Design of Experiments SUTD 06/04/2016 1
|
|
- Alaina Tyler
- 5 years ago
- Views:
Transcription
1 Design of Experiments SUTD 06/04/2016 1
2 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 design 6. More about DOE SUTD 06/04/2016 2
3 1. Introduction Observing a system or process while it is in operation is an important part of the learning process, and is an integral part of understanding and learning about how systems and processes work. To understand what happens to a process when you change certain input factors, you have to do more than just watch you actually have to change the factors. This means that to really understand cause-and-effect relationships in a system you must deliberately change the input variables to the system and observe the changes in the system output that these changes to the inputs produce. In other words, you need to conduct experiments on the system. SUTD 06/04/2016 3
4 1. Introduction Investigators perform experiments in virtually all fields of inquiry, usually to discover something about a particular process or system. Each experimental run is a test. Experiment: A test or series of runs in which purposeful changes are made to the input variables of a process or system so that we may observe and identify the reasons for changes that may be observed in the output response. This module is about planning and conducting experiments; and about analyzing the resulting data so that valid and objective conclusions are obtained. SUTD 06/04/2016 4
5 1. Introduction x i Controllable factors Inputs Process Output y Uncontrollable factors z j SUTD 06/04/2016 5
6 1. Introduction Objectives of the experiment: Determining which variables are most influential on the response y. Determining where to set the influential x so that y is almost always near the desired nominal value. Determining where to set the influential x so that variability in y is small. Determining where to set the influential x so that the effects of the uncontrollable variables z are minimized. SUTD 06/04/2016 6
7 1. Introduction You are a (lazy) golf player who doesn t enjoy practicing. But you also want to lower your score. SUTD 06/04/2016 7
8 1. Introduction Factors that may affect your score: Oversized vs regular-sized driver Balata vs three-piece ball Walk vs ride Water vs sth else Morning vs afternoon game Cool vs hot day Metal vs soft golf shoe spike Windy vs calm day SUTD 06/04/2016 8
9 1. Introduction Preliminary elimination Use your experience to eliminate the factors that will not affect the response (output) significantly. Engineers, scientists, and business analysts, often must make these types of decisions about some of the factors they are considering in real experiments. Based on your long experience with golf, you decide that the effects of the last four factors on your score are very small. SUTD 06/04/2016 9
10 1. Introduction Factors that may affect your score: Oversized vs regular-sized driver O vs R Balata vs three-piece ball B vs T Walk vs ride W vs R Water vs sth else W vs SE Morning vs afternoon game Cool vs hot day Metal vs soft golf shoe spike Windy vs calm day SUTD 06/04/
11 1. Introduction A. Best Guess Approach Suppose that a maximum of eight rounds of golf can be played over the course of the experiment (resource constraints). You decide to start with the following combination: O B R W resulting score: 87 During the round, you notice several wayward shots with the big driver, so you decide to play another round with the regular-sized driver. R B R W resulting score: 89 This approach could be continued almost indefinitely. SUTD 06/04/
12 1. Introduction A. Best Guess Approach Best-guess approach is frequently used in practice by engineers and scientists. It often works reasonably well, because the experimenters often have a great deal of technical or theoretical knowledge of the system they are studying, as well as considerable practical experience. The best-guess approach has at least two disadvantages: 1. Bad guess: Suppose the initial best-guess does not produce the desired results. Now the experimenter has to take another guess at the correct combination of factor levels. This could continue for a long time, without any guarantee of success. 2. Good guess: Suppose the initial best-guess produces an acceptable result. Now the experimenter is tempted to stop testing, although there is no guarantee that the best solution has been found. SUTD 06/04/
13 1. Introduction B. One-factor-at-a-time (OFAT) The OFAT method consists of selecting a starting point, or baseline set of levels, for each factor, and then successively varying each factor over its range with the other factors held constant at the baseline level. After all tests are performed, a series of graphs are usually constructed showing how the response variable is affected by varying each factor with all other factors held constant. SUTD 06/04/
