1 statistical design of experiments bits pilani, november 19 2006 ~ shilpa gupta (97a4)...
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Statistical Design of Experiments
BITS 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 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
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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
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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