Sizing Mixture (RSM) Designs for Adequate Precision via Fraction of Design Space (FDS)
|
|
- Beverly Pope
- 6 years ago
- Views:
Transcription
1 for Adequate Precision via Fraction of Design Space (FDS) Pat Whitcomb Stat-Ease, Inc fax Gary W. Oehlert School of Statistics University it of Minnesota Copyright 009 Stat-Ease, Inc. Review Power to size factorial designs Precision in place of power Introduce FDS Sizing designs for precision Two component mixture Three component constrained mixture Sizing designs to detect a difference Summary 1
2 Sizing Factorial Designs During screening and characterization the emphasis is on identifying factor effects. What are the important design factors? For this purpose power is an ideal metric to evaluate design suitability. Screening Known Factors Unknown Factors Screening Ti Trivial i many Characterization Vital few Factor effects and interactions no Curvature? yes Optimization Response Surface methods Verification Confirm? no Backup Celebrate! yes 3 Copyright 009 Stat-Ease, Inc. Factorial Design Power 3 Full Factorial Δ= and σ=1 One Factor R B: B A: A R1 10 Δ A: A 4
3 Factorial Design Power 3 Full Factorial Δ= and σ=1 Power is reported at a 5.0% alpha level to detect the specified signal/noise ratio. Recommended power is at least 80%. R1 Signal (delta) =.00 Noise (sigma) = 1.00 Signal/Noise (delta/sigma) =.00 A B C 57. % 57. % 57. % The following three slides dig into how power is computed. 5 Copyright 009 Stat-Ease, Inc. One Replicate of 3 Full Factorial C = (X T X) -1 matrix The design determines the standard error of the coefficient: C = β β β = = i i i t-value i = SE ( βi ) ciiσˆ ( 0.15) σˆ 6 3
4 NonCentrality Parameter 3 Full Factorial Δ= and σ=1 The reference t distribution assumes the null hypothesis of Δ = 0. The noncentrality parameter (.88) defines the t distribution under the alternate hypothesis of Δ =. noncentrality = i = β ii i cˆ σ Δi cˆ σ ii 1 = ( )( 1 ) 1 = = Copyright 009 Stat-Ease, Inc. Factorial Design Power 3 Full Factorial Δ= and σ=1 noncentral t α=0.05,df=4 with noncentrality parameter of Power = 57.% 8 4
5 Factorial Design Power Two Replicates of 3 Full Factorial Δ= and σ=1 Power is reported at a 5.0% alpha level to detect the specified signal/noise ratio. Recommended power is at least 80%. R1 Signal (delta) =.00 Noise (sigma) = 1.00 Signal/Noise (delta/sigma) =.00 A B C 95.6 % 95.6 % 95.6 % 9 Copyright 009 Stat-Ease, Inc. Two Replicates of 3 Full Factorial C = (X T X) -1 matrix The design determines the standard error of the coefficient: C = β β β i i i t-value i = = = SE ( βi ) ciiσˆ ( 0.065) σˆ 10 5
6 NonCentrality Parameter Two Replicates of 3 Full Factorial Δ= and σ=1 noncentrality = i = Δ i i = iiσ cˆ iiσ β cˆ 1 ( 0.065)( 1) 1 = = Copyright 009 Stat-Ease, Inc. Factorial Design Power Two Replicates of 3 Full Factorial Δ= and σ=1 noncentral t α=0.05,df=1 with noncentrality parameter of Power = 95.6% 1 6
7 Conclusions Factorial DOE During screening and characterization (factorials) emphasis is on identifying factor effects. What are the important design factors? For this purpose power is an ideal metric to evaluate design suitability. 13 Copyright 009 Stat-Ease, Inc. Review Power to size factorial designs Precision in place of power Introduce FDS Sizing designs for precision Two component mixture Three component constrained mixture Sizing designs to detect a difference Summary 14 7
8 Sizing Response Surface Designs Screening Known Factors Unknown Factors Screening Ti Trivial i many When the goal is optimization emphasis is the fitted surface. How well does the surface represent true behavior? For this purpose precision (FDS) is an good metric to evaluate design suitability. (Assuming model adequacy; i.e. insignificant lack of fit.) Vital few Characterization Factor effects and interactions no Curvature? yes Optimization Response Surface methods Verification no Confirm? Backup Celebrate! yes 15 Copyright 009 Stat-Ease, Inc. Design Properties 1. Estimate the designed for polynomial well.. Give sufficient information to allow a test for lack of fit. Have more unique design points than coefficients i in the model. Provide an estimate of pure error. 3. Remain insensitive to outliers, influential values and bias from model misspecification. 4. Be robust to errors in control of the component levels. 5. Provide a check on model assumptions, e.g., normality of errors. 6. Generate useful information throughout the region of interest, i.e., provide a good distribution of Var Yˆ σ. ( ) 7. Do not contain an excessively large number of trials. 16 8
9 FDS Fraction of Design Space Fraction of Design Space: Calculates the volume of the design space having a prediction variance (PV) less than or equal to a specified value. The ratio of this volume to the total volume of the design volume is the fraction of design space. Produces a single plot showing the cumulative fraction of the design space on the x-axis (from zero to one) versus the PV on the y-axis. 17 Copyright 009 Stat-Ease, Inc. FDS PV Fraction of Design Space Prediction Variance: ( ) var yˆ ( ) 0 PV x = = x T ( X T X ) 1 x s PV is a function of: x 0 the location in the design space (i.e. the x coordinates for all model terms). X the experimental design (i.e. where the runs are in the design space). 18 9
10 FDS StdErr Fraction of Design Space Prediction standard error of the expected value: var ( y ˆ 0 ) 1 T T PV ( x ) = = x ( X X ) x s StdErr x s yˆ 0 ( ) = = PV ( x ) 0 0 s 19 Copyright 009 Stat-Ease, Inc. FDS Plot Fraction of Design Space 1. Pick random points in the design space.. Calculate the standard error of the expected value SEŷ0 T T 1 = x0 ( X X) x0 s 3. Plot the standard error as a fraction of the design space. StdErr Mean FDS Graph Fraction of Design Space 0 10
