The 2 k Factorial Design. Dr. Mohammad Abuhaiba 1
|
|
- Meryl Ryan
- 5 years ago
- Views:
Transcription
1 The 2 k Factorial Design Dr. Mohammad Abuhaiba 1
2 HoweWork Assignment Due Tuesday 1/6/ , 6.2, 6.17, 6.18, 6.19 Dr. Mohammad Abuhaiba 2
3 Design of Engineering Experiments The 2 k Factorial Design Special case of the general factorial design; k factors, all at two levels The two levels are usually called low and high (they could be either quantitative or qualitative) It provides the smallest number of runs with which k factors can be studied in a complete factorial design. Assumptions: Factors are fixed Completely randomized designs Usual normality assumptions are satisfied Response is nearly linear over the range of the factors levels chosen Dr. Mohammad Abuhaiba 3
4 The 2 2 Design Chemical Process Example Study the effect of concentration of reactant and amount of catalyst on conversion in a chemical process. - and + denote low and high levels of a factor Low and high are arbitrary terms Geometrically, the four runs form the corners of a square Factors can be quantitative or qualitative Dr. Mohammad Abuhaiba 4
5 Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery Dr. Mohammad Abuhaiba 5
6 Analysis Procedure for a Factorial Design Estimate factor effects Formulate model Statistical testing (ANOVA) Refine the model Analyze residuals (graphical) Interpret results Dr. Mohammad Abuhaiba 6
7 Estimation of Factor Effects ab a b (1) 1 A y y 2n [ ab a b (1)] A A 2n 2n ab b a (1) 1 B y y 2n [ ab b a (1)] B B 2n 2n ab (1) a b 1 AB 2n [ ab (1) a b] 2n 2n Dr. Mohammad Abuhaiba 7
8 Sum of Squares Sum of squares for any contrast can be computed from Eq SS T is given by Eq. 6-9 SS T has 4n-1 DOF Contrast A ab a b (1) Contrast ab b a (1) B Contrast ab (1) a b SS SS SS AB A B AB ab a b (1) 4n ab b a (1) 4n ab (1) a b 4n Dr. Mohammad Abuhaiba 8
9 Statistical Testing - ANOVA Dr. Mohammad Abuhaiba 9
10 Standards Order (Yates) Effests (1) a b ab A B AB Treatment combination Factorial effect I A B AB (1) a b ab Dr. Mohammad Abuhaiba 10
11 The Regression Model y x x o x 1 is a coded variable that represents reactant concentration x 2 is a coded variable that represents amount of catlyst Relationship between natural variables and cosed variables is given by: x con x con x /2 high conlow x 1 grand average x x /2 o 1 2 A B /2 /2 x 2 con high high high con x x x low cat cat cat x cat x cat low low /2 /2 Dr. Mohammad Abuhaiba 11
12 The Response Surface Dr. Mohammad Abuhaiba 12
13 Residuals and Diagnostic Checking Dr. Mohammad Abuhaiba 13
14 The 2 3 Factorial Design Dr. Mohammad Abuhaiba 14
15 Effects in The 2 3 Factorial Design A y y A B y y B C y y C See Eqs 6-11 to 6.17 for the factors' effects A B C Dr. Mohammad Abuhaiba 15
16 Table of and + Signs for the 2 3 Factorial Design Dr. Mohammad Abuhaiba 16
17 Properties of the Table Except for column I, every column has an equal number of + and signs The sum of the product of signs in any two columns is zero Multiplying any column by I leaves that column unchanged (identity element) The product of any two columns yields a column in the table: AB AB 2 ABBC AB C AC Orthogonal design Orthogonality is an important property shared by all factorial designs Each effect has a single DOF Sum of squares for any effect is: Dr. Mohammad Abuhaiba SS Contrast 2 8n 17
18 Example of a 2 3 Factorial Design Example 6-1: The Fill Height Experiment Run Coded Factors Fill Height Deviation Factor Levels A B C Replicate 1 Replicate 2 Low (-1) High (+1) A (%) B (psi) C (bpm) Dr. Mohammad Abuhaiba 18
19 Example of a 2 3 Factorial Design Estimation of Factor Effects ANOVA Model Coefficients Full Model Remove non-significant factors Model Coefficients Reduced Model The AB interaction is significant at about 10%. Thus, there is some mild interaction between carbonation and pressure. Run the process at low pressure and high line speed. Reduce variablity in carbonation by controlling temperature more precisely Dr. Mohammad Abuhaiba 19
20 Model Summary Statistics for Reduced Model R 2 and adjusted R 2 R R 2 SS SS Model T 2 SS E / Adj 1 SST / DOF DOF R 2 for prediction (based on PRESS) R 2 Pred 1 E T PRESS SS T Dr. Mohammad Abuhaiba 20
21 Model Summary Statistics (pg. 222) Standard error of model coefficients (full model) 2 ˆ ˆ MS E se( ) V ( ) k k n2 n2 Confidence interval on model coefficients ˆ t se( ˆ) ˆ t se( ˆ) / 2, df / 2, df E E Dr. Mohammad Abuhaiba 21