14 1. Introduction Baseline: O B W W Optimal combination wrt OFAT: R (B or T) R - W SUTD 06/04/
15 1. Introduction The major disadvantage of the OFAT strategy is that it fails to consider any possible interaction between the factors. An interaction is the failure of one factor to produce the same effect on the response at different levels of another factor. OFAT experiments are always less efficient than other methods based on a statistical approach to design. SUTD 06/04/
16 1. Introduction Correct approach? 2 4 factorial design (4 factors each having 2 levels) This experiment requires 16 runs. A 10-factor experiment would require 1024 runs. Fortunately, we can use fractional factorial designs. SUTD 06/04/
17 1. Introduction 2 2 factorial design with 2 replications: = 8 rounds D = B = = 3.25 = 0.75 DB = = 0.25 Statistical tests show that: D is significant, B and DB are not. SUTD 06/04/
18 1. Introduction Guidelines for designing an experiment: 1. Recognition of and statement of the problem 2. Selection of the response variable 3. Choice of factors, levels, and ranges 4. Choice of experimental design 5. Performing the experiment 6. Statistical analysis of the data 7. Conclusions and recommendation SUTD 06/04/
19 1. Introduction Guidelines for designing an experiment: 1. Recognition of and statement of the problem 2. Selection of the response variable 3. Choice of factors, levels, and ranges 4. Choice of experimental design Factor screening or characterization Optimization Confirmation Discovery Robustness 5. Performing the experiment 6. Statistical analysis of the data 7. Conclusions and recommendation SUTD 06/04/
20 1. Introduction Guidelines for designing an experiment: 1. Recognition of and statement of the problem 2. Selection of the response variable 3. Choice of factors, levels, and ranges 4. Choice of experimental design 5. Performing the experiment 6. Statistical analysis of the data The experimenter should be certain that this variable really provides useful information about the process under study. It is usually critically important to identify issues related to the method to measure the response variables before conducting the experiment. 7. Conclusions and recommendation SUTD 06/04/
21 1. Introduction Guidelines for designing an experiment: 1. Recognition of and statement of the problem 2. Selection of the response variable 3. Choice of factors, levels, and ranges 4. Choice of experimental design 5. Performing the experiment Design factors vs Nuisance factors Nuisance factors: Controllable (batches) Uncontrollable (humidity in environment) Noise 6. Statistical analysis of the data 7. Conclusions and recommendation SUTD 06/04/
22 1. Introduction Guidelines for designing an experiment: 1. Recognition of and statement of the problem 2. Selection of the response variable 3. Choice of factors, levels, and ranges 4. Choice of experimental design Consideration of Sample size (number of replicates) Selection of a suitable run order Determination of whether or not blocking or other randomization restrictions are involved. 5. Performing the experiment 6. Statistical analysis of the data 7. Conclusions and recommendation SUTD 06/04/
23 1. Introduction Guidelines for designing an experiment: 1. Recognition of and statement of the problem Errors in experimental procedure at this stage will usually destroy experimental validity. 2. Selection of the response variable 3. Choice of factors, levels, and ranges 4. Choice of experimental design 5. Performing the experiment 6. Statistical analysis of the data 7. Conclusions and recommendation SUTD 06/04/
24 1. Introduction Guidelines for designing an experiment: 1. Recognition of and statement of the problem 2. Selection of the response variable 3. Choice of factors, levels, and ranges 4. Choice of experimental design 5. Performing the experiment Statistical methods should be used to analyze the data so that results and conclusions are objective rather than judgmental in nature. Often we find that simple graphical methods play an important role in data analysis and interpretation. 6. Statistical analysis of the data 7. Conclusions and recommendation SUTD 06/04/
25 1. Introduction Guidelines for designing an experiment: 1. Recognition of and statement of the problem 2. Selection of the response variable 3. Choice of factors, levels, and ranges Once the data have been analyzed, the experimenter must draw practical conclusions about the results and recommend a course of action. 4. Choice of experimental design 5. Performing the experiment 6. Statistical analysis of the data 7. Conclusions and recommendation SUTD 06/04/
