Engineering Statistics Handbook Episode 8 Part 2 pps

Block plots for main and interaction effects factor 1 and factor 2 effects are consistent over all combinations of the other factors.. The Yates analysis shows that the factor 1 and fa

5.6.1.11 Conclusions and Next Step

3 Plots for interaction effects

1 Generate a dex interaction

effects matrix plot.

1 The dex interaction effects matrix plot does not show any major

interaction effects.

4 Block plots for main and interaction effects

factor 1 and factor 2 effects are consistent over all

combinations of the other factors.

5 Estimate main and interaction effects

1 Perform a Yates fit to estimate the

main effects and interaction effects.

1 The Yates analysis shows that the factor 1 and factor 2 main effects are significant, and the interaction for factors 2 and 3 is at the

boundary of statistical significance.

6 Model selection

1 Generate half-normal

probability plots of the effects.

2 Generate a Youden plot of the

effects.

1 The probability plot indicates that the model should include main effects for factors 1 and 2.

2 The Youden plot indicates that the model should include main effects for factors 1 and 2.

7 Model validation

1 Compute residuals and predicted values

from the partial model suggested by

the Yates analysis.

2 Generate residual plots to validate

the model.

1 Check the link for the values of the residual and predicted values.

2 The residual plots do not indicate any major problems with the model using main effects for factors 1 and 2.

5.6.1.12 Work This Example Yourself

8 Dex contour plot

1 Generate a dex contour plot using

factors 1 and 2.

1 The dex contour plot shows X1 = -1 and X2 = +1 to be the best settings.

5.6.1.12 Work This Example Yourself

5 Process Improvement

5.6 Case Studies

5.6.2 Sonoluminescent Light Intensity Case Study

5.6.2.1 Background and Data

Background

and

Motivation

Sonoluminescence is the process of turning sound energy into light An ultrasonic horn is used to resonate a bubble of air in a medium, usually water The bubble is ultrasonically compressed and then collapses to light-emitting plasma

In the general physics community, sonoluminescence studies are being carried out to characterize it, to understand it, and to uncover its

practical uses An unanswered question in the community is whether sonoluminescence may be used for cold fusion

NIST's motive for sonoluminescent investigations is to assess its suitability for the dissolution of physical samples, which is needed in the production of homogeneous Standard Reference Materials (SRMs)

It is believed that maximal dissolution coincides with maximal energy and maximal light intensity The ultimate motivation for striving for maximal dissolution is that this allows improved determination of alpha-and beta-emitting radionuclides in such samples

The objectives of the NIST experiment were to determine the important factors that affect sonoluminescent light intensity and to ascertain optimal settings of such factors that will predictably achieve high intensities An original list of 49 factors was reduced, based on physics reasons, to the following seven factors: molarity (amount of solute), solute type, pH, gas type in the water, water depth, horn depth, and flask clamping

Time restrictions caused the experiment to be about one month, which

in turn translated into an upper limit of roughly 20 runs A 7-factor, 2-level fractional factorial design (Resolution IV) was constructed and run The factor level settings are given below

Eva Wilcox and Ken Inn of the NIST Physics Laboratory conducted this experiment during 1999 Jim Filliben of the NIST Statistical

Engineering Division performed the analysis of the experimental data 5.6.2.1 Background and Data

Variable,

Factor

Variables,

and

Factor-Level

Settings

This experiment utilizes the following response and factor variables

Response Variable (Y) = The sonoluminescent light intensity.

1

Factor 1 (X1) = Molarity (amount of Solute) The coding is -1 for

0.10 mol and +1 for 0.33 mol

2

Factor 2 (X2) = Solute type The coding is -1 for sugar and +1 for

glycerol

3

Factor 3 (X3) = pH The coding is -1 for 3 and +1 for 11.

