What to do when there is no internal estimate of error? One approach to the analysis is to assume that certain high-order interactions are negligible and combine their mean squares to estimate the error. Occasionally real high-order interactions occur, then use the Q-Q Normal Plot (or half-normal plot) of the estimates of the effects. The effect that are negligible will tend to fall along a straight line whereas significant effects will not lie along the straight line.
10 Normal Q-Q plot 10 A D A:D tem$y 0 5 C B A:B:D B:C A:B:C:D A:B:C B:DA:B ACD A:C:D C:D B:C:D -5 A:C tem$x
11
12 Analysis of Variance Table Response: FRate Df Sum Sq Mean Sq F value Pr(>F) A e-05 C D A:C e-05 A:D C:D A:C:D Residuals
13 scoefsa scoefsc scoefsd levels(factor(a)) C levels(factor(c)) D levels(factor(d)) mean of pilotp lant[, 5] mean of pilotp lant[, 5] A -1 1 A
14 Final Model Checking Normal Q-Q Plot -6-6 Sample Quantiles resid r Theoretical Quantiles fitted
15 Model Interpretation Response Surface Plots contour plot with stirring rate D=1 contour plot with temperature A= C 0.0 D A C With concentration at either the low or high level, high temperature and high stirring rate results in high filtration rates
16 The Drilling Experiment Example 6-3, pg. 237 A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill
17 Half Normal Probability Plot of Effects The Drilling Experiment
18 Residual Analysis Normal Q-Q Plot Sample Qua resid ntiles Theoretical Quantiles fitted
19 Residual Plots 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 * y = y λ Transformations are typically performed to Stabilize variance Induce normality Simplify the model More Statistics tutorial at
20 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)
21 Box-Cox Transformation ihood log-likel % 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 λ
22 Effect Estimates Following the Log Transformation ts Effect D C B ts Effect Three main effects are large No indication of large C interaction effects D What happened to the interactions? B Half Normal plot Half Normal plot
23 ANOVA Following the Log Transformation
24 Following the Log Transformation Normal Q-Q Plot Sa ample Quantiles resid Theoretical Quantiles fitted
25 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
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