11 Simplex-Lattice Augmented and 4 Replicates A: A FDS Graph StdErr Mean B: B StdErr of Design 1.00 C: C Fraction of Design Space Mix section 1 Copyright 009 Stat-Ease, Inc. Review Power to size factorial designs Precision in place of power Introduce FDS Sizing designs for precision Two component mixture Three component constrained mixture Sizing designs to detect a difference Summary 11
12 Two Component Mixture Linear Model Two Component Mix 7 6 R Actual A Actual B Copyright 009 Stat-Ease, Inc. Two Component Mixture Linear Model The solid center line is the fitted model; ŷ is the expected value or mean prediction. The curved dotted lines are the computer generated confidence limits, or the actual precision. d Is the half-width of the desired confidence interval, or the desired precision. It is used to create the outer straight lines. R Two Component Mix y ˆ ± d Note: The actual precision of the fitted value depends on where we are predicting. 3 Actual A Actual B
13 Two Component Mixture Linear Model Evaluate the standard error as a fraction of design space for this design. 7 Two Component Mix 6 R Actual A 0 Actual B Sizing Mixture (RSM) Designs 5 Copyright 009 Stat-Ease, Inc. Click on Evaluate and then Graphs : Two Component Mixture Linear Model 6 13
14 Two Component Mixture Linear Model Want a linear surface to represent the true response value within ± 0.64 with 95% confidence. The overall standard deviation for this response is Enter: d = s = 0.55 a (α) = = Copyright 009 Stat-Ease, Inc. Two Component Mixture Linear Model Only 53% of the design space is precise enough to predict the mean within ±
15 Define Precision Half Width of the Confidence Interval Statistical detail Confidence interval on the expected value: The mean response is estimated and the precision of the estimate is quantified by a confidence interval: α, df ( y ) y ˆ ± t sˆ We will use the half-width of the confidence interval (d) to define the precision p desired: d d = tα, ( sˆ) or sˆ = df y y t α, df Mix section 4 9 Copyright 009 Stat-Ease, Inc. What Precision is Needed? Mean Confidence Interval Half-Width Statistical detail Input Half-width of confidence interval: d Estimate of standard deviation: s yˆ ± d d t s ( ˆ ) = α, df y ( ) 1 T T ˆ = 0 X X y 0 s s x X X x syˆ 1 T T StdErr Mean( FDS) = = x ( X X ) x s 0 0 Mix section
16 Two Component Mixture Linear Model Statistical detail Want a linear surface to represent the true response value within ± 0.64 with 95% confidence. The overall standard deviation for this response is For 95% confidence t =.571, d = 0.64 & s = 0.55 s yˆ.05,5 d 0.64 = = = t α, df ( ) StdErr Mean FDS s y ˆ 0.5 = = = 0.45 s 0.55 Mix section 4 31 Copyright 009 Stat-Ease, Inc. FDS StdErr Mean Plot Fraction of Design Space Statistical detail 1. Pick random points in the design space.. Calculate the standard error of the expected value SE S ( ) ŷ 1 0 T T = x0 X X x0 3. Plot the standard error as a fraction of the design space. StdErr FDS Graph Mix section 4 Fraction of Design Space 3 16
17 Two Component Mixture Linear Model 53% is precise enough, i.e. 53% has a StdErr 0.45 d = 0.64 s = 0.55 α = Copyright 009 Stat-Ease, Inc. Two Component Mixture Linear Model Two Component Mix 53% of the design space has the desired precision, i.e. is inside the solid straight (red) lines % R1 5 y ˆ ± d 4 3 Actual A Actual B
18 Sizing for Precision FDS? How good is good enough? Rules of thumb: For exploration want FDS 80% For verification want FDS 90% or higher! What can be done to improve precision? Manage expectations; i.e. increase d Decrease noise; i.e. decrease s Increase risk of Type I error; i.e. increase alpha Increase the number of runs in the design 35 Copyright 009 Stat-Ease, Inc. Review Power to size factorial designs Precision in place of power Introduce FDS Sizing designs for precision Two component mixture Three component constrained mixture Sizing designs to detect a difference Summary 36 18
19 Mixture Simplex Quadratic Model ŷ = 4A + 4B+ 8C+ 16AC Most of the action occurs on the A-C edge because of the AC coefficient of 16. The quadratic coefficient of 16 means that the response is 4 units higher at A=0.5, C=0.5 than one would expect with linear blending. R A (1) B (0) C (1) C (0) A (0) B (1) 37 Copyright 009 Stat-Ease, Inc. Mixture Simplex Quadratic Model ŷ = 4A + 4B+ 8C+ 16AC This illustrates the quadratic blending along the A-C edge as C increases from zero to one β = 4 A β = 4 AC β C = C 38 19
20 Mixture Constrained Design for Quadratic Model 0.0 A B C 0.1 Total = 1.0 A: A B: B StdErr of Design C: C 39 Copyright 009 Stat-Ease, Inc. Mixture Constrained Quadratic Model (1 replicate) Power at 5 % alpha level for effect of Term StdErr* VIF R i -Squared 1 s s 3 s A % 5.0 % 5.0 % B % 5.0 % 5.0 % C % 5.0 % 5.0 % AB % 67.3 % 94.7 % AC % 5.0 % 5.0 % BC % 5.0 % 5.0 % *Basis Std. Dev. = 1.0 Power is bottomed out at alpha! 40 0
21 Define Precision Half Width of the Confidence Interval 6. Generate useful information throughout the region of interest. Question: Will predictions, using the quadratic model from this design, be precise enough for our purposes? To know the truth requires an infinite number of runs; most likely this will exceed our budget. So the question is how precisely do we need to estimate the response? The trade off is more precision requires more runs. 41 Copyright 009 Stat-Ease, Inc. Define Precision Half Width of the Confidence Interval Confidence interval on the expected value: The mean response is estimated and the precision of the estimate is quantified by a confidence interval: α, df ( y ) y ˆ ± t sˆ We will use half-width of the confidence interval (d) to define the precision desired: d d = tα, ( sˆ) or sˆ = df y y t α, df 4 1
22 Input Half-width of confidence interval: d Estimate of standard deviation: s What Precision is Needed? Confidence Interval Half-Width yˆ d = t α, df T ( ) 1 T s = s x X X x yˆ 0 0 syˆ 1 T T StdErr Mean( FDS) = = x ( X X ) x s s Copyright 009 Stat-Ease, Inc. Mixture Constrained Quadratic Model (1 replicate) Want quadratic surface to represent the true response value within ±10 with 95% confidence. The standard deviation for this response is 7.8. For 95% confidence t =.365, d = 10 & s = 7.8 s yˆ t α, df.05,7 d 10 = = = ( ) StdErr Mean FDS s y ˆ 4.3 = = = 0.54 s
