Business process improvement_15 doc

Initial Plots/Main Effects We can make the following conclusions based on the ordered data plot.. We can make the following conclusions based on the dex scatter plot.Important Factors: F

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5 Process Improvement

5.5 Advanced topics

5.5.9 An EDA approach to experimental design

5.5.9.10 DEX contour plot

5.5.9.10.6 How to Interpret: Optimal Curve

100 curve This is the optimal response curve.

5.5.9.10.6 How to Interpret: Optimal Curve

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5.5 Advanced topics

5.5.9 An EDA approach to experimental design

5.5.9.10 DEX contour plot

5.5.9.10.7 How to Interpret: Optimal Setting

estimated contour surface is identical to "nature's" response surface In reality, the

plotted contour curves are truth estimates based on the available (and "noisy") n =

8 data values We are confident of the contour curves in the vicinity of the data points (the four corner points on the chart), but as we move away from the corner

points, our confidence in the contour curves decreases Thus the point on the Y =

100 optimal response curve that is "most likely" to be valid is the one that is closest to a corner point Our objective then is to locate that "near-point".

Defective

springs

example

In terms of the defective springs contour plot, we draw a line from the best corner,

(+,+), outward and perpendicular to the Y = 90, Y = 95, and Y = 100 contour curves The Y = 100 intersection yields the "nearest point" on the optimal

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the original (uncoded) units"? That is, what does (X1=1.5, X3=1.3) correspond to

in the units of the original data?

To deduce his, we need to refer back to the original (uncoded) factors in this problem They were:

Coded Factor

Oven Temperature (1450 and 1600 degrees)

Diagram To determine the corresponding setting for (X1=1.5, X3=1.3), we thus refer to the

following diagram, which mimics a scatter plot of response averages oven temperature (OT) on the horizontal axis and quench temperature (QT) on the vertical axis:

5.5.9.10.7 How to Interpret: Optimal Setting

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The "X" on the chart represents the "near point" setting on the optimal curve.

Optimal

setting for

X1 (oven

temperature)

To determine what "X" is in uncoded units, we note (from the graph) that a linear

transformation between OT and X1 as defined by

and so X1 = +2, say, would be at oven temperature OT = 1675:

| -| -| -| X1: -1 0 +1 +2 OT: 1450 1525 1600 1675

and hence the optimal X1 setting of 1.5 must be at

OT = 1600 + 0.5*(1675-1600) = 1637.5

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setting for

X3 (quench

temperature)

Similarly, from the graph we note that a linear transformation between quench

temperature QT and coded factor X3 as specified by

and so X3 = +2, say, would be quench temperature = 145:

| -| -| -| X3: -1 0 +1 +2 QT: 70 95 120 145

Hence, the optimal X3 setting of 1.3 must be at

and finally, including the best setting of the fixed X2 factor (carbon concentration CC) of X2 = -1 (CC = 5%), we thus have the final, complete recommended

optimal settings for all three factors:

coded : (X1 = +1.5, X2 = -1.0, X3 = +1.3)

uncoded: (OT = 1637.5, CC = 7%, QT = 127.5)

If we were to run another experiment, this is the point (based on the data) that we would set oven temperature, carbon concentration, and quench temperature with the hope/goal of achieving 100% acceptable springs.

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Options for

next step

In practice, we could either

collect a single data point (if money and time are an issue) at this recommended setting and see how close to 100% we achieve, or

1

collect two, or preferably three, (if money and time are less of an issue) replicates at the center point (recommended setting).

2

if money and time are not an issue, run a 22 full factorial design with center

point The design is centered on the optimal setting (X1 = +1,5, X3 = +1.3) with one overlapping new corner point at (X1 = +1, X3 = +1) and with new corner points at (X1,X3) = (+1,+1), (+2,+1), (+1,+1.6), (+2,+1.6) Of these

four new corner points, the point (+1,+1) has the advantage that it overlaps with a corner point of the original design.