22 The General 2 k Factorial Design There will be k main effects, and k two-factor interactions 2 k three-factor interactions 3 1 k factor interaction Dr. Mohammad Abuhaiba 22
23 The General 2 k Factorial Design Analysis Procedure for a 2 k Design 1. Estimate factor effects 2. Form initial model 3. ANOVA 4. Refine model 5. Analyze residuals 6. Interpret results Dr. Mohammad Abuhaiba 23
24 The General 2 k Factorial Design ANOVA Contrast AB... K ( a 1)( b 1)...( k 1) The sign in each set of parentheses is negative if the factor is included in the effect and positive if the factor is not included. Examples: 2 3 and 2 5 designs Effects and sum of squares are estimated as 2 AB... K Contrast k AB n2 1 SS k Contrast n2... K AB... K AB... K Dr. Mohammad Abuhaiba 24 2
25 Unreplicated 2 k Factorial Designs 2 k factorial designs with one observation at each treatment combination An unreplicated 2 k factorial design is also sometimes called a single replicate Risks: If there is only one observation at each corner, is there a chance of unusual response observations spoiling the results? Modeling noise? Dr. Mohammad Abuhaiba 25
26 Unreplicated 2 k Factorial Designs Spacing of Factor Levels in the If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data More aggressive spacing is usually best Dr. Mohammad Abuhaiba 26
27 Unreplicated 2 k Factorial Designs Lack of replication causes potential problems in statistical testing Replication admits an internal estimate of error With no replication, fitting the full model results in zero degrees of freedom for error Potential solutions to this problem Pooling high-order interactions to estimate error Normal probability plotting of effects (Daniels, 1959) Dr. Mohammad Abuhaiba 27
28 Example of an Unreplicated 2 k Design A 2 4 factorial was used to investigate the effects of four factors on the filtration rate of a resin The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate Experiment was performed in a pilot plant Process engineer is interested in maximizing filteration rate. Factor C currently at the high level Would reduce formaldehyde concentration as much as possible. Dr. Mohammad Abuhaiba 28
29 The Resin Plant Experiment Dr. Mohammad Abuhaiba 29
30 The Resin Plant Experiment Dr. Mohammad Abuhaiba 30
31 Contrast Constants for the 2 k Design A B AB C AC BC ABC D AD BD ABD CD ACD BCD ABCD {1} a b ab c ac bc abc d ad bd abd cd acd bcd abcd Dr. Mohammad Abuhaiba 31
32 Estimates of the Effects Dr. Mohammad Abuhaiba 32
33 The Normal Probability Plot of Effects Dr. Mohammad Abuhaiba 33
34 The Half Normal Plot of Effects A plot of the absolute value of the effect estimates against their cummulative normal probabilities. Figure 6-15 The straight line always passes through the origin and should also pass close to the fiftieth percentile data value. Dr. Mohammad Abuhaiba 34
35 Main Effects and Interactions Dr. Mohammad Abuhaiba 35
36 Design Projection Example 6-2 Because B is not significant and all interactions involving B are negligible, we may discard B from the experiment so that the design becomes a 2 3 factorial in A, C, and D with two replicates. ANOVA for the 2 3 design is shown in Table 6.13 By projecting the single replicate of the 2 4 into a replicated 2 3, we now have both an estimate of the ACD interaction and an estimate of error based on what is sometimes called hidden replication. Dr. Mohammad Abuhaiba 36
37 Design Projection General Case If we have a single replicate of 2 k design, and if h (h<k) factors are negligible and can be dropped, then the original data correspond to a full two-level factorial in the remaining k h factors with 2 h replicates Dr. Mohammad Abuhaiba 37
38 ANOVA Summary for the Model Dr. Mohammad Abuhaiba 38
39 The Regression Model Dr. Mohammad Abuhaiba 39
40 Model Residuals Dr. Mohammad Abuhaiba 40
41 Model Interpretation Response Surface Plots With concentration at either the low or high level, high temperature and high stirring rate results in high filtration rates Dr. Mohammad Abuhaiba 41
42 Example 6-3: The Drilling Experiment A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill Dr. Mohammad Abuhaiba 42
43 The Drilling Experiment Normal Probability Plot of Effects Dr. Mohammad Abuhaiba 43
44 DESIGN-EXPERT Plot adv._rate Residuals Residuals vs. Predicted Predicted Residual Plots Dr. Mohammad Abuhaiba 44