23 Mixture Constrained Quadratic Model (1 replicate) Only 58% of the design space has StdErr 0.54 d = 10 s= 7.8 α = Copyright 009 Stat-Ease, Inc. Mixture Constrained Quadratic Model ( replicates) Power at 5 % alpha level for effect of Term StdErr* VIF R i -Squared 1 s s 3 s A % 5.0 % 5.0 % B % 5.0 % 5.0 % C % 5.0 % 5.0 % AB % 96.5 % 99.9 % AC % 5.0 % 5.0 % BC % 5.0 % 5.0 % *Basis Std. Dev. = 1.0 Adding a replicate does little to increase power! 46 3
24 Mixture Constrained Why Does Power Not Increase? Quadratic, linear or combinations of them model response. Model Coefficients are Correlated! 14 1 Linear 10 8 Quadratic Constraint C Individual coefficients can not be resolved! 47 Copyright 009 Stat-Ease, Inc. Mixture Constrained Quadratic Model ( replicates) Want quadratic surface to represent the true response value within ± 10 with 95% confidence. The overall standard deviation for this response is 7.8. For 95% confidence t =.086, d = 10 & s = 7.8 s yˆ t α, df.05, 0 d 10 = = = ( ) StdErr Mean FDS s y ˆ 4.79 = = = 0.61 s
25 Mixture Constrained Quadratic Model ( replicates) 100% of the design space has StdErr Copyright 009 Stat-Ease, Inc. Precision Depends On The size of the confidence interval half-width (d): A larger half-width (d) increases the FDS. The size of the experimental error σ: A smaller s increases the FDS. The α risk chosen: A larger α increases the FDS. Choose design appropriate to the problem: Size the design for the precision required. 50 5
26 Conclusions Factorial DOE During screening and characterization (factorials) emphasis is on identifying factor effects. What are the important design factors? For this purpose power is an ideal metric to evaluate design suitability. Mixture Design and Response Surface Methods When the goal is optimization (usually the case for mixture design & RSM) emphasis is on the fitted surface. How well does the surface represent true behavior? For this purpose precision (FDS) is a good metric to evaluate design suitability. 51 Copyright 009 Stat-Ease, Inc. Review Power to size factorial designs Precision in place of power Introduce FDS Sizing designs for precision Two component mixture Three component constrained mixture Sizing designs to detect a difference Summary 5 6
27 Detecting a Difference of Δ wherever it may occur Δ What is the probability of finding a difference Δ if it occurs between any two points in the design space? 53 Copyright 009 Stat-Ease, Inc. FPDS StdErr Diff Plot Fraction of Paired Design Space 1. Pick random pairs of points in the design space.. Calculate the standard error of the difference SE ( ) ( ) ( ) ( ) yˆ ˆ y T T T T T T = x1 X X x1 + x X X x x1 X X x s 3. Plot the standard error as a fraction of the design space..00 FPDS Graph StdErr Diff Fraction of Paired Design Space 54 7
28 Revisit Mixture Constrained Design for Quadratic Model 0.0 A B C 0.1 Total = 1.0 A: A B: B StdErr of Design C: C 55 Copyright 009 Stat-Ease, Inc. Mixture Constrained Quadratic Model (1 replicate) Want to detect a difference of 10 on a quadratic surface with 95% confidence. The overall standard deviation this for response is 7.8. For 95% confidence t =.365, Δ= 10 & s = 7.8 s Δ.05,7 Δ 10 = = = 4.3 t α, df ( ) StdErr Diff FPDS sδ 4.3 = = = 0.54 s
29 Mixture Constrained Quadratic Model (1 replicate) Only 37% of the design space has StdErr 0.54 Δ = 10 s= 7.8 α = Copyright 009 Stat-Ease, Inc. Mixture Constrained Quadratic Model ( replicates) Want to detect a difference of 10 on a quadratic surface with 95% confidence. The overall standard deviation this for response is 7.8. For 95% confidence t =.086, Δ= 10 & s = 7.8 s Δ t α, df.05, 0 Δ 10 = = = ( ) StdErr Diff FPDS sδ 4.79 = = = 0.61 s
30 Mixture Constrained Quadratic Model ( replicates) 86% of the design space has StdErr Copyright 009 Stat-Ease, Inc. Detecting a Difference of Δ Depends On The size of the difference (Δ): A larger difference (Δ) increases the FPDS. The size of the experimental error σ: A smaller σ increases the FPDS The α risk chosen: A larger α increases the FPDS. Choose design appropriate to the problem: Size the design to detect t a difference of interest. t 60 30
31 Conclusions Factorial DOE During screening and characterization (factorials) emphasis is on identifying factor effects. What are the important design factors? For this purpose power is an ideal metric to evaluate design suitability. Mixture Design and Response Surface Methods When the goal is to detect a change in the response emphasis is on differences over the fitted surface. Does a difference D occur in the design space? For this purpose precision (FPDS) is a good metric to evaluate design suitability. 61 Copyright 009 Stat-Ease, Inc. Review Power to size factorial designs Precision in place of power Introduce FDS Sizing designs for precision Two component mixture Three component constrained mixture Sizing designs to detect a difference Summary 6 31
32 Summary Two criteria covered: 1. Precision of expected value, StdErr Mean: Use the StdErr Mean when the purpose of the DOE is to quantify shape; i.e. functional design (e.g. optimization).. Precision to detect a difference, StdErr Diff: Use the StdErr Diff when the purpose of the DOE is to detect t change in the design space. 63 Copyright 009 Stat-Ease, Inc. Summary Two criteria not covered: 1. Precision of predicted value, StdErr Pred: Use the StdErr Pred when the purpose of the DOE is to develop a model for prediction.. Precision to form a tolerance interval, TI Multiplier: Use the TI Multiplier when the purpose of the DOE is to form a tolerance interval at a point in the design space. (new in Design-Expert version 8) 64 3
