IV.3 Designs to Minimize IV.3 Designs to Minimize VariabilityVariability

BackgroundBackground An ExampleAn Example

– Design StepsDesign Steps– TransformationsTransformations– The AnalysisThe Analysis

A Case StudyA Case Study

BackgroundBackgroundAccuracy/PrecisionAccuracy/Precision

Factors Can Affect Response Variable Factors Can Affect Response Variable by Either by Either – Changing Its Average Value (Accuracy)Changing Its Average Value (Accuracy)– Changing Its Variation (Precision) orChanging Its Variation (Precision) or– BOTHBOTH

BackgroundBackgroundExample 4 - Example I.2.3 RevisitedExample 4 - Example I.2.3 Revisited

1

2

Which Factors AffectWhich Factors Affect– Accuracy?Accuracy?– Precision?Precision?

BackgroundBackgroundAnalysis for Changes in VariabilityAnalysis for Changes in Variability

For studying Variability, we can use For studying Variability, we can use ALL the designs, ALL the ideas that ALL the designs, ALL the ideas that we used when studying changes in we used when studying changes in mean response level.mean response level.

However,However,– Smaller Variability is Smaller Variability is ALWAYSALWAYS better better– We We MUSTMUST work with replicated work with replicated

experimentsexperiments– We will need to transform the response We will need to transform the response ss

Example 5Example 5Mounting an Integrated Circuit on SubstrateMounting an Integrated Circuit on Substrate

Figure 5 - Factor LevelFigure 5 - Factor LevelLochner and Matar - Figure 5.11Lochner and Matar - Figure 5.11

Response: bond strengthResponse: bond strength

Factor Low Level High LevelA. Adhesive Type D2A H-1-EB. Conductor Material Copper NickelC. Cure Time (at 90C) 90 min. 120 min.D. I.C. Post Coating Tin Silver

Example 5 - Design StepsExample 5 - Design StepsSelecting the DesignSelecting the Design

Figure 6 - The Experimental DesignFigure 6 - The Experimental DesignLochner and Matar - Figure 5.12Lochner and Matar - Figure 5.12

1. Select an appropriate experimental 1. Select an appropriate experimental designdesign

StandardOrder

AdhesiveType

ConductorMaterial

CureTime

I.C.Post

Coating1 D2A copper 90 tin2 D2A copper 120 silver3 D2A nickel 90 silver4 D2A nickel 120 tin5 H-1-E copper 90 silver6 H-1-E copper 120 tin7 H-1-E nickel 90 tin8 H-1-E nickel 120 silver

Example 5 - Design StepsExample 5 - Design StepsReplication and RandomizationReplication and Randomization

2. 2. Determine number of replicates to be usedDetermine number of replicates to be used– Consider at Least 5 (up to 10)Consider at Least 5 (up to 10)– In Example 5: In Example 5:

5 replicates, 40 trials5 replicates, 40 trials

3. 3. Randomize order of Randomize order of ALLALL trials trials– Replicates Run Sequentially Often Have Less Replicates Run Sequentially Often Have Less

Variation Than True Process VariationVariation Than True Process Variation– This May Be Inconvenient!This May Be Inconvenient!

Example 5 - Design StepsExample 5 - Design StepsCollecting the DataCollecting the DataFigure 7 - The DataFigure 7 - The Data

Lochner and Matar - Figure 5.13Lochner and Matar - Figure 5.13 4. Perform experiment; record data4. Perform experiment; record data 5. Group data for each factor level combination 5. Group data for each factor level combination

and calculate and calculate ss..

StandardOrder y1 y2 y3 y4 y5 y s Log(s)

1 73.0 73.2 72.8 72.2 76.2 73.48 1.57 0.1962 87.7 86.4 86.9 87.9 86.4 87.06 0.71 -0.1493 80.5 81.4 82.6 81.3 82.1 81.58 0.80 -0.0974 79.8 77.8 81.3 79.8 78.2 79.38 1.41 0.1495 85.2 85.0 80.4 85.2 83.6 83.88 2.06 0.3146 78.0 75.5 83.1 81.2 79.9 79.54 2.93 0.4677 78.4 72.8 80.5 78.4 67.9 75.60 5.17 0.7138 90.2 87.4 92.9 90.0 91.1 90.32 1.99 0.299

Example 5 - Design StepsExample 5 - Design StepsThe AnalysisThe Analysis

6. Calculate logarithms of standard 6. Calculate logarithms of standard deviations obtained in 5. Record deviations obtained in 5. Record these.these.

7. Analyze log s as the response.7. Analyze log s as the response.

TransformationsTransformationsWhy transform s?Why transform s?

If the data follow a bell-shaped curve, then If the data follow a bell-shaped curve, then so do the cell means and the factor effects so do the cell means and the factor effects for the means. However, the cell standard for the means. However, the cell standard deviations and factor effects of the deviations and factor effects of the standard deviations do not follow a bell-standard deviations do not follow a bell-shaped curve.shaped curve.

If we plot such data on our normal plotting If we plot such data on our normal plotting paper, we would obtain a graph that paper, we would obtain a graph that indicates important or unusual factor effects indicates important or unusual factor effects in the absence of any in the absence of any realreal effect. The log effect. The log transformation ‘normalizes’ the data.transformation ‘normalizes’ the data.