45 Residual Plots The residual plots indicate that there are problems with the equality of variance assumption Employ a transformation on the response Power family transformations are widely used * y y Transformations are typically performed to Stabilize variance Induce normality Simplify the model Dr. Mohammad Abuhaiba 45
46 Selecting a Transformation Empirical selection of lambda Prior (theoretical) knowledge or experience can often suggest the form of a transformation Dr. Mohammad Abuhaiba 46
47 Effect Estimates Following Log Transformation Three main effects are large No indication of large interaction effects Dr. Mohammad Abuhaiba 47
48 ANOVA Following Log Transformation Dr. Mohammad Abuhaiba 48
49 Following Log Transformation Dr. Mohammad Abuhaiba 49
50 Addition of Center Points to a 2 k Design Based on the idea of replicating some of the runs in a factorial design Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models: First-order model (interaction) Second-order model 0 0 k k k y x x x i i ij i j i1 i1 ji k k k k 2 i i ij i j ii i i1 i1 ji i1 y x x x x Dr. Mohammad Abuhaiba 50
51 y F y C The hypotheses are: SS Pure Quad H H 0 1 no "curvature" k : 0 i1 k : 0 i1 ii ii nfnc ( yf yc ) n n This sum of squares has a single degree of freedom F C 2 Dr. Mohammad Abuhaiba 51
52 Example 6-6 Refer to the original experiment shown in Table Suppose that four center points are added to this experiment, and at the points x1=x2 =x3=x4=0 the four observed filtration rates were 73, 75, 66, and 69. The average of these four center points is 70.75, and the average of the 16 factorial runs is Since are very similar, we suspect that there is no strong curvature present. nc 4 Usually between 3 and 6 center points will work well Design-Expert provides the analysis, including the F-test for pure quadratic curvature Dr. Mohammad Abuhaiba 52
53 ANOVA for Example 6-6 Dr. Mohammad Abuhaiba 53
54 If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response surface model Dr. Mohammad Abuhaiba 54
55 Practical Use of Center Points (pg. 250) Use current operating conditions as the center point Check for abnormal conditions during the time the experiment was conducted Check for time trends Use center points as the first few runs when there is little or no information available about the magnitude of error Center points and qualitative factors? Dr. Mohammad Abuhaiba 55
56 Center Points and Qualitative Factors Dr. Mohammad Abuhaiba 56
Design 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 informationUnreplicated 2 k Factorial Designs
Unreplicated 2 k Factorial Designs These are 2 k factorial designs with one observation at each corner of the cube An unreplicated 2 k factorial design is also sometimes called a single replicate of the
More informationDesign and Analysis of
Design and Analysis of Multi-Factored Experiments Module Engineering 7928-2 Two-level Factorial Designs L. M. Lye DOE Course 1 The 2 k Factorial Design Special case of the general factorial design; k factors,
More information4. The 2 k Factorial Designs (Ch.6. Two-Level Factorial Designs)
4. The 2 k Factorial Designs (Ch.6. Two-Level Factorial Designs) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University Introduction to 2 k Factorial Designs Special case of the general factorial
More informationDesign and Analysis of Multi-Factored Experiments
Design and Analysis of Multi-Factored Experiments Two-level Factorial Designs L. M. Lye DOE Course 1 The 2 k Factorial Design Special case of the general factorial design; k factors, all at two levels
More information4. Design of Experiments (DOE) (The 2 k Factorial Designs)
4. Design of Experiments (DOE) (The 2 k Factorial Designs) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 1 Example: Golfing How to improve my score in Golfing? Practice!!! Other than
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 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 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 information2 k, 2 k r and 2 k-p Factorial Designs
2 k, 2 k r and 2 k-p Factorial Designs 1 Types of Experimental Designs! Full Factorial Design: " Uses all possible combinations of all levels of all factors. n=3*2*2=12 Too costly! 2 Types of Experimental
More informationSuppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks.