26 2 k Factorial Design SUTD 06/04/
27 2. 2 k Factorial Design Geometric Coding and Labels (2 3 factorial design) 8 combination of the factors, or treatments. Run A B C Labels 1 (1) 2 + a 3 + b ab 5 + c ac bc abc SUTD 06/04/
28 2. 2 k Factorial Design Algebraic Signs for Calculating Effects Labels I A B AB C AC BC ABC (1) a b ab c ac bc abc A B = AB, AB B = AB 2 = A A = 1 4n 1 + a b + ab c + ac bc + abc SUTD 06/04/
29 2. 2 k Factorial Design Geometric View SUTD 06/04/
30 2. 2 k Factorial Design A chemical product is produced in a pressure vessel. A factorial experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate (FR) of this product. The four factors are: A. Temperature (Temp) B. Pressure (Pres) C. Concentration of formaldehyde (Conc) D. Stirring rate (SR) Objectives: The process engineer is interested in maximizing the filtration rate. Current rates are around 75 gal/h. The process currently uses the conc. of formaldehyde, at the high level. The engineer would like to reduce the formaldehyde concentration as much as possible; but has been unable to do so because it always results in lower filtration rates. SUTD 06/04/
31 2. 2 k Factorial Design Each factor has 2 levels. We use a 2 4 factorial design. We carry out a single run for each of the 16 combinations. SUTD 06/04/
32 2. 2 k Factorial Design Regression Model y = β 0 + β A x A + β B x B + β C x C + β D x D + β AB x A x B + β AC x A x C + β AD x A x D + β BC x B x C + β B x B x D + β CD x C x D + β ABC x A x B x C + β ABD x A x B x D + β ACD x A x C x D + β BCD x B x C x D + β ABCD x A x B x C x D SUTD 06/04/
33 2. 2 k Factorial Design Minitab Exercise: Section 2 in DOE-Minitab.pdf SUTD 06/04/
34 2. 2 k Factorial Design Pareto Chart Normal Plot SUTD 06/04/
35 2. 2 k Factorial Design Fitted Regression Model Try all factors at high levels: Temp = Press = Conc = SR = +1 FR = Observed value: 96 SUTD 06/04/
36 2. 2 k Factorial Design Main Effects Plot If we consider only the main effects, we would run all factors at high levels. SUTD 06/04/
37 2. 2 k Factorial Design Interaction Plot SUTD 06/04/
38 2. 2 k Factorial Design Temp (A) effect is: very small when C is high, very large when C is low. best: C = 1, A = +1 SR (D) effect is: little when A is low, large when A is high. Best combination: A = +1, C = 1, D = +1 Note that B is not significant. SUTD 06/04/
39 2. 2 k Factorial Design Risks with Unreplicated Factorial Design Error variance is not estimable. We can never be entirely certain that the experimental error is small. A good practice in these types of experiments is to spread out the factor levels aggressively. One approach to the analysis of an unreplicated factorial is to assume that certain highorder interactions are negligible and combine their mean squares to estimate the error. This is an appeal to the sparsity of effects principle; that is, most systems are dominated by some of the main effects and low-order interactions, and most high-order interactions are negligible. Appropriate when you are confident that experimental error is small: Computer simulations with no random numbers: same inputs always result in the same outputs (experimental error = 0). SUTD 06/04/
40 Choice of Sample Size SUTD 06/04/
41 3. Choice of Sample Size What is the appropriate sample size (number of replications)? Depends on desired power of the experimentation, overall variance of the response, desired accuracy. SUTD 06/04/
42 3. Choice of Sample Size Power: β = Pr Type II error = Pr fail to reject H 0 H 0 is false power = 1 β As power increases, the chance to catch a significant effect increases. Effects: The minimum difference between ± levels of the factors that you want to detect. Minimum detectable change (MDC) size. SUTD 06/04/
43 3. Choice of Sample Size As the effects increase, the power increases. As the sample size get larger, the power of the test increases. Minitab Exercise: Section 3 in DOE-Minitab.pdf SUTD 06/04/
44 3. Choice of Sample Size SUTD 06/04/
45 2 k p Fractional Factorial Design SUTD 06/04/
46 4. 2 k p Fractional Factorial Design Fractional Design with defining relation I = ABC Labels I A B AB C AC BC ABC (1) a b ab c ac bc abc SUTD 06/04/
47 4. 2 k p Fractional Factorial Design Fractional Design with defining relation I = ABC Labels I A B C a + + b + + c + + abc Number of runs is reduced from 8 to 4, with the cost of aliasing some effects. Resolving the aliases: A = A I = A ABC = A 2 BC = BC B = B I = B ABC = AB 2 C = AC C = C I = C ABC = ABC 2 = AB This means: A = A + BC B = B + AC C = C + AB I = I + ABC SUTD 06/04/