3

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5.6 Case Studies

Contents The purpose of this section is to illustrate the analysis of designed

experiments with data collected from experiments run at the National Institute of Standards and Technology and SEMATECH A secondary goal is to give the reader an opportunity to run the analyses in real-time using the Dataplot software package

Eddy current probe sensitivity study

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Background and data

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5.6 Case Studies

5.6.1 Eddy Current Probe Sensitivity Case Study

5.6.1.1 Background and Data

Background The data for this case study is a subset of a study performed by

Capobianco, Splett, and Iyer Capobianco was a member of the NIST Electromagnetics Division and Splett and Iyer were members of the NIST Statistical Engineering Division at the time of this study.

The goal of this project is to develop a nondestructive portable device for detecting cracks and fractures in metals A primary application would be the detection of defects in airplane wings The internal mechanism of the detector would be for sensing crack-induced changes in the detector's electromagnetic field, which would in turn result in changes in the impedance level of the detector This change of impedance is termed

"sensitivity" and it is a sub-goal of this experiment to maximize such sensitivity as the detector is moved from an unflawed region to a flawed region on the metal.

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Data Used

in the

Analysis

There were three detector wiring component factors under consideration:

X1 = Number of wire turns

1

X2 = Wire winding distance

2

X3 = Wire guage

3

Since the maximum number of runs that could be afforded timewise and

costwise in this experiment was n = 10, a 23 full factoral experiment

(involving n = 8 runs) was chosen With an eye to the usual monotonicity

assumption for 2-level factorial designs, the selected settings for the three factors were as follows:

X1 = Number of wire turns : -1 = 90, +1 = 180

1

X2 = Wire winding distance: -1 = 0.38, +1 = 1.14

2

X3 = Wire guage : -1 = 40, +1 = 48

3

The experiment was run with the 8 settings executed in random order The following data resulted.

Y X1 X2 X3 Probe Number Winding Wire Run Impedance of Turns Distance Guage Sequence

1.70 -1 -1 -1 2

4.57 +1 -1 -1 8

0.55 -1 +1 -1 3

3.39 +1 +1 -1 6

1.51 -1 -1 +1 7

4.59 +1 -1 +1 1

0.67 -1 +1 +1 4

4.29 +1 +1 +1 5

Note that the independent variables are coded as +1 and -1 These represent the low and high settings for the levels of each variable.

Factorial designs often have 2 levels for each factor (independent variable) with the levels being coded as -1 and +1 This is a scaling of the data that can simplify the analysis If desired, these scaled values can

be converted back to the original units of the data for presentation.

5.6.1.1 Background and Data

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5.6 Case Studies

5.6.1.2 Initial Plots/Main Effects

We can make the following conclusions based on the ordered data plot

Important Factors: The 4 highest response values have X1 = + while the 4 lowest response values have X1 = - This implies factor 1 is the most important factor When X1 = -, the - values of X2 are higher than the + values of X2 Similarly, when X1 = +, the - values of X2 are higher than the + values of X2 This implies X2 is important, but less so than X1 There

is no clear pattern for X3.

1

Best Settings: In this experiment, we are using the device as a detector, and so high

sensitivities are desirable Given this, our first pass at best settings yields (X1 = +1, X2 = -1, X3 = either).

2

5.6.1.2 Initial Plots/Main Effects

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

Important Factors: Factor 1 (Number of Turns) is clearly important When X1 = -1, all 4 senstivities are low, and when X1 = +1, all 4 sensitivities are high Factor 2 (Winding Distance) is less important The 4 sensitivities for X2 = -1 are slightly higher, as a group, than the 4 sensitivities for X2 = +1 Factor 3 (Wire Gage) does not appear to be important

at all The sensitivity is about the same (on the average) regardless of the settings for X3.

1

Best Settings: In this experiment, we are using the device as a detector, so high sensitivities

are desirable Given this, our first pass at best settings yields (X1 = +1, X2 = -1, X3 =

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factor X1 is clearly important, X2 is somewhat important, and X3 is probably not important.

The dex mean plot shows the main effects This provides probably the easiest to interpertindication of the important factors

We can make the following conclusions from the dex mean plot

Important Factors:

X1 (effect = large: about 3 ohms) X2 (effect = moderate: about -1 ohm) X3 (effect = small: about 1/4 ohm)

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of Plots

All of these plots are used primarily to detect the most important factors Because it plots asummary statistic rather than the raw data, the dex mean plot shows 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

5.6.1.2 Initial Plots/Main Effects

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We can make the following conclusions from the dex interaction effects plot.