58 2. 2 factorials in 2 blocks Suppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks. Some more algebra: If two effects are confounded with
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 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 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 information19. Blocking & confounding
146 19. Blocking & confounding Importance of blocking to control nuisance factors - day of week, batch of raw material, etc. Complete Blocks. This is the easy case. Suppose we run a 2 2 factorial experiment,
More information20g g g Analyze the residuals from this experiment and comment on the model adequacy.
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 3.11. A pharmaceutical
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/terms.
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 information23. Fractional factorials - introduction
173 3. Fractional factorials - introduction Consider a 5 factorial. Even without replicates, there are 5 = 3 obs ns required to estimate the effects - 5 main effects, 10 two factor interactions, 10 three
More informationChapter 11: Factorial Designs
Chapter : Factorial Designs. Two factor factorial designs ( levels factors ) This situation is similar to the randomized block design from the previous chapter. However, in addition to the effects within
More informationFractional Factorial Designs
k-p Fractional Factorial Designs Fractional Factorial Designs If we have 7 factors, a 7 factorial design will require 8 experiments How much information can we obtain from fewer experiments, e.g. 7-4 =
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 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 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/terms.
More informationLecture 10: 2 k Factorial Design Montgomery: Chapter 6
Lecture 10: 2 k Factorial Design Montgomery: Chapter 6 Page 1 2 k Factorial Design Involving k factors Each factor has two levels (often labeled + and ) Factor screening experiment (preliminary study)
More informationInstitutionen för matematik och matematisk statistik Umeå universitet November 7, Inlämningsuppgift 3. Mariam Shirdel
Institutionen för matematik och matematisk statistik Umeå universitet November 7, 2011 Inlämningsuppgift 3 Mariam Shirdel (mash0007@student.umu.se) Kvalitetsteknik och försöksplanering, 7.5 hp 1 Uppgift
More information6. Fractional Factorial Designs (Ch.8. Two-Level Fractional Factorial Designs)
6. Fractional Factorial Designs (Ch.8. Two-Level Fractional Factorial Designs) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 1 Introduction to The 2 k-p Fractional Factorial Design
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 informationStrategy of Experimentation II
LECTURE 2 Strategy of Experimentation II Comments Computer Code. Last week s homework Interaction plots Helicopter project +1 1 1 +1 [4I 2A 2B 2AB] = [µ 1) µ A µ B µ AB ] +1 +1 1 1 +1 1 +1 1 +1 +1 +1 +1
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 informationUnit 6: Fractional Factorial Experiments at Three Levels
Unit 6: Fractional Factorial Experiments at Three Levels Larger-the-better and smaller-the-better problems. Basic concepts for 3 k full factorial designs. Analysis of 3 k designs using orthogonal components
More informationCS 5014: Research Methods in Computer Science
Computer Science Clifford A. Shaffer Department of Computer Science Virginia Tech Blacksburg, Virginia Fall 2010 Copyright c 2010 by Clifford A. Shaffer Computer Science Fall 2010 1 / 254 Experimental
More informationST3232: Design and Analysis of Experiments
Department of Statistics & Applied Probability 2:00-4:00 pm, Monday, April 8, 2013 Lecture 21: Fractional 2 p factorial designs The general principles A full 2 p factorial experiment might not be efficient
More informationFractional Factorial Designs
Fractional Factorial Designs ST 516 Each replicate of a 2 k design requires 2 k runs. E.g. 64 runs for k = 6, or 1024 runs for k = 10. When this is infeasible, we use a fraction of the runs. As a result,
More informationIE 361 Exam 3 (Form A)
December 15, 005 IE 361 Exam 3 (Form A) Prof. Vardeman This exam consists of 0 multiple choice questions. Write (in pencil) the letter for the single best response for each question in the corresponding
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 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 informationHigher Order Factorial Designs. Estimated Effects: Section 4.3. Main Effects: Definition 5 on page 166.