48 4. 2 k p Fractional Factorial Design Geometric View If the experimenter removes any of the three factors from the analysis, the design will project into a full factorial 2 2 design. SUTD 06/04/
49 4. 2 k p Fractional Factorial Design Improving Online Learning A university that specializes in online learning wants to improve the effectiveness of its 8-week course. Statistics group identifies 7 factors to test, each at 2 levels: Code Factor - + A Textbook Current New B Readings No Yes C Homework 3 Hours 5 Hours D Software Current New E Sessions 3 per week 4 per week F Review No Yes G Lecture Notes No Yes SUTD 06/04/
50 4. 2 k p Fractional Factorial Design Improving Online Learning Full factorial design requires 2 7 = 128 runs for a single replication. Assigning students to 128 sections does not seem possible. You decide to use III saturated design. 8 runs for each replication. Each of the 8 runs defines characteristics of a section. 10 students are randomly assigned to each of the 8 sections. At the end of the course, each student takes a final exam (response). SUTD 06/04/
51 4. 2 k p Fractional Factorial Design Improving Online Learning Regression Model: y = β 0 + β A x A + β B x B + β C x C + β D x D + β E x E + β F x F + β G x G Only the main effects can be estimated. Actually, they are confounded with interactions, and we know the alias structure. SUTD 06/04/
52 4. 2 k p Fractional Factorial Design Improving Online Learning Generators: D = AB, E = AC, F = BC, G = ABC Or equivalently: I = ABD, I = ACE, I = BCF, I = ABCG Two-level aliases of A: A = B AB = BD (first generator) A = C AC = CE (second generator) A = BC ABC = FG (third and fourth generators) A = A + BD + CE + FG + higher-order-interactions SUTD 06/04/
53 4. 2 k p Fractional Factorial Design Improving Online Learning SUTD 06/04/
54 4. 2 k p Fractional Factorial Design Improving Online Learning Section Book Read HW SW Sess Rev LecN 1 Current No 3 hr New 4 /wk Yes No 2 New No 3 hr Current 3 /wk Yes Yes 3 Current Yes 3 hr Current 4 /wk No Yes 4 New Yes 3 hr New 3 /wk No No 5 Current No 5 hr New 3 /wk No Yes 6 New No 5 hr Current 4 /wk No No 7 Current Yes 5 hr Current 3 /wk Yes No 8 New Yes 5 hr New 4 /wk Yes Yes SUTD 06/04/
55 4. 2 k p Fractional Factorial Design Minitab Exercise: Section 4 in DOE-Minitab.pdf SUTD 06/04/
56 4. 2 k p Fractional Factorial Design Improving Online Learning SUTD 06/04/
57 4. 2 k p Fractional Factorial Design Model Adequacy Checking Normal Probability Plot should resemble a straight line. Histogram should resemble normal cdf. Versus Fits should not be structureless. Versus Order should not have a trend. SUTD 06/04/
58 4. 2 k p Fractional Factorial Design SUTD 06/04/
59 4. 2 k p Fractional Factorial Design Best option: Use new textbook (A = +1), make 3 sessions per week (E = 1). SUTD 06/04/
60 4. 2 k p Fractional Factorial Design However note that: A = A + BD + CE + FG + E = E + AC + BG + DF + The conclusion (best option is A = +1, E = 1) relies on the assumption that: BD CE FG AC BG DF 0 What if some of the listed two-level interactions are not negligible? SUTD 06/04/
61 Folding Over SUTD 06/04/
62 5. Folding Over You are suspicious with these interpretations and the course is about to be offered again. You plan to run a follow-up experiment. Again with 8 sections, and 10 students (replications) in each section. However, this time you change the settings in the sections. In particular, you fold over factor A (textbook), which appeared to be the most significant factor in the initial analysis. SUTD 06/04/
63 5. Folding Over Folding over factor A means reversing the signs in column A while leaving the other columns unchanged SUTD 06/04/
64 4. 2 k p Fractional Factorial Design Improving Online Learning Section Book Read HW SW Sess Rev LecN 1 Current No 3 hr New 4 /wk Yes No 2 New No 3 hr Current 3 /wk Yes Yes 3 Current Yes 3 hr Current 4 /wk No Yes 4 New Yes Section 3 hr Book New Read 3 /wk HW No SW No Sess Rev LecN 5 Current No 9 5 hr New 3 No /wk 3 No hr New Yes 4 /wk Yes No 6 New No 10 5 hr Current 4 No /wk 3 No hr Current No 3 /wk Yes Yes 7 Current Yes 11 5 hr Current New 3 Yes /wk 3 Yes hr Current No 4 /wk No Yes 8 New Yes 12 5 hr Current New 4 Yes /wk 3 Yes hr New Yes 3 /wk No No 13 New No 5 hr New 3 /wk No Yes 14 Current No 5 hr Current 4 /wk No No 15 New Yes 5 hr Current 3 /wk Yes No 16 Current Yes 5 hr New 4 /wk Yes Yes SUTD 06/04/
65 5. Folding Over By folding over factor A, we isolate the main effect A from all two and threelevel interactions. Moreover, all two-level interactions including A will be isolated from other twolevel interactions. We will be more confident with the estimates of the main effect of A and all its twolevel interactions. Minitab Exercise: Section 5 in DOE-Minitab.pdf SUTD 06/04/
