DOE Shilpagupta 111906

8/2/2019 DOE Shilpagupta 111906

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Statistical Design of

ExperimentsBITS Pilani, November 19 2006

~ Shilpa Gupta (97A4)

[email protected]


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Quiz Design of Experiments

Did you attend the lecture on Design of Experiment part I ?

_______

Control chart help in distinguishing two types of ________

over time - ____________ and ___________ Difference between Control Charts and Design of

Experiments?

Three types of experimentation strategies are

____________, ______________, ______________


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Outline

Motivation for conducting Experiments Types of Experiments Applications of Experimental Designs Guidelines for Experimental Design

Choice of Factor and levels

Basic Principles Randomization

Replication Blocking

Factorial Design

Fractional Factorial Design Other Designs Research Topics References


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Objective is to optimize y,

Increase yield Decrease the number of defects

Reduced variability and closer conformance to

nominal

Reduced development time

Reduced overall costs Interested in determining:

x variables which are most influential on response y.

where to set influential xs so that y is near nominal

requirement.

where to set influential xs so that variability in y is

small.

where to set influential xs so that effects of

uncontrollable variables z are minimized.

Why study a process..?

Model of a System or a Process


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Design of Experiment

Series of changes made to input variables to observechanges in the output response

Three approaches

Best Guess approach - No guarantee of success.

One factor at a time (OFAT) - Fails to consider interaction

effects

Statistical Design of Experiments planning to gather data

that can be analyzed using statistical methods resulting in valid

and objective conclusions

Sophisticated QC tool and hence leads to significant gains in the

process as compared to the other tools


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Guidelines for Experimental Design*

* Coleman, D. E, and Montgomery, D. C. (1993), A Systematic Approach to Planning for a DesignedIndustrial Experiment, Technometrics, 35, pp 1-27


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Choice of Factor and Levels

Design FactorsHeld-constant

Allowed-to-vary

Nuisance FactorsControllable e.g.

Blocking

Uncontrollable e.g.

analysis of covariance

Noise e.g.

Robust design


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Principles

Blocking

Randomization

Replication


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Example#4

A product development engineer is interested ininvestigating the tensile strength of a new syntheticfiber that will be used to make cloth for mens shirt.

The engineer knows from past experience that the

strength of the fiber is affected by the weightpercentage of cotton content in the blend ofmaterials for the fiber. The engineer suspectsincreasing the cotton content will increase thestrength. The cotton content ranges from 10-40%. So

the engineer decides to test at 5 treatmentlevels:15, 20, 25, 30, 35


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Basic Principles Replication, Randomization and Blocking

Replication

Repetition of basic experiment and NOT repeatedmeasurements

Obtain an estimate of error

More precise estimate of the error (incase of mean)

Example: Take 5 replicates,

pick the runs randomly

Single replicate experiments Combine higher order interactions to obtainan estimate of error

Cotton

Weight

Percenta

ge

Experimental Run Number

Rep

1

Rep2 Rep

3

Rep 4 Rep 5

15

20

25

30

35


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Randomization Averaging out the effect of nuisance parameters

Suppose the 25 runs were not randomized, i.e. all 5 runs at15% were tested first followed by 5 runs at 20% and so on. If

the tensile strength testing machine exhibits warm-up effectwhich means the longer it is on, the lower tensile strengthreadings will be. This warmup effect will contaminate thetensile strength data and destroy the validity of theexperiment.