4

Factor 4 (X4) = Gas type in water The coding is -1 for helium

and +1 for air

5

Factor 5 (X5) = Water depth The coding is -1 for half and +1 for

full

6

Factor 6 (X6) = Horn depth The coding is -1 for 5 mm and +1 for

10 mm

7

Factor 7 (X7) = Flask clamping The coding is -1 for unclamped

and +1 for clamped

8

This data set has 16 observations It is a 27-3 design with no center points

Goal of the

Experiment

This case study demonstrates the analysis of a 27-3 fractional factorial experimental design The goals of this case study are:

Determine the important factors that affect the sonoluminescent light intensity Specifically, we are trying to maximize this intensity

1

Determine the best settings of the seven factors so as to maximize the sonoluminescent light intensity

2

Data

Used in

the

Analysis

The following are the data used for this analysis This data set is given in Yates order

Y X1 X2 X3 X4 X5 X6 X7

Light Solute Gas Water Horn Flask

Intensity Molarity type pH Type Depth Depth Clamping

80.6 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0

-1.0 66.1 1.0 -1.0 -1.0 -1.0 -1.0 1.0 1.0

59.1 -1.0 1.0 -1.0 -1.0 1.0 -1.0 1.0

68.9 1.0 1.0 -1.0 -1.0 1.0 1.0 -1.0

5.6.2.1 Background and Data

75.1 -1.0 -1.0 1.0 -1.0 1.0 1.0 1.0

373.8 1.0 -1.0 1.0 -1.0 1.0 -1.0 -1.0

66.8 -1.0 1.0 1.0 -1.0 -1.0 1.0 -1.0

79.6 1.0 1.0 1.0 -1.0 -1.0 -1.0 1.0

114.3 -1.0 -1.0 -1.0 1.0 1.0 1.0 -1.0

84.1 1.0 -1.0 -1.0 1.0 1.0 -1.0 1.0

68.4 -1.0 1.0 -1.0 1.0 -1.0 1.0 1.0

88.1 1.0 1.0 -1.0 1.0 -1.0 -1.0 -1.0

78.1 -1.0 -1.0 1.0 1.0 -1.0 -1.0 1.0

327.2 1.0 -1.0 1.0 1.0 -1.0 1.0 -1.0

77.6 -1.0 1.0 1.0 1.0 1.0 -1.0 -1.0

61.9 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Reading

Data into

Dataplot

These data can be read into Dataplot with the following commands SKIP 25

READ INN.DAT Y X1 TO X7 5.6.2.1 Background and Data

Plot the

Data: Dex

Scatter Plot

The next step in the analysis is to generate a dex scatter plot

Conclusions

from the

DEX

Scatter Plot

We can make the following conclusions based on the dex scatter plot

Important Factors: Again, two points dominate the plot For X1, X2, X3, and X7, these two points emanate from the same setting, (+, -, +, -), while for X4, X5, and X6 they emanate from different settings We conclude that X1, X2, X3, and X7 are potentially important, while X4, X5, and X6 are probably not important.

1

Best Settings: Our first pass at best settings yields (X1 = +, X2 = -, X3 = +, X4 = either, X5

= either, X6 = either, X7 = -).

2

Check for

Main

Effects: Dex

Mean Plot

The dex mean plot is generated to more clearly show the main effects:

5.6.2.2 Initial Plots/Main Effects

from the

DEX Mean

Plot

We can make the following conclusions from the dex mean plot

Important Factors:

X2 (effect = large: about -80) X7 (effect = large: about -80) X1 (effect = large: about 70) X3 (effect = large: about 65) X6 (effect = small: about -10) X5 (effect = small: between 5 and 10) X4 (effect = small: less than 5)

1

Best Settings: Here we step through each factor, one by one, and choose the setting that yields the highest average for the sonoluminescent light intensity:

(X1,X2,X3,X4,X5,X6,X7) = (+,-,+,+,+,-,-)