Important Factors: Looking for the plots that have the steepest lines (that is, largesteffects), we note that:

X1 (number of turns) is the most important effect: estimated effect = -3.1025;

2

5.6.1.3 Interaction Effects

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5.6 Case Studies

5.6.1.4 Main and Interaction Effects: Block Plots

Block Plots Block plots are a useful adjunct to the dex mean plot and the dex interaction effects plot to

confirm the importance of factors, to establish the robustness of main effect conclusions, and todetermine the existence of interactions Specifically,

The first plot below answers the question: Is factor 1 important? If factor 1 is important, is

this importance robust over all 4 settings of X2 and X3?

1

The second plot below answers the question: Is factor 2 important? If factor 2 is important,

is this importance robust over all 4 settings of X1 and X3?

2

The third plot below answers the question: Is factor 3 important? If factor 3 is important, is

this importance robust over all 4 settings of X1 and X2?

3

For block plots, it is the height of the bars that is important, not the relative positioning of eachbar Hence we focus on the size and internals of the blocks, not "where" the blocks are onerelative to another

5.6.1.4 Main and Interaction Effects: Block Plots

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Relative Importance of Factors: All of the bar heights in plot 1 (turns) are greater than thebar heights in plots 2 and 3 Hence, factor 1 is more important than factors 2 and 3.

1

Statistical Significance: In plot 1, looking at the levels within each bar, we note that theresponse for level 2 is higher than level 1 in each of the 4 bars By chance, this happenswith probability 1/(24) = 1/16 = 6.25% Hence, factor 1 is near-statistically significant atthe 5% level Similarly, for plot 2, level 1 is greater than level 2 for all 4 bars Hence,factor 2 is near-statistically significant For factor 3, there is not consistent ordering withinall 4 bars and hence factor 3 is not statistically significant Rigorously speaking then,factors 1 and 2 are not statistically significant (since 6.25% is not < 5%); on the other handsuch near-significance is suggestive to the analyst that such factors may in fact be

important, and hence warrant further attention

Note that the usual method for determining statistical significance is to perform an analysis

of variance (ANOVA) ANOVA is based on normality assumptions If these normalityassumptions are in fact valid, then ANOVA methods are the most powerful method fordetermining statistical signficance The advantage of the block plot method is that it isbased on less rigorous assumptions than ANOVA At an exploratory stage, it is useful toknow that our conclusions regarding important factors are valid under a wide range ofassumptions

2

Interactions: For factor 1, the 4 bars do not change height in any systematic way and hence

there is no evidence of X1 interacting with either X2 or X3 Similarly, there is no evidence

of interactions for factor 2

3

5.6.1.4 Main and Interaction Effects: Block Plots

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5.6 Case Studies

5.6.1.5 Estimate Main and Interaction Effects

Effects

Estimation

Although the effect estimates were given on the dex interaction plot on a previous page, they can also be estimated quantitatively.

The full model for the 23 factorial design is

Data from factorial designs with two levels can be analyzed using the Yates technique, which is described in Box, Hunter, and Hunter The Yates technique utilizes the

special structure of these designs to simplify the computation and presentation of the fit.

Dataplot

Output

Dataplot generated the following output for the Yates analysis.

(NOTE DATA MUST BE IN STANDARD ORDER) NUMBER OF OBSERVATIONS = 8 NUMBER OF FACTORS = 3

NO REPLICATION CASE

PSEUDO-REPLICATION STAND DEV = 0.20152531564E+00 PSEUDO-DEGREES OF FREEDOM = 1

(THE PSEUDO-REP STAND DEV ASSUMES ALL

3, 4, 5, -TERM INTERACTIONS ARE NOT REAL, BUT MANIFESTATIONS OF RANDOM ERROR)

STANDARD DEVIATION OF A COEF = 0.14249992371E+00 (BASED ON PSEUDO-REP ST DEV.)

GRAND MEAN = 0.26587500572E+01 GRAND STANDARD DEVIATION = 0.17410624027E+01

99% CONFIDENCE LIMITS (+-) = 0.90710897446E+01 95% CONFIDENCE LIMITS (+-) = 0.18106349707E+01

5.6.1.5 Estimate Main and Interaction Effects

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