Higher Order Factorial Designs Estimated Effects: Section 4.3 Main Effects: Definition 5 on page 166. Without A effects, we would fit values with the overall mean. The main effects are how much we need
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 informationLecture 12: 2 k Factorial Design Montgomery: Chapter 6
Lecture 12: 2 k Factorial Design Montgomery: Chapter 6 1 Lecture 12 Page 1 2 k Factorial Design Involvingk factors: each has two levels (often labeled+and ) Very useful design for preliminary study Can
More informationDesign and Analysis of Experiments
Design and Analysis of Experiments Part VII: Fractional Factorial Designs Prof. Dr. Anselmo E de Oliveira anselmo.quimica.ufg.br anselmo.disciplinas@gmail.com 2 k : increasing k the number of runs required
More informationUnit 9: Confounding and Fractional Factorial Designs
Unit 9: Confounding and Fractional Factorial Designs STA 643: Advanced Experimental Design Derek S. Young 1 Learning Objectives Understand what it means for a treatment to be confounded with blocks Know
More informationCSCI 688 Homework 6. Megan Rose Bryant Department of Mathematics William and Mary
CSCI 688 Homework 6 Megan Rose Bryant Department of Mathematics William and Mary November 12, 2014 7.1 Consider the experiment described in Problem 6.1. Analyze this experiment assuming that each replicate
More informationCS 5014: Research Methods in Computer Science. Experimental Design. Potential Pitfalls. One-Factor (Again) Clifford A. Shaffer.
Department of Computer Science Virginia Tech Blacksburg, Virginia Copyright c 2015 by Clifford A. Shaffer Computer Science Title page Computer Science Clifford A. Shaffer Fall 2015 Clifford A. Shaffer
More information5. Blocking and Confounding
5. Blocking and Confounding Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 1 Why Blocking? Blocking is a technique for dealing with controllable nuisance variables Sometimes, it is
More informationFactorial designs (Chapter 5 in the book)
Factorial designs (Chapter 5 in the book) Ex: We are interested in what affects ph in a liquide. ph is the response variable Choose the factors that affect amount of soda air flow... Choose the number
More informationChapter 4: Randomized Blocks and Latin Squares
Chapter 4: Randomized Blocks and Latin Squares 1 Design of Engineering Experiments The Blocking Principle Blocking and nuisance factors The randomized complete block design or the RCBD Extension of the
More informationExperimental design (DOE) - Design
Experimental design (DOE) - Design Menu: QCExpert Experimental Design Design Full Factorial Fract Factorial This module designs a two-level multifactorial orthogonal plan 2 n k and perform its analysis.
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 informationStrategy of Experimentation III
LECTURE 3 Strategy of Experimentation III Comments: Homework 1. Design Resolution A design is of resolution R if no p factor effect is confounded with any other effect containing less than R p factors.