2

5.6.2.2 Initial Plots/Main Effects

of Plots

All of the above three plots are used primarily to determine the most important factors Because it plots a summary statistic rather than the raw data, the dex mean plot shows the ordering of the main effects most clearly However, it is still recommended to generate either the ordered data plot or the dex scatter plot (or both) Since these plot the raw data, they can sometimes reveal features of the data that might be masked by the dex mean plot

In this case, the ordered data plot and the dex scatter plot clearly show two dominant points This feature would not be obvious if we had generated only the dex mean plot

Interpretation-wise, the most important factor X2 (solute) will, on the average, change the light

intensity by about 80 units regardless of the settings of the other factors The other factors are interpreted similarly

In terms of the best settings, note that the ordered data plot, based on the maximum response value, yielded

+, , +, , +, , -Note that a consensus best value, with "." indicating a setting for which the three plots disagree, would be

+, , +, , +, ,

-Note that the factor for which the settings disagree, X4, invariably defines itself as an

"unimportant" factor

5.6.2.2 Initial Plots/Main Effects

from the

DEX

Interaction

Effects Plot

We make the following conclusions from the dex interaction effects plot

Important Factors: Looking for the plots that have the steepest lines (that is, the largest effects), and noting that the legends on each subplot give the estimated effect, we have that

The diagonal plots are the main effects The important factors are: X2, X7, X1, and

X3 These four factors have |effect| > 60 The remaining three factors have |effect| <

10

❍

The off-diagonal plots are the 2-factor interaction effects Of the 21 2-factor interactions, 9 are nominally important, but they fall into three groups of three:

X1*X3, X4*X6, X2*X7 (effect = 70)

■

X2*X3, X4*X5, X1*X7 (effect approximately 63.5)

■

X1*X2, X5*X6, X3*X7 (effect = -59.6)

■

All remaining 2-factor interactions are small having an |effect| < 20 A virtue of the interaction effects matrix plot is that the confounding structure of this Resolution IV

design can be read off the plot In this case, the fact that X1*X3, X4*X6, and X2*X7

all have effect estimates identical to 70 is not a mathematical coincidence It is a reflection of the fact that for this design, the three 2-factor interactions are

confounded This is also true for the other two sets of three (X2*X3, X4*X5, X1*X7, and X1*X2, X5*X6, X3*X7).

❍

1

Best Settings: Reading down the diagonal plots, we select, as before, the best settings "on the average":

(X1,X2,X3,X4,X5,X6,X7) = (+,-,+,+,+,-,-) For the more important factors (X1, X2, X3, X7), we note that the best settings (+, -, +, -)

are consistent with the best settings for the 2-factor interactions (cross-products):

X1: +, X2: with X1*X2: -X1: +, X3: + with X1*X3: + X1: +, X7: with X1*X7: X2: , X3: + with X2*X3: -X2: -, X7: - with X2*X7: + X3: +, X7: with X3*X7:

-2

5.6.2.3 Interaction Effects

from the

Block Plots

We can make the following conclusions from the block plots

Relative Importance of Factors: Because of the expanded vertical axis, due to the two

"outliers", the block plot is not particularly revealing Block plots based on alternatively scaled data (e.g., LOG(Y)) would be more informative

1

5.6.2.4 Main and Interaction Effects: Block Plots

from the

Youden plot

We can make the following conclusions from the Youden plot

In the upper left corner are the interaction term X1*X3 and the main effects X1 and X3.

1

In the lower right corner are the main effects X2 and X7 and the interaction terms X2*X3 and X1*X2.

2

The remaining terms are clustered in the center, which indicates that such effects have averages that are similar (and hence the effects are near zero), and so such effects are relatively unimportant

3

On the far right of the plot, the confounding structure is given (e.g., 13: 13+27+46), which

suggests that the information on X1*X3 (on the plot) must be tempered with the fact that

X1*X3 is confounded with X2*X7 and X4*X6.

4

5.6.2.5 Important Factors: Youden Plot