TransformationsTransformationsDistributions and Normal Probability Plots of sDistributions and Normal Probability Plots of s22

and Log(sand Log(s22))

Sampling from Normal

(n=5 sigma=1)

0.00

0.10

0.20

0.0 3.0 6.0 9.0 12.0

Sampling from Normal

(n=5 sigma=1)

0.00

0.30

0.60

-1.0 0.0 1.0 2.0 3.0

40 observations of s

-2.5

0.0

2.5

0.0 2.0 4.0 6.0 8.0

xxx xx

x xxxxxxx xx xxxx

x xxxxxxx

xx x x xx xxx xx

xx2

40 observations of Log s

-2.5

0.0

2.5

-0.80 0.00 0.80 1.60 2.40

xxx xx

x xxxxx xx xx xxxx

x xxxxxxx

xx x x xxxxx xx

xx

2

Example 5 - AnalysisExample 5 - AnalysisFigure 8 - Response Table for MeanFigure 8 - Response Table for Mean

Lochner and Matar - Figure 5.14Lochner and Matar - Figure 5.14

  y A B C AB AC BC D

Standard Order

Bond Strength

Adhesive Type

Conductor Material

Cure Time      

IC Post Coating

1 73.48 -1 -1 -1 1 1 1 -1

2 83.88 1 -1 -1 -1 -1 1 1

3 81.58 -1 1 -1 -1 1 -1 1

4 75.6 1 1 -1 1 -1 -1 -1

5 87.06 -1 -1 1 1 -1 -1 1

6 79.54 1 -1 1 -1 1 -1 -1

7 79.38 -1 1 1 -1 -1 1 -1

8 90.32 1 1 1 1 1 1 1

Sum 650.84 7.84 2.92 21.76 2.08 -1 3.28 34.84

Divisor 8 4 4 4 4 4 4 4

Effect 81.355 1.96 0.73 5.44 0.52 -0.25 0.82 8.71

Example 5 - AnalysisExample 5 - AnalysisFigure 9 - Response Table for Log(s)Figure 9 - Response Table for Log(s)

Lochner and Matar - Figure 5.15Lochner and Matar - Figure 5.15

  y A B C AB AC BC D

Standard Order Log(s)

Adhesive Type

Conductor Material

Cure Time      

IC Post Coating

1 0.196 -1 -1 -1 1 1 1 -1

2 0.314 1 -1 -1 -1 -1 1 1

3 -0.097 -1 1 -1 -1 1 -1 1

4 0.713 1 1 -1 1 -1 -1 -1

5 -0.149 -1 -1 1 1 -1 -1 1

6 0.467 1 -1 1 -1 1 -1 -1

7 0.149 -1 1 1 -1 -1 1 -1

8 0.299 1 1 1 1 1 1 1

Sum 1.892 1.694 0.236 -0.36 0.226 -0.162 0.024 -1.158

Divisor 8 4 4 4 4 4 4 4

Effect 0.2365 0.4235 0.059 -0.09 0.0565 -0.041 0.006 -0.2895

Example 5 - AnalysisExample 5 - AnalysisFigure 10 - Effects Normal Probability Plot for Figure 10 - Effects Normal Probability Plot for

MeanMean

9876543210

.999

.99

.95

.80

.50

.20

.05

.01

.001

Effects

A CD

What Factor What Factor Settings Favorably Settings Favorably Affect the Mean?Affect the Mean?

What Factor What Factor Settings Favorably Settings Favorably Affect the Mean?Affect the Mean?

Example 5 - AnalysisExample 5 - AnalysisFigure 11 - Effects Normal Probability Plot for Figure 11 - Effects Normal Probability Plot for

Log(s)Log(s)Lochner and Matar - Figure 5.16Lochner and Matar - Figure 5.16

0.40.30.20.10.0-0.1-0.2-0.3

.999

.99

.95

.80

.50

.20

.05

.01

.001

Effects

A

D

What Factor What Factor Settings Favorably Settings Favorably Affect Variability?Affect Variability?

What Factor What Factor Settings Favorably Settings Favorably Affect Variability?Affect Variability?

Example 5 - InterpretationExample 5 - Interpretation

Silver IC post coating Silver IC post coating increasesincreases bond bond strength strength andand decreasesdecreases variation in variation in bond strength.bond strength.

Adhesive D2A decreases variation in Adhesive D2A decreases variation in bond strength.bond strength.

120-minute cure time increases bond 120-minute cure time increases bond strength.strength.

Case Study 1Case Study 1Filling Weight of Dry Soup Mix - Factors and Filling Weight of Dry Soup Mix - Factors and

ResponseResponseFactor Low High

A: Number of Ports 1 3B: Cooling Method Water cooled Air cooledC: Mixing Time (secs) 60 80D: Batch weight (lbs) 1500 2000E: Delay (days) 1 7

Response: Weight of packet contents (oz)

A B C D E Log(s)1 1 1 1 2 -.01321 1 1 2 1 .26721 1 2 1 1 .07191 1 2 2 2 -.11921 2 1 1 1 .05311 2 1 2 2 -.10791 2 2 1 2 .16731 2 2 2 1 .03742 1 1 1 1 .23042 1 1 2 2 -.20762 1 2 1 2 -.00882 1 2 2 1 .32222 2 1 1 2 .09692 2 1 2 1 .13352 2 2 1 1 .10722 2 2 2 2 .0414

Case Study 1Case Study 1Filling Weight of Dry Soup Mix - Effects TableFilling Weight of Dry Soup Mix - Effects Table

Interpret This DataInterpret This Data– Determine the Determine the

Important EffectsImportant Effects– Do the Interaction Do the Interaction

Tables and Plots for Tables and Plots for Significant InteractionsSignificant Interactions

Interpret This DataInterpret This Data– Determine the Determine the

Important EffectsImportant Effects– Do the Interaction Do the Interaction

Tables and Plots for Tables and Plots for Significant InteractionsSignificant Interactions

Effect ValueA 0.04482B -0.00175C 0.02088D -0.04222E -0.17175

AB 0.01245AC 0.03132AD 0.00818AE -0.04610BC 0.02355BD -0.03780BE 0.13837CD 0.02828CE 0.05725DE -0.11665