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 informationRandomized Blocks, Latin Squares, and Related Designs. Dr. Mohammad Abuhaiba 1
Randomized Blocks, Latin Squares, and Related Designs Dr. Mohammad Abuhaiba 1 HomeWork Assignment Due Sunday 2/5/2010 Solve the following problems at the end of chapter 4: 4-1 4-7 4-12 4-14 4-16 Dr. Mohammad
More informationChapter 13 Experiments with Random Factors Solutions
Solutions from Montgomery, D. C. (01) Design and Analysis of Experiments, Wiley, NY Chapter 13 Experiments with Random Factors Solutions 13.. An article by Hoof and Berman ( Statistical Analysis of Power
More informationContents. 2 2 factorial design 4
Contents TAMS38 - Lecture 10 Response surface methodology Lecturer: Zhenxia Liu Department of Mathematics - Mathematical Statistics 12 December, 2017 2 2 factorial design Polynomial Regression model First
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 informationTMA4267 Linear Statistical Models V2017 (L19)
TMA4267 Linear Statistical Models V2017 (L19) Part 4: Design of Experiments Blocking Fractional factorial designs Mette Langaas Department of Mathematical Sciences, NTNU To be lectured: March 28, 2017
More informationTWO-LEVEL FACTORIAL EXPERIMENTS: BLOCKING. Upper-case letters are associated with factors, or regressors of factorial effects, e.g.
STAT 512 2-Level Factorial Experiments: Blocking 1 TWO-LEVEL FACTORIAL EXPERIMENTS: BLOCKING Some Traditional Notation: Upper-case letters are associated with factors, or regressors of factorial effects,
More informationSMA 6304 / MIT / MIT Manufacturing Systems. Lecture 10: Data and Regression Analysis. Lecturer: Prof. Duane S. Boning
SMA 6304 / MIT 2.853 / MIT 2.854 Manufacturing Systems Lecture 10: Data and Regression Analysis Lecturer: Prof. Duane S. Boning 1 Agenda 1. Comparison of Treatments (One Variable) Analysis of Variance
More informationSession 3 Fractional Factorial Designs 4
Session 3 Fractional Factorial Designs 3 a Modification of a Bearing Example 3. Fractional Factorial Designs Two-level fractional factorial designs Confounding Blocking Two-Level Eight Run Orthogonal Array
More information1. Review of Lecture level factors Homework A 2 3 experiment in 16 runs with no replicates
Lecture 3.1 1. Review of Lecture 2.2 2-level factors Homework 2.2.1 2. A 2 3 experiment 3. 2 4 in 16 runs with no replicates Lecture 3.1 1 2 k Factorial Designs Designs with k factors each at 2 levels
More informationChapter 5 Introduction to Factorial Designs
Chapter 5 Introduction to Factorial Designs 5. Basic Definitions and Principles Stud the effects of two or more factors. Factorial designs Crossed: factors are arranged in a factorial design Main effect:
More informationConfounding and fractional replication in 2 n factorial systems
Chapter 20 Confounding and fractional replication in 2 n factorial systems Confounding is a method of designing a factorial experiment that allows incomplete blocks, i.e., blocks of smaller size than the
More informationa) Prepare a normal probability plot of the effects. Which effects seem active?
Problema 8.6: R.D. Snee ( Experimenting with a large number of variables, in experiments in Industry: Design, Analysis and Interpretation of Results, by R. D. Snee, L.B. Hare, and J. B. Trout, Editors,
More information10.0 REPLICATED FULL FACTORIAL DESIGN
10.0 REPLICATED FULL FACTORIAL DESIGN (Updated Spring, 001) Pilot Plant Example ( 3 ), resp - Chemical Yield% Lo(-1) Hi(+1) Temperature 160 o 180 o C Concentration 10% 40% Catalyst A B Test# Temp Conc
More informationReference: CHAPTER 7 of Montgomery(8e)
Reference: CHAPTER 7 of Montgomery(8e) 60 Maghsoodloo BLOCK CONFOUNDING IN 2 k FACTORIALS (k factors each at 2 levels) It is often impossible to run all the 2 k observations in a 2 k factorial design (or
More informationLecture 11: Blocking and Confounding in 2 k design
Lecture 11: Blocking and Confounding in 2 k design Montgomery: Chapter 7 Page 1 There are n blocks Randomized Complete Block 2 k Design Within each block, all treatments (level combinations) are conducted.
