1 Statistical Design of Experiments BITS Pilani, November ~ Shilpa Gupta (97A4)

2 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  ____________, ______________, ______________

3 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

4 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

5 Difference between SPC and DOE Statistical Process-Control Methods  Passive  Information about the health of the process Designed Experiments  Active  Information to make improvements to the process

6 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

7 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

8 Choice of Factor and Levels Design Factors  Held-constant  Allowed-to-vary Nuisance Factors  Controllable – e.g. Blocking  Uncontrollable – e.g. analysis of covariance  Noise – e.g. Robust design

9 Principles Blocking Randomization Replication

10 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

11 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 Percenta ge Experimental Run Number Rep 1 Rep2Rep 3 Rep 4Rep

12 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

13 Example - Demonstrate ANOVA Tensile Strength experiment Cotton Weight Percen tage Observation TotalAverageRep 1Rep2Rep 3Rep 4Rep

14 Box Plot

15 Analysis Steps Effects Model Hypothesis Test Statistic obtained by partitioning the total sum of squares Critical region

16 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

17 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.

18 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

19 Design Space x1 x2

20 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 2 4 = 16 experimental runs Fractional Factorials – fewer combinations of the factors are examined  Half fraction of 2 4 = = 8 experimental runs  Sparsity of Effects principle -> higher order interactions are not significant

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

23 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

24 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

25 Design Types Full Factorials Fractional Factorials Central Composite Design (Box-Wilson) Geometric Design First order with Interactions Second order models

26 Mixture Design Mixture Design or simplex designs  Factors are components of a mixture  design space is constrained x1x1 x2x2 x3x3

27 Nested Design Nested Designs or hierarchical designs Supplier 1 Batch 1Batch 2Batch 3 Supplier 2 Batch 1Batch 2Batch 3

28 Split Plot Origins in Agriculture  Restriction on Randomization  Hard to change factors and easy to change factors a1a2a3 b1 b2 b3 b1 b2 b3 b1

29 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 “Response Surface Methodology: A Retrospective and Literature Survey”, Journal of Quality Technology, 36, pp

30 Reference Basic Concepts and Examples  Mitra, A. “ Fundamentals of Quality Control and Improvement, 2 nd Edition, Prentice Hall.  Montgomery, D. C. “Design and Analysis of Experiments”, 6 th Edition, Wiley, New York. Advanced Experimental Designs  Myers, R. H., Montgomery, D. C. “Response Surface Methodology” 2 nd Edition, Wiley, New York

31 QUESTIONS