33 References [1] Gary W. Oehlert (000), A First Course in Design and Analysis of Experiments, W. H. Freeman and Company. [] Douglas C. Montgomery (005), 6 th edition, Design and Analysis of Experiments, John Wiley and Sons, Inc. [3] Gary Oehlert and Patrick Whitcomb (001), Sizing Fixed Effects for Computing Power in Experimental Designs, Quality and Reliability Engineering International, July 7, 001 [4] Alyaa R. Zahran, Christine M. Anderson-Cook and Raymond H. Myers, Fraction of Design Space to Assess Prediction, Journal of Quality Technology, Vol. 35, No. 4, October 003 [5] Heidi B. Goldfarb, Christine M. Anderson-Cook, Connie M. Borror and Douglas C. Montgomery, Fraction of Design Space plots for Assessing Mixture and Mixture-Process Designs, Journal of Quality Technology, Vol. 36, No., October 004. [6] Steven DE Gryze, Ivan Langhans and Martina Vandebroek, Using the Intervals for Prediction: A Tutorial on Tolerance Intervals for Ordinary Least-Squares Regression, Chemometrics and Intelligent Laboratory Systems, 87 (007) [7] Raymond H. Myers, Douglas C. Montgomery and Christine M. Anderson-Cook (009), 3 rd edition, Response Surface Methodology, John Wiley and Sons, Inc. [8] John A. Cornell (00), 3 rd edition, Experiments with Mixtures, John Wiley and Sons, Inc. [9] Wendell F. Smith (005), Experimental Design for Formulation, ASA-SIAM Series on Statistics and Applied probability. 65 Copyright 009 Stat-Ease, Inc. Thank You for Attending To obtain a copy of my slides: Give me your business card Send a request to pat@statease.com t t 66 33
2015 Stat-Ease, Inc. Practical Strategies for Model Verification
Practical Strategies for Model Verification *Presentation is posted at www.statease.com/webinar.html There are many attendees today! To avoid disrupting the Voice over Internet Protocol (VoIP) system,
More information2017 Stat-Ease, Inc. Practical DOE Tricks of the Trade. Presentation is posted at
Practical DOE Tricks of the Trade Presentation is posted at www.statease.com/webinar.html There are many attendees today! To avoid disrupting the Voice over Internet Protocol (VoIP) system, I will mute
More informationResponse Surface Methodology
Response Surface Methodology Process and Product Optimization Using Designed Experiments Second Edition RAYMOND H. MYERS Virginia Polytechnic Institute and State University DOUGLAS C. MONTGOMERY Arizona
More information2010 Stat-Ease, Inc. Dual Response Surface Methods (RSM) to Make Processes More Robust* Presented by Mark J. Anderson (
Dual Response Surface Methods (RSM) to Make Processes More Robust* *Posted at www.statease.com/webinar.html Presented by Mark J. Anderson (Email: Mark@StatEase.com ) Timer by Hank Anderson July 2008 Webinar
More informationResponse Surface Methodology:
Response Surface Methodology: Process and Product Optimization Using Designed Experiments RAYMOND H. MYERS Virginia Polytechnic Institute and State University DOUGLAS C. MONTGOMERY Arizona State University
More informationSizing Fixed Effects for Computing Power in Experimental Designs
Sizing Fixed Effects for Computing Power in Experimental Designs Gary W. Oehlert School of Statistics University of Minnesota Pat Whitcomb Stat-Ease, Inc. March 2, 2001 Corresponding author: Gary W. Oehlert
More informationAnalysis of Variance and Design of Experiments-II
Analysis of Variance and Design of Experiments-II MODULE VIII LECTURE - 36 RESPONSE SURFACE DESIGNS Dr. Shalabh Department of Mathematics & Statistics Indian Institute of Technology Kanpur 2 Design for
More informationRESPONSE SURFACE METHODOLOGY
RESPONSE SURFACE METHODOLOGY WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Iain
More informationThe Model Building Process Part I: Checking Model Assumptions Best Practice (Version 1.1)
The Model Building Process Part I: Checking Model Assumptions Best Practice (Version 1.1) Authored by: Sarah Burke, PhD Version 1: 31 July 2017 Version 1.1: 24 October 2017 The goal of the STAT T&E COE
More informationWhat If There Are More Than. Two Factor Levels?
What If There Are More Than Chapter 3 Two Factor Levels? Comparing more that two factor levels the analysis of variance ANOVA decomposition of total variability Statistical testing & analysis Checking
More informationThe Model Building Process Part I: Checking Model Assumptions Best Practice
The Model Building Process Part I: Checking Model Assumptions Best Practice Authored by: Sarah Burke, PhD 31 July 2017 The goal of the STAT T&E COE is to assist in developing rigorous, defensible test
More informationCHAPTER 6 A STUDY ON DISC BRAKE SQUEAL USING DESIGN OF EXPERIMENTS
134 CHAPTER 6 A STUDY ON DISC BRAKE SQUEAL USING DESIGN OF EXPERIMENTS 6.1 INTRODUCTION In spite of the large amount of research work that has been carried out to solve the squeal problem during the last
More informationDr. Maddah ENMG 617 EM Statistics 11/28/12. Multiple Regression (3) (Chapter 15, Hines)
Dr. Maddah ENMG 617 EM Statistics 11/28/12 Multiple Regression (3) (Chapter 15, Hines) Problems in multiple regression: Multicollinearity This arises when the independent variables x 1, x 2,, x k, are
More informationResponse Surface Methodology in Application of Optimal Manufacturing Process of Axial-Flow Fans Adopted by Modern Industries
American Journal of Theoretical and Applied Statistics 018; 7(6): 35-41 http://www.sciencepublishinggroup.com/j/ajtas doi: 10.11648/j.ajtas.0180706.16 ISSN: 36-8999 (Print); ISSN: 36-9006 (Online) Response
More informationResponse Surface Methodology: Process and Product Optimization Using Designed Experiments, 3rd Edition
Brochure More information from http://www.researchandmarkets.com/reports/705963/ Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 3rd Edition Description: Identifying
More informationAny of 27 linear and nonlinear models may be fit. The output parallels that of the Simple Regression procedure.