More informationMATH602: APPLIED STATISTICS
MATH602: APPLIED STATISTICS Dr. Srinivas R. Chakravarthy Department of Science and Mathematics KETTERING UNIVERSITY Flint, MI 48504-4898 Lecture 10 1 FRACTIONAL FACTORIAL DESIGNS Complete factorial designs
More informationConfounding and Fractional Replication in Factorial Design
ISSN -580 (Paper) ISSN 5-05 (Online) Vol.6, No.3, 016 onfounding and Fractional Replication in Factorial esign Layla. hmed epartment of Mathematics, ollege of Education, University of Garmian, Kurdistan
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analysis of Variance and Design of Experiment-I MODULE IX LECTURE - 38 EXERCISES Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Example (Completely randomized
More informationGeometry Problem Solving Drill 08: Congruent Triangles
Geometry Problem Solving Drill 08: Congruent Triangles Question No. 1 of 10 Question 1. The following triangles are congruent. What is the value of x? Question #01 (A) 13.33 (B) 10 (C) 31 (D) 18 You set
More informationTWO-LEVEL FACTORIAL EXPERIMENTS: REGULAR FRACTIONAL FACTORIALS
STAT 512 2-Level Factorial Experiments: Regular Fractions 1 TWO-LEVEL FACTORIAL EXPERIMENTS: REGULAR FRACTIONAL FACTORIALS Bottom Line: A regular fractional factorial design consists of the treatments
More informationDOE Wizard Screening Designs
DOE Wizard Screening Designs Revised: 10/10/2017 Summary... 1 Example... 2 Design Creation... 3 Design Properties... 13 Saving the Design File... 16 Analyzing the Results... 17 Statistical Model... 18
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 MACHINABILITY MODELS WITH THREE INDEPENDENT VARIABLES
CHAPTER 6 MACHINABILITY MODELS WITH THREE INDEPENDENT VARIABLES 6.1 Introduction It has been found from the literature review that not much research has taken place in the area of machining of carbon silicon
More informationSTAT451/551 Homework#11 Due: April 22, 2014
STAT451/551 Homework#11 Due: April 22, 2014 1. Read Chapter 8.3 8.9. 2. 8.4. SAS code is provided. 3. 8.18. 4. 8.24. 5. 8.45. 376 Chapter 8 Two-Level Fractional Factorial Designs more detail. Sequential
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 informationFRACTIONAL FACTORIAL
FRACTIONAL FACTORIAL NURNABI MEHERUL ALAM M.Sc. (Agricultural Statistics), Roll No. 443 I.A.S.R.I, Library Avenue, New Delhi- Chairperson: Dr. P.K. Batra Abstract: Fractional replication can be defined
More informationChapter 30 Design and Analysis of
Chapter 30 Design and Analysis of 2 k DOEs Introduction This chapter describes design alternatives and analysis techniques for conducting a DOE. Tables M1 to M5 in Appendix E can be used to create test
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 informationBlocks are formed by grouping EUs in what way? How are experimental units randomized to treatments?
VI. Incomplete Block Designs A. Introduction What is the purpose of block designs? Blocks are formed by grouping EUs in what way? How are experimental units randomized to treatments? 550 What if we have
More informationUSE OF COMPUTER EXPERIMENTS TO STUDY THE QUALITATIVE BEHAVIOR OF SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS
USE OF COMPUTER EXPERIMENTS TO STUDY THE QUALITATIVE BEHAVIOR OF SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS Seshadev Padhi, Manish Trivedi and Soubhik Chakraborty* Department of Applied Mathematics
More informationSoo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11: Figure 12: Figure 13:
1.0 ial Experiment Design by Block... 3 1.1 ial Experiment in Incomplete Block... 3 1. ial Experiment with Two Blocks... 3 1.3 ial Experiment with Four Blocks... 5 Example 1... 6.0 Fractional ial Experiment....1
More informationTwo-Level Fractional Factorial Design
Two-Level Fractional Factorial Design Reference DeVor, Statistical Quality Design and Control, Ch. 19, 0 1 Andy Guo Types of Experimental Design Parallel-type approach Sequential-type approach One-factor
More informationR version ( ) Copyright (C) 2009 The R Foundation for Statistical Computing ISBN
Math 3080 1. Treibergs Bread Wrapper Data: 2 4 Incomplete Block Design. Name: Example April 26, 2010 Data File Used in this Analysis: # Math 3080-1 Bread Wrapper Data April 23, 2010 # Treibergs # # From