Restriction on randomization call for specialized

designs Randomized complete block design and Latin Squares

Split Plot Design Hard to change factors

Nested or Hierarchical Design

Basic Principles Replication, Randomization and Blocking


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Example - Demonstrate ANOVA

Tensile Strength experiment

Cotton

Weight

Percen

tage

Observation

Total Average Rep 1 Rep2 Rep 3 Rep 4 Rep 5

15 49 9.8 7 7 15 11 9

20 77 15.4 12 17 12 18 18

25 88 17.6 14 18 18 19 19

30 108 21.6 19 25 22 19 23

35 54 10.8 7 10 11 15 11

376 15.04


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Box Plot

Tensile

Strength

3530252015

25

20

15

10

5

Boxplot of 15, 20, 25, 30, 35


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Analysis Steps Effects Model

Hypothesis

Test Statistic obtained by partitioning the total sum of squares

Critical region

1,2,...,,

1,2,...,ij i ij

i ay

j nm t e

== + +

=

0 1 2: 0

: atleast one is 0

a

a

H

H

t t t= = = =

L

( ) ( ) ( )22 2

.. . .. .

1 1 1 1 1

T Treatments Error

a n a a n

i i ij i

i j i i j

SS SS SS

y y n y y y y= = = = =

= +

- = - + -

2

2

~

~

Treatments

Error

TreatmentsTreatments DoF

Treatments

ErrorDoF

Error

SSMS

DOF

SSMSE

DOF

c

c

=

=

1 , ,T reat ment s E rro r

Treatments

DOF DOF

Error

MSTest Statistic F

MSa-

= =


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Checking assumptions

Assumptions Independence

Constant Variance

Errors are distributed Normal with mean zero

Linear relationship

Residual Plots

Normal Probability Plot

Residuals versus Fitted

Residuals vs. Time order


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Basic Principles Randomization, Replication and Blocking

Blocking Creating homogeneous conditions for subset of

experiments

Improve the precision by eliminating the variabilitydue to nuisance factor (factors that are influential butnot of interest and can be observed but notcontrolled)

Sum of Squares of Block account for the variability

due to blocks Example:

Suppose each replication was done on a separate dayand atmospheric temperature is nuisance factor. Useblocking.


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Experimental Designs

Features of a desirable design Reasonable distribution of data points

Allows lack of fit to be estimated

Allows experiments to be performed in blocks

Allows designs of higher order to be built up sequentially

Provides an internal estimate of error Provides precise estimates of the model coefficients

Provides good profile of the prediction variance

Provides robustness against outliers

Does not require large runs

Does not require too many levels of the independent factors Ensure simplicity of calculation of the model parameters


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Design Space

x1

x2


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Factorial Design

All factors are varied together Full factorials all combinations of the factors

are tested in each replicate

If we have 4 factors at 2 levels => we have 24

= 16experimental runs

Fractional Factorials fewer combinations of the

factors are examined Half fraction of 24 = 24-1 = 8 experimental runs

Sparsity of Effects principle -> higher order

interactions are not significant


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Generator ABC

Defining relationship, I = ABC Alias, e.g. [A] = A + BC, [B] = B + AC


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Design Resolution

Resolution III design - Main effects arealiased with two - factor interactions (FI)

Resolution IV design 2 FI are aliased

with 2 FI

Resolution V Design 2 FI are aliased

with 3 FI


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Analysis Procedure for Factorial Designs

Estimate Factor Effects

Form Preliminary Model

Test for significance of factor effects Analyze residuals

Refine Model, if necessary

Interpret results


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Research Opportunities in Design ofExperiments*

Design for computer experiments

Response surface designs for cases involving

randomization restriction

Model robust designs

Designs for non - normal response

Design, analysis and optimization of multiple responses

Second order designs involving categorical factors

* Myers, R. H. , Montgomery, D. C., Vining, G. G, Borror, C. and Kowalski, S. M. 2004. Response

Surface Methodology: A Retrospective and Literature Survey, Journal of Quality Technology, 36, pp53 - 77


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Reference

Basic Concepts and Examples Mitra, A. Fundamentals of Quality Control and Improvement, 2nd

Edition, Prentice Hall.

Montgomery, D. C. Design and Analysis of Experiments, 6th

Edition, Wiley, New York.

Advanced Experimental Designs

Myers, R. H., Montgomery, D. C. Response Surface Methodology

2nd Edition, Wiley, New York


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QUESTIONS

DOE Shilpagupta 111906

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