66 5. Folding Over SUTD 06/04/
67 5. Folding Over Model Adequacy SUTD 06/04/
68 5. Folding Over Previous interpretation: The alias structure was: Book (A) and Session (E) are significant factors. E = E + AC + BG + DF SUTD 06/04/
69 5. Folding Over SUTD 06/04/
70 5. Folding Over SUTD 06/04/
71 5. Folding Over With the current book, more homework is better. With the new one, less homework results in higher scores. New book performs always better than the current one. Best option: use new textbook (A = +1), assign less homework (C = 1). SUTD 06/04/
72 More about DoE SUTD 06/04/
73 6. More about DOE a) Model Adequacy Checking (Bad Examples) b) Quadratic Curvature c) Blocking d) Response Surface Methods (Optimization) SUTD 06/04/
74 6. a) Model Adequacy Checking Normal Probability Plot SUTD 06/04/
75 6. a) Model Adequacy Checking Residuals versus Fits SUTD 06/04/
76 6. a) Model Adequacy Checking Residuals versus Observation (Run) Order SUTD 06/04/
77 6. b) Quadratic Curvature 2-level factorial design assumes linearity. Perfect linearity is unnecessary, and the 2 k system will work quite well even when the linearity assumption holds only very approximately. In fact, the model is capable of representing some curvature when interaction terms are added to the model (results from the twisting of the plane induced by the interaction terms). First-order model: SUTD 06/04/
78 6. b) Quadratic Curvature In some situations, the curvature in the response function will not be adequately modeled by the first-order model. In such cases, a logical model to consider is the second-order response surface model: where β jj represents quadratic effects. SUTD 06/04/
79 6. b) Quadratic Curvature Checking the existence of quadratic effects There is a method of replicating certain points in a 2 k factorial that will provide protection against curvature from second-order effects as well as allow an independent estimate of error to be obtained. The method consists of adding center points to the 2 k design. SUTD 06/04/
80 6. b) Quadratic Curvature Checking the presence of quadratic effects Adding replicates at the center points: If y F y C the center points lie on or near the plane passing through the factorial points, there is no quadratic curvature. If y F y C quadratic curvature is present. SUTD 06/04/
81 6. b) Quadratic Curvature Fitting the second-order model A simple and highly effective solution to this problem is to augment the 2 k with axial runs (± 2). The resulting design, called a central composite design, can now be used to fit the second-order model. SUTD 06/04/
82 6. b) Quadratic Curvature Fitting the second-order model: Central Composite Design SUTD 06/04/
83 6. c) Blocking Suppose an experimenter is investigating the effect of the concentration of the reactant (factor A with levels 15% and 25%), and the amount of catalyst (factor B with levels 1 pound and 2 pounds) on the yield (response) in a chemical process. SUTD 06/04/
84 6. c) Blocking 2 2 factorial design with 3 replicates is used (4 3 = 12 runs). Suppose only 4 experimental trials can be made from a single batch of raw material (the experimenter will use 3 batches to complete the experiment). What if the raw material affects the yield? Raw material is not a design factor since the experimenter does not want to measure its affect on the yield. However, existence of its effect may be misleading in the analysis. In such cases, we use blocking, to estimate the effect of the blocks (raw material) and cancel out its effect from our factor analysis. SUTD 06/04/
85 6. d) Response Surface Methods Suppose that two factors that affect the response y of your design: the width of the product w, the thickness of your product t. These are your design variables. You have the constraints that: w 10, 100 centimeters t 5, 10 millimetres How do you find w and t that maximizes y? SUTD 06/04/
86 6. d) Response Surface Methods Response surface methodology, or RSM, is a collection of mathematical, and statistical techniques useful for the modeling and analysis of problems in which a response of interest is influenced by several variables and the objective is to optimize this response. SUTD 06/04/
87 6. d) Response Surface Methods For example, suppose that a chemical engineer wishes to find the levels of temperature x 1, and pressure x 2 that maximize the yield y of a process. The process yield is a function of the levels of temperature and pressure, say y = f x 1, x 2 + ε If we denote the expected response by E y = f x 1, x 2 represented by = η, then the surface η = f x 1, x 2 is called a response surface. SUTD 06/04/