STATGRAPHICS Rev. 9/13/213 Calibration Models Summary... 1 Data Input... 3 Analysis Summary... 5 Analysis Options... 7 Plot of Fitted Model... 9 Predicted Values... 1 Confidence Intervals... 11 Observed
More informationDesign & Analysis of Experiments 7E 2009 Montgomery
Chapter 5 1 Introduction to Factorial Design Study the effects of 2 or more factors All possible combinations of factor levels are investigated For example, if there are a levels of factor A and b levels
More information" M A #M B. Standard deviation of the population (Greek lowercase letter sigma) σ 2
Notation and Equations for Final Exam Symbol Definition X The variable we measure in a scientific study n The size of the sample N The size of the population M The mean of the sample µ The mean of the
More informationDesign of Engineering Experiments Chapter 5 Introduction to Factorials
Design of Engineering Experiments Chapter 5 Introduction to Factorials Text reference, Chapter 5 page 170 General principles of factorial experiments The two-factor factorial with fixed effects The ANOVA
More informationPolynomial Regression
Polynomial Regression Summary... 1 Analysis Summary... 3 Plot of Fitted Model... 4 Analysis Options... 6 Conditional Sums of Squares... 7 Lack-of-Fit Test... 7 Observed versus Predicted... 8 Residual Plots...
More informationMeasuring System Analysis in Six Sigma methodology application Case Study
Measuring System Analysis in Six Sigma methodology application Case Study M.Sc Sibalija Tatjana 1, Prof.Dr Majstorovic Vidosav 1 1 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije
More informationDesign of Engineering Experiments Part 5 The 2 k Factorial Design
Design of Engineering Experiments Part 5 The 2 k Factorial Design Text reference, Special case of the general factorial design; k factors, all at two levels The two levels are usually called low and high
More informationCost optimisation by using DoE
Cost optimisation by using DoE The work of a paint formulator is strongly dependent of personal experience, since most properties cannot be predicted from mathematical equations The use of a Design of
More informationTaguchi Method and Robust Design: Tutorial and Guideline
Taguchi Method and Robust Design: Tutorial and Guideline CONTENT 1. Introduction 2. Microsoft Excel: graphing 3. Microsoft Excel: Regression 4. Microsoft Excel: Variance analysis 5. Robust Design: An Example
More information2016 Stat-Ease, Inc.
What s New in Design-Expert version 10 Presentation is posted at www.statease.com/webinar.html There are many attendees today! To avoid disrupting the Voice over Internet Protocol (VoIP) system, I will
More informationAn Optimal Split-Plot Design for Performing a Mixture-Process Experiment
Science Journal of Applied Mathematics and Statistics 217; 5(1): 15-23 http://www.sciencepublishinggroup.com/j/sjams doi: 1.11648/j.sjams.21751.13 ISSN: 2376-9491 (Print); ISSN: 2376-9513 (Online) An Optimal
More information3.4. A computer ANOVA output is shown below. Fill in the blanks. You may give bounds on the P-value.
3.4. A computer ANOVA output is shown below. Fill in the blanks. You may give bounds on the P-value. One-way ANOVA Source DF SS MS F P Factor 3 36.15??? Error??? Total 19 196.04 Completed table is: One-way
More informationOPTIMIZATION OF FIRST ORDER MODELS
Chapter 2 OPTIMIZATION OF FIRST ORDER MODELS One should not multiply explanations and causes unless it is strictly necessary William of Bakersville in Umberto Eco s In the Name of the Rose 1 In Response
More informationChapter 6 The 2 k Factorial Design Solutions
Solutions from Montgomery, D. C. (004) Design and Analysis of Experiments, Wiley, NY Chapter 6 The k Factorial Design Solutions 6.. A router is used to cut locating notches on a printed circuit board.
More informationSoftware Sleuth Solves Engineering Problems
1 Software Sleuth Solves Engineering Problems Mark J. Anderson and Patrick J. Whitcomb Stat-Ease, Inc. 2021 East Hennepin Ave, Suite 191, Minneapolis, MN 55413 Telephone: 612/378-9449, Fax: 612/378-2152
More informationChapter 5 Introduction to Factorial Designs Solutions
Solutions from Montgomery, D. C. (1) Design and Analysis of Experiments, Wiley, NY Chapter 5 Introduction to Factorial Designs Solutions 5.1. The following output was obtained from a computer program that
More informationNonlinear Regression. Summary. Sample StatFolio: nonlinear reg.sgp
Nonlinear Regression Summary... 1 Analysis Summary... 4 Plot of Fitted Model... 6 Response Surface Plots... 7 Analysis Options... 10 Reports... 11 Correlation Matrix... 12 Observed versus Predicted...
More informationDESIGN AND ANALYSIS OF EXPERIMENTS Third Edition
DESIGN AND ANALYSIS OF EXPERIMENTS Third Edition Douglas C. Montgomery ARIZONA STATE UNIVERSITY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore Contents Chapter 1. Introduction 1-1 What
More informationReference: Chapter 6 of Montgomery(8e) Maghsoodloo
Reference: Chapter 6 of Montgomery(8e) Maghsoodloo 51 DOE (or DOX) FOR BASE BALANCED FACTORIALS The notation k is used to denote a factorial experiment involving k factors (A, B, C, D,..., K) each at levels.