More informationCHAPTER 5 NON-LINEAR SURROGATE MODEL
96 CHAPTER 5 NON-LINEAR SURROGATE MODEL 5.1 INTRODUCTION As set out in the research methodology and in sequent with the previous section on development of LSM, construction of the non-linear surrogate
More informationAPPENDIX 1. Binodal Curve calculations
APPENDIX 1 Binodal Curve calculations The weight of salt solution necessary for the mixture to cloud and the final concentrations of the phase components were calculated based on the method given by Hatti-Kaul,
More informationCOM111 Introduction to Computer Engineering (Fall ) NOTES 6 -- page 1 of 12
COM111 Introduction to Computer Engineering (Fall 2006-2007) NOTES 6 -- page 1 of 12 Karnaugh Maps In this lecture, we will discuss Karnaugh maps (K-maps) more formally than last time and discuss a more
More informationContents. TAMS38 - Lecture 8 2 k p fractional factorial design. Lecturer: Zhenxia Liu. Example 0 - continued 4. Example 0 - Glazing ceramic 3
Contents TAMS38 - Lecture 8 2 k p fractional factorial design Lecturer: Zhenxia Liu Department of Mathematics - Mathematical Statistics Example 0 2 k factorial design with blocking Example 1 2 k p fractional
More informationMean Comparisons PLANNED F TESTS
Mean Comparisons F-tests provide information on significance of treatment effects, but no information on what the treatment effects are. Comparisons of treatment means provide information on what the treatment
More informationProcess/product optimization using design of experiments and response surface methodology
Process/product optimization using design of experiments and response surface methodology Mikko Mäkelä Sveriges landbruksuniversitet Swedish University of Agricultural Sciences Department of Forest Biomaterials
More informationDesign of Engineering Experiments Part 2 Basic Statistical Concepts Simple comparative experiments
Design of Engineering Experiments Part 2 Basic Statistical Concepts Simple comparative experiments The hypothesis testing framework The two-sample t-test Checking assumptions, validity Comparing more that
More informationResponse Surface Methodology
Response Surface Methodology Bruce A Craig Department of Statistics Purdue University STAT 514 Topic 27 1 Response Surface Methodology Interested in response y in relation to numeric factors x Relationship
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 informationLec 5: Factorial Experiment
November 21, 2011 Example Study of the battery life vs the factors temperatures and types of material. A: Types of material, 3 levels. B: Temperatures, 3 levels. Example Study of the battery life vs the
More informationThe hypergeometric distribution - theoretical basic for the deviation between replicates in one germination test?
Doubt is the beginning, not the end, of wisdom. ANONYMOUS The hypergeometric distribution - theoretical basic for the deviation between replicates in one germination test? Winfried Jackisch 7 th ISTA Seminar
More informationChapter 14 Repeated-Measures Designs
Chapter 14 Repeated-Measures Designs [As in previous chapters, there will be substantial rounding in these answers. I have attempted to make the answers fit with the correct values, rather than the exact
More informationLecture 9: Factorial Design Montgomery: chapter 5
Lecture 9: Factorial Design Montgomery: chapter 5 Page 1 Examples Example I. Two factors (A, B) each with two levels (, +) Page 2 Three Data for Example I Ex.I-Data 1 A B + + 27,33 51,51 18,22 39,41 EX.I-Data
More informationIE 361 EXAM #3 FALL 2013 Show your work: Partial credit can only be given for incorrect answers if there is enough information to clearly see what you were trying to do. There are two additional blank
More informationChap The McGraw-Hill Companies, Inc. All rights reserved.
11 pter11 Chap Analysis of Variance Overview of ANOVA Multiple Comparisons Tests for Homogeneity of Variances Two-Factor ANOVA Without Replication General Linear Model Experimental Design: An Overview
More informationSolutions to Exercises
1 c Atkinson et al 2007, Optimum Experimental Designs, with SAS Solutions to Exercises 1. and 2. Certainly, the solutions to these questions will be different for every reader. Examples of the techniques
More information