88 6. d) Response Surface Methods SUTD 06/04/
89 6. d) Response Surface Methods RSM is a sequential procedure Often, when we are at a point on the response surface that is remote from the optimum, such as the current operating conditions; there is little curvature in the system and the first-order model will be appropriate. Our objective here is to lead the experimenter rapidly and efficiently along a path of improvement toward the general vicinity of the optimum. SUTD 06/04/
90 6. d) Response Surface Methods RSM is a sequential procedure Once the region of the optimum has been found, a more elaborate model, such as the second-order model, may be employed, and an analysis may be performed to locate the optimum. The analysis of a response surface can be thought of as climbing a hill, where the top of the hill represents the point of maximum response. SUTD 06/04/
91 References Montgomery, D.C., (2013). Design and Analysis of Experiments - 8th Edition, John Wiley and Sons Inc. Ledolter, J. and Swersey, A.J., (2007). Testing 1 2 3: Experimental Design with Applications in Marketing and Service Operations, Stanford University Press. SUTD 06/04/
92 Final Exercise List the design variables, factors and response(s) related with your project. Identify also the alternative values that each design variable can attain. Considering your constraints (time, budget, etc), try to identify the maximum number of experiments you can carry out. Identify a 2-level factorial design (full factorial, fractional) and the number of replicates that best fits your situation. Check the power of your experimental design using an arbitrary estimate on the standard deviation. Try to identify any factors that need to be blocked. SUTD 06/04/
Design 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 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 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 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 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 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 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 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 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 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 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 informationDesign of Experiments
Design of Experiments D R. S H A S H A N K S H E K H A R M S E, I I T K A N P U R F E B 19 TH 2 0 1 6 T E Q I P ( I I T K A N P U R ) Data Analysis 2 Draw Conclusions Ask a Question Analyze data What to
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 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 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 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 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 information2.830J / 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 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 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 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 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 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 informationReference: Chapter 8 of Montgomery (8e)
Reference: Chapter 8 of Montgomery (8e) 69 Maghsoodloo Fractional Factorials (or Replicates) For Base 2 Designs As the number of factors in a 2 k factorial experiment increases, the number of runs (or
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 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 informationDesign and Analysis of Multi-Factored Experiments
Design and Analysis of Multi-Factored Experiments Fractional Factorial Designs L. M. Lye DOE Course 1 Design of Engineering Experiments The 2 k-p Fractional Factorial Design Motivation for fractional factorials
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
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 informationDESIGN OF EXPERIMENT ERT 427 Response Surface Methodology (RSM) Miss Hanna Ilyani Zulhaimi
+ DESIGN OF EXPERIMENT ERT 427 Response Surface Methodology (RSM) Miss Hanna Ilyani Zulhaimi + Outline n Definition of Response Surface Methodology n Method of Steepest Ascent n Second-Order Response Surface
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 informationHow To: Analyze a Split-Plot Design Using STATGRAPHICS Centurion
How To: Analyze a SplitPlot Design Using STATGRAPHICS Centurion by Dr. Neil W. Polhemus August 13, 2005 Introduction When performing an experiment involving several factors, it is best to randomize the