More informationDesign of Experiments SUTD - 21/4/2015 1
Design of Experiments SUTD - 21/4/2015 1 Outline 1. Introduction 2. 2 k Factorial Design Exercise 3. Choice of Sample Size Exercise 4. 2 k p Fractional Factorial Design Exercise 5. Follow-up experimentation
More information7. Response Surface Methodology (Ch.10. Regression Modeling Ch. 11. Response Surface Methodology)
7. Response Surface Methodology (Ch.10. Regression Modeling Ch. 11. Response Surface Methodology) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 1 Introduction Response surface methodology,
More informationFactorial designs. Experiments
Chapter 5: Factorial designs Petter Mostad mostad@chalmers.se Experiments Actively making changes and observing the result, to find causal relationships. Many types of experimental plans Measuring response
More informationAssessing Model Adequacy
Assessing Model Adequacy A number of assumptions were made about the model, and these need to be verified in order to use the model for inferences. In cases where some assumptions are violated, there are
More informationUseful Numerical Statistics of Some Response Surface Methodology Designs
Journal of Mathematics Research; Vol. 8, No. 4; August 20 ISSN 19-9795 E-ISSN 19-9809 Published by Canadian Center of Science and Education Useful Numerical Statistics of Some Response Surface Methodology
More informationDesign of Experiments SUTD 06/04/2016 1
Design of Experiments SUTD 06/04/2016 1 Outline 1. Introduction 2. 2 k Factorial Design 3. Choice of Sample Size 4. 2 k p Fractional Factorial Design 5. Follow-up experimentation (folding over) with factorial
More informationOptimize Your Process-Optimization Efforts
1 Optimize Your Process-Optimization Efforts Highly efficient, statistically based methods can identify the vital few factors that affect process efficiency and product quality. Mark J. Anderson and Patrick
More information14.0 RESPONSE SURFACE METHODOLOGY (RSM)
4. RESPONSE SURFACE METHODOLOGY (RSM) (Updated Spring ) So far, we ve focused on experiments that: Identify a few important variables from a large set of candidate variables, i.e., a screening experiment.
More informationCHAPTER 4 EXPERIMENTAL DESIGN. 4.1 Introduction. Experimentation plays an important role in new product design, manufacturing
CHAPTER 4 EXPERIMENTAL DESIGN 4.1 Introduction Experimentation plays an important role in new product design, manufacturing process development and process improvement. The objective in all cases may be
More informationCOST-EFFECTIVE AND INFORMATION-EFFICIENT ROBUST DESIGN FOR OPTIMIZING PROCESSES AND ACCOMPLISHING SIX SIGMA OBJECTIVES
COST-EFFECTIVE AND INFORMATION-EFFICIENT ROBUST DESIGN FOR OPTIMIZING PROCESSES AND ACCOMPLISHING SIX SIGMA OBJECTIVES Mark J. Anderson Shari L. Kraber Principal Consultant Stat-Ease, Inc. Stat-Ease, Inc.
More informationGage repeatability & reproducibility (R&R) studies are widely used to assess measurement system
QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL Qual. Reliab. Engng. Int. 2008; 24:99 106 Published online 19 June 2007 in Wiley InterScience (www.interscience.wiley.com)..870 Research Some Relationships
More informationBox-Cox Transformations
Box-Cox Transformations Revised: 10/10/2017 Summary... 1 Data Input... 3 Analysis Summary... 3 Analysis Options... 5 Plot of Fitted Model... 6 MSE Comparison Plot... 8 MSE Comparison Table... 9 Skewness
More informationAddition of Center Points to a 2 k Designs Section 6-6 page 271
to a 2 k Designs Section 6-6 page 271 Based on the idea of replicating some of the runs in a factorial design 2 level designs assume linearity. If interaction terms are added to model some curvature results
More informationCost Penalized Estimation and Prediction Evaluation for Split-Plot Designs
Cost Penalized Estimation and Prediction Evaluation for Split-Plot Designs Li Liang Virginia Polytechnic Institute and State University, Blacksburg, VA 24060 Christine M. Anderson-Cook Los Alamos National
More informationOne-sided and two-sided t-test
One-sided and two-sided t-test Given a mean cancer rate in Montreal, 1. What is the probability of finding a deviation of > 1 stdev from the mean? 2. What is the probability of finding 1 stdev more cases?
More informationResponse Surface Methodology IV
LECTURE 8 Response Surface Methodology IV 1. Bias and Variance If y x is the response of the system at the point x, or in short hand, y x = f (x), then we can write η x = E(y x ). This is the true, and
More informationInference for Regression Inference about the Regression Model and Using the Regression Line
Inference for Regression Inference about the Regression Model and Using the Regression Line PBS Chapter 10.1 and 10.2 2009 W.H. Freeman and Company Objectives (PBS Chapter 10.1 and 10.2) Inference about
More informationIntroduc)on to the Design and Analysis of Experiments. Violet R. Syro)uk School of Compu)ng, Informa)cs, and Decision Systems Engineering
Introduc)on to the Design and Analysis of Experiments Violet R. Syro)uk School of Compu)ng, Informa)cs, and Decision Systems Engineering 1 Complex Engineered Systems What makes an engineered system complex?
