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Statistical Design of Experiments

BITS Pilani, November 19 2006

~ Shilpa Gupta (97A4)gupta.shilpa@gmail.com

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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 x’s so that y is near nominal

requirement. where to set influential x’s so that variability in y is

small. where to set influential x’s 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 observe changes 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 Designed Industrial 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 in investigating the tensile strength of a new synthetic fiber that will be used to make cloth for men’s shirt. The engineer knows from past experience that the strength of the fiber is affected by the weight percentage of cotton content in the blend of materials for the fiber. The engineer suspects increasing the cotton content will increase the strength. The cotton content ranges from 10-40%. So the engineer decides to test at 5 treatment levels: 15, 20, 25, 30, 35

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Basic Principles – Replication, Randomization and Blocking Replication

Repetition of basic experiment and NOT repeated measurements

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 obtain

an estimate of error

Cotton Weight Percentage

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 at 15% were tested first followed by 5 runs at 20% and so on. If the tensile strength testing machine exhibits warm-up effect which means the longer it is on, the lower tensile strength readings will be. This warm –up effect will contaminate the tensile strength data and destroy the validity of the experiment.

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 experimentCotton Weight Percentage

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

Tensi

le S

trength

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

ì =ïï= + + íï =ïî

0 1 2: 0

: atleast one is 0a

a

H

H

t t t= = = =

¹

L

( ) ( ) ( )22 2

.. . .. .1 1 1 1 1

T Treatments Error

a n a a n

i i ij ii j i i j

SS SS SS

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

= +

- = - + -å å å å å

2

2

~

~

Treatments

Error

TreatmentsTreatments DoF

Treatments

ErrorDoF

Error

SSMS

DOF

SSMSE

DOF

c

c

=

=

1 , ,Treatments Error

TreatmentsDOF DOF

Error

MSTest Statistic F

MS a-= =

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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 variability

due to nuisance factor (factors that are influential but not of interest and can be observed but not controlled)

Sum of Squares of Block – account for the variability due to blocks

Example: Suppose each replication was done on a separate day

and atmospheric temperature is nuisance factor. Use blocking.

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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 = 16

experimental 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 are

aliased 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 of Experiments*

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, pp 53 - 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