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 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 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 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 Process and Product Optimization Using Designed Experiments Second Edition RAYMOND H. MYERS Virginia Polytechnic Institute and State University DOUGLAS C. MONTGOMERY Arizona
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 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 informationExperimental design (KKR031, KBT120) Tuesday 11/ :30-13:30 V
Experimental design (KKR031, KBT120) Tuesday 11/1 2011-8:30-13:30 V Jan Rodmar will be available at ext 3024 and will visit the examination room ca 10:30. The examination results will be available for
More informationIENG581 Design and Analysis of Experiments INTRODUCTION
Experimental Design IENG581 Design and Analysis of Experiments INTRODUCTION Experiments are performed by investigators in virtually all fields of inquiry, usually to discover something about a particular
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 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 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 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 information9 Correlation and Regression
9 Correlation and Regression SW, Chapter 12. Suppose we select n = 10 persons from the population of college seniors who plan to take the MCAT exam. Each takes the test, is coached, and then retakes the
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 12 - Part I: Correlation Analysis
ST coursework due Friday, April - Chapter - Part I: Correlation Analysis Textbook Assignment Page - # Page - #, Page - # Lab Assignment # (available on ST webpage) GOALS When you have completed this lecture,
More informationSizing Mixture (RSM) Designs for Adequate Precision via Fraction of Design Space (FDS)
for Adequate Precision via Fraction of Design Space (FDS) Pat Whitcomb Stat-Ease, Inc. 61.746.036 036 fax 61.746.056 pat@statease.com Gary W. Oehlert School of Statistics University it of Minnesota 61.65.1557
More informationQuestion. Hypothesis testing. Example. Answer: hypothesis. Test: true or not? Question. Average is not the mean! μ average. Random deviation or not?
Hypothesis testing Question Very frequently: what is the possible value of μ? Sample: we know only the average! μ average. Random deviation or not? Standard error: the measure of the random deviation.
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 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 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 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 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 informationObjective Experiments Glossary of Statistical Terms
Objective Experiments Glossary of Statistical Terms This glossary is intended to provide friendly definitions for terms used commonly in engineering and science. It is not intended to be absolutely precise.
More informationIntroduction to Design of Experiments
Introduction to Design of Experiments Jean-Marc Vincent and Arnaud Legrand Laboratory ID-IMAG MESCAL Project Universities of Grenoble {Jean-Marc.Vincent,Arnaud.Legrand}@imag.fr November 20, 2011 J.-M.
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 informationDesign of Engineering Experiments Chapter 8 The 2 k-p Fractional Factorial Design
Design of Engineering Experiments Chapter 8 The 2 k-p Fractional Factorial Design Text reference, Chapter 8 Motivation for fractional factorials is obvious; as the number of factors becomes large enough
More information, (1) e i = ˆσ 1 h ii. c 2016, Jeffrey S. Simonoff 1
Regression diagnostics As is true of all statistical methodologies, linear regression analysis can be a very effective way to model data, as along as the assumptions being made are true. For the regression
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 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 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 informationDesign and Analysis of Experiments 8E 2012 Montgomery
1 Why do Fractional Factorial Designs Work? The sparsity of effects principle There may be lots of factors, but few are important System is dominated by main effects, low-order interactions The projection
More informationFractional Factorials
Fractional Factorials Bruce A Craig Department of Statistics Purdue University STAT 514 Topic 26 1 Fractional Factorials Number of runs required for full factorial grows quickly A 2 7 design requires 128
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 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 informationRegression, part II. I. What does it all mean? A) Notice that so far all we ve done is math.