More informationUnit 10: Simple Linear Regression and Correlation
Unit 10: Simple Linear Regression and Correlation Statistics 571: Statistical Methods Ramón V. León 6/28/2004 Unit 10 - Stat 571 - Ramón V. León 1 Introductory Remarks Regression analysis is a method for
More informationBusiness Statistics. Lecture 10: Course Review
Business Statistics Lecture 10: Course Review 1 Descriptive Statistics for Continuous Data Numerical Summaries Location: mean, median Spread or variability: variance, standard deviation, range, percentiles,
More informationSimple Linear Regression for the Climate Data
Prediction Prediction Interval Temperature 0.2 0.0 0.2 0.4 0.6 0.8 320 340 360 380 CO 2 Simple Linear Regression for the Climate Data What do we do with the data? y i = Temperature of i th Year x i =CO
More informationLecture 3: Inference in SLR
Lecture 3: Inference in SLR STAT 51 Spring 011 Background Reading KNNL:.1.6 3-1 Topic Overview This topic will cover: Review of hypothesis testing Inference about 1 Inference about 0 Confidence Intervals
More informationPrediction of Bike Rental using Model Reuse Strategy
Prediction of Bike Rental using Model Reuse Strategy Arun Bala Subramaniyan and Rong Pan School of Computing, Informatics, Decision Systems Engineering, Arizona State University, Tempe, USA. {bsarun, rong.pan}@asu.edu
More informationChemometrics Unit 4 Response Surface Methodology
Chemometrics Unit 4 Response Surface Methodology Chemometrics Unit 4. Response Surface Methodology In Unit 3 the first two phases of experimental design - definition and screening - were discussed. In
More informationBasic Statistics. 1. Gross error analyst makes a gross mistake (misread balance or entered wrong value into calculation).
Basic Statistics There are three types of error: 1. Gross error analyst makes a gross mistake (misread balance or entered wrong value into calculation). 2. Systematic error - always too high or too low
More informationOutline. PubH 5450 Biostatistics I Prof. Carlin. Confidence Interval for the Mean. Part I. Reviews
Outline Outline PubH 5450 Biostatistics I Prof. Carlin Lecture 11 Confidence Interval for the Mean Known σ (population standard deviation): Part I Reviews σ x ± z 1 α/2 n Small n, normal population. Large
More informationRidge Regression. Summary. Sample StatFolio: ridge reg.sgp. STATGRAPHICS Rev. 10/1/2014
Ridge Regression Summary... 1 Data Input... 4 Analysis Summary... 5 Analysis Options... 6 Ridge Trace... 7 Regression Coefficients... 8 Standardized Regression Coefficients... 9 Observed versus Predicted...
More information1 Introduction to Minitab
1 Introduction to Minitab Minitab is a statistical analysis software package. The software is freely available to all students and is downloadable through the Technology Tab at my.calpoly.edu. When you
More informationBusiness Statistics. Lecture 9: Simple Regression
Business Statistics Lecture 9: Simple Regression 1 On to Model Building! Up to now, class was about descriptive and inferential statistics Numerical and graphical summaries of data Confidence intervals
More informationRegression Analysis. Simple Regression Multivariate Regression Stepwise Regression Replication and Prediction Error EE290H F05
Regression Analysis Simple Regression Multivariate Regression Stepwise Regression Replication and Prediction Error 1 Regression Analysis In general, we "fit" a model by minimizing a metric that represents
More informationLinear Regression. Simple linear regression model determines the relationship between one dependent variable (y) and one independent variable (x).
Linear Regression Simple linear regression model determines the relationship between one dependent variable (y) and one independent variable (x). A dependent variable is a random variable whose variation
More informationy response variable x 1, x 2,, x k -- a set of explanatory variables
11. Multiple Regression and Correlation y response variable x 1, x 2,, x k -- a set of explanatory variables In this chapter, all variables are assumed to be quantitative. Chapters 12-14 show how to incorporate
More informationDesign of Experiments (DOE) Instructor: Thomas Oesterle
1 Design of Experiments (DOE) Instructor: Thomas Oesterle 2 Instructor Thomas Oesterle thomas.oest@gmail.com 3 Agenda Introduction Planning the Experiment Selecting a Design Matrix Analyzing the Data Modeling
More information2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/term
More informationSimple linear regression
Simple linear regression Biometry 755 Spring 2008 Simple linear regression p. 1/40 Overview of regression analysis Evaluate relationship between one or more independent variables (X 1,...,X k ) and a single
More informationSTATISTICS 110/201 PRACTICE FINAL EXAM
STATISTICS 110/201 PRACTICE FINAL EXAM Questions 1 to 5: There is a downloadable Stata package that produces sequential sums of squares for regression. In other words, the SS is built up as each variable
More informationFormal Statement of Simple Linear Regression Model
Formal Statement of Simple Linear Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters X i is a known constant, the value of the predictor
More informationChapter 6 The 2 k Factorial Design Solutions
Solutions from Montgomery, D. C. () Design and Analysis of Experiments, Wiley, NY Chapter 6 The k Factorial Design Solutions 6.. An engineer is interested in the effects of cutting speed (A), tool geometry
More informationChapter 4 - Mathematical model
Chapter 4 - Mathematical model For high quality demands of production process in the micro range, the modeling of machining parameters is necessary. Non linear regression as mathematical modeling tool
More information-However, this definition can be expanded to include: biology (biometrics), environmental science (environmetrics), economics (econometrics).