Regression, part II I. What does it all mean? A) Notice that so far all we ve done is math. 1) One can calculate the Least Squares Regression Line for anything, regardless of any assumptions. 2) But, if
More informationTWO-LEVEL FACTORIAL EXPERIMENTS: IRREGULAR FRACTIONS
STAT 512 2-Level Factorial Experiments: Irregular Fractions 1 TWO-LEVEL FACTORIAL EXPERIMENTS: IRREGULAR FRACTIONS A major practical weakness of regular fractional factorial designs is that N must be a
More informationThe One-Quarter Fraction
The One-Quarter Fraction ST 516 Need two generating relations. E.g. a 2 6 2 design, with generating relations I = ABCE and I = BCDF. Product of these is ADEF. Complete defining relation is I = ABCE = BCDF
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 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 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 informationChapter 7. Inference for Distributions. Introduction to the Practice of STATISTICS SEVENTH. Moore / McCabe / Craig. Lecture Presentation Slides
Chapter 7 Inference for Distributions Introduction to the Practice of STATISTICS SEVENTH EDITION Moore / McCabe / Craig Lecture Presentation Slides Chapter 7 Inference for Distributions 7.1 Inference 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 informationLecture 14. Analysis of Variance * Correlation and Regression. The McGraw-Hill Companies, Inc., 2000
Lecture 14 Analysis of Variance * Correlation and Regression Outline Analysis of Variance (ANOVA) 11-1 Introduction 11-2 Scatter Plots 11-3 Correlation 11-4 Regression Outline 11-5 Coefficient of Determination
More informationLecture 14. Outline. Outline. Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA)
Outline Lecture 14 Analysis of Variance * Correlation and Regression Analysis of Variance (ANOVA) 11-1 Introduction 11- Scatter Plots 11-3 Correlation 11-4 Regression Outline 11-5 Coefficient of Determination
More informationBusiness Statistics. Lecture 5: Confidence Intervals
Business Statistics Lecture 5: Confidence Intervals Goals for this Lecture Confidence intervals The t distribution 2 Welcome to Interval Estimation! Moments Mean 815.0340 Std Dev 0.8923 Std Error Mean
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 informationAnalysis of Covariance. The following example illustrates a case where the covariate is affected by the treatments.
Analysis of Covariance In some experiments, the experimental units (subjects) are nonhomogeneous or there is variation in the experimental conditions that are not due to the treatments. For example, a
More informationeqr014 Lenth s Method for the Analysis of Unreplicated Experiments
eqr014 Lenth s Method for the Analysis of Unreplicated Experiments Russell V. Lenth Department of Statistics and Actuarial Science The University of Iowa Iowa City, IA USA 52242 Voice 319-335-0814 FAX
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 informationSwarthmore Honors Exam 2012: Statistics
Swarthmore Honors Exam 2012: Statistics 1 Swarthmore Honors Exam 2012: Statistics John W. Emerson, Yale University NAME: Instructions: This is a closed-book three-hour exam having six questions. You may
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 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 informationMathematics Practice Test 2
Mathematics Practice Test 2 Complete 50 question practice test The questions in the Mathematics section require you to solve mathematical problems. Most of the questions are presented as word problems.
More informationPRODUCT QUALITY IMPROVEMENT THROUGH RESPONSE SURFACE METHODOLOGY : A CASE STUDY
PRODUCT QULITY IMPROVEMENT THROUGH RESPONSE SURFCE METHODOLOGY : CSE STUDY HE Zhen, College of Management and Economics, Tianjin University, China, zhhe@tju.edu.cn, Tel: +86-22-8740783 ZHNG Xu-tao, College
More informationHomework 04. , not a , not a 27 3