Chemometrics Application of mathematical, statistical, graphical or symbolic methods to maximize chemical information. -However, this definition can be expanded to include: biology (biometrics), environmental
More informationSix Sigma Black Belt Study Guides
Six Sigma Black Belt Study Guides 1 www.pmtutor.org Powered by POeT Solvers Limited. Analyze Correlation and Regression Analysis 2 www.pmtutor.org Powered by POeT Solvers Limited. Variables and relationships
More informationOn Consistency of Tests for Stationarity in Autoregressive and Moving Average Models of Different Orders
American Journal of Theoretical and Applied Statistics 2016; 5(3): 146-153 http://www.sciencepublishinggroup.com/j/ajtas doi: 10.11648/j.ajtas.20160503.20 ISSN: 2326-8999 (Print); ISSN: 2326-9006 (Online)
More informationSTK4900/ Lecture 5. Program
STK4900/9900 - Lecture 5 Program 1. Checking model assumptions Linearity Equal variances Normality Influential observations Importance of model assumptions 2. Selection of predictors Forward and backward
More informationAlternative analysis of 6-Nitro BIPS behaviour based on factorial design
Malaysian Journal of Mathematical Sciences 10(S) February: 431 442 (2016) Special Issue: The 3 rd International Conference on Mathematical Applications in Engineering 2014 (ICMAE 14) MALAYSIAN JOURNAL
More informationInference for Regression Simple Linear Regression
Inference for Regression Simple Linear Regression IPS Chapter 10.1 2009 W.H. Freeman and Company Objectives (IPS Chapter 10.1) Simple linear regression p Statistical model for linear regression p Estimating
More informationInferences for Regression
Inferences for Regression An Example: Body Fat and Waist Size Looking at the relationship between % body fat and waist size (in inches). Here is a scatterplot of our data set: Remembering Regression In
More informationLINEAR REGRESSION ANALYSIS. MODULE XVI Lecture Exercises
LINEAR REGRESSION ANALYSIS MODULE XVI Lecture - 44 Exercises Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Exercise 1 The following data has been obtained on
More informationApplication Note. The Optimization of Injection Molding Processes Using Design of Experiments
The Optimization of Injection Molding Processes Using Design of Experiments Problem Manufacturers have three primary goals: 1) produce goods that meet customer specifications; 2) improve process efficiency
More informationWe are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors
We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists 3,500 108,000 1.7 M Open access books available International authors and editors Downloads Our
More informationThe simple linear regression model discussed in Chapter 13 was written as
1519T_c14 03/27/2006 07:28 AM Page 614 Chapter Jose Luis Pelaez Inc/Blend Images/Getty Images, Inc./Getty Images, Inc. 14 Multiple Regression 14.1 Multiple Regression Analysis 14.2 Assumptions of the Multiple
More informationBusiness Statistics. Lecture 10: Correlation and Linear Regression
Business Statistics Lecture 10: Correlation and Linear Regression Scatterplot A scatterplot shows the relationship between two quantitative variables measured on the same individuals. It displays the Form
More informationAnswer Keys to Homework#10
Answer Keys to Homework#10 Problem 1 Use either restricted or unrestricted mixed models. Problem 2 (a) First, the respective means for the 8 level combinations are listed in the following table A B C Mean
More informationCHAPTER EIGHT Linear Regression
7 CHAPTER EIGHT Linear Regression 8. Scatter Diagram Example 8. A chemical engineer is investigating the effect of process operating temperature ( x ) on product yield ( y ). The study results in the following
More informationClasses of Second-Order Split-Plot Designs
Classes of Second-Order Split-Plot Designs DATAWorks 2018 Springfield, VA Luis A. Cortés, Ph.D. The MITRE Corporation James R. Simpson, Ph.D. JK Analytics, Inc. Peter Parker, Ph.D. NASA 22 March 2018 Outline
More informationThe 2 k Factorial Design. Dr. Mohammad Abuhaiba 1
The 2 k Factorial Design Dr. Mohammad Abuhaiba 1 HoweWork Assignment Due Tuesday 1/6/2010 6.1, 6.2, 6.17, 6.18, 6.19 Dr. Mohammad Abuhaiba 2 Design of Engineering Experiments The 2 k Factorial Design Special
More informationAssignment 9 Answer Keys
Assignment 9 Answer Keys Problem 1 (a) First, the respective means for the 8 level combinations are listed in the following table A B C Mean 26.00 + 34.67 + 39.67 + + 49.33 + 42.33 + + 37.67 + + 54.67
More informationResponse surface methodology: advantages and challenges
100 FORUM The forum series invites readers to discuss issues and suggest possible improvements that can help in developing new methods towards the advancing of current existing processes and applied methods
More informationDrilling Mathematical Models Using the Response Surface Methodology
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:5 No:06 9 Drilling Mathematical Models Using the Response Surface Methodology Panagiotis Kyratsis, Nikolaos Taousanidis, Apostolos
More informationAn Introduction to Design of Experiments
An Introduction to Design of Experiments Douglas C. Montgomery Regents Professor of Industrial Engineering and Statistics ASU Foundation Professor of Engineering Arizona State University Bradley Jones
More information2.830 Homework #6. April 2, 2009
2.830 Homework #6 Dayán Páez April 2, 2009 1 ANOVA The data for four different lithography processes, along with mean and standard deviations are shown in Table 1. Assume a null hypothesis of equality.
More informationChapter 9. Correlation and Regression
Chapter 9 Correlation and Regression Lesson 9-1/9-2, Part 1 Correlation Registered Florida Pleasure Crafts and Watercraft Related Manatee Deaths 100 80 60 40 20 0 1991 1993 1995 1997 1999 Year Boats in
More informationTroubleshooting experiments-stripped FOR DOWNLOAD:Metal Tech Speaking Template USE THIS.qxd 3/8/ :21 AM Page 1
Troubleshooting experiments-stripped FOR DOWNLOAD:Metal Tech Speaking Template USE THIS.qxd 3/8/202 0:2 AM Page BY PATRICK VALENTINE, NORTH AMERICAN PCB AND EP&F BUSINESS MANAGER, OMG ELECTRONIC CHEMICALS,
More informationMultiple Linear Regression for the Salary Data
Multiple Linear Regression for the Salary Data 5 10 15 20 10000 15000 20000 25000 Experience Salary HS BS BS+ 5 10 15 20 10000 15000 20000 25000 Experience Salary No Yes Problem & Data Overview Primary
More informationA Quality-by-Design Methodology for Rapid HPLC Column and Solvent Selection
A Quality-by-Design Methodology for Rapid HPLC Column and Solvent Selection Introduction High Performance Liquid Chromatography (HPLC) method development work typically involves column screening followed
More informationOne-Way Repeated Measures Contrasts
Chapter 44 One-Way Repeated easures Contrasts Introduction This module calculates the power of a test of a contrast among the means in a one-way repeated measures design using either the multivariate test
More informationExam C Solutions Spring 2005
Exam C Solutions Spring 005 Question # The CDF is F( x) = 4 ( + x) Observation (x) F(x) compare to: Maximum difference 0. 0.58 0, 0. 0.58 0.7 0.880 0., 0.4 0.680 0.9 0.93 0.4, 0.6 0.53. 0.949 0.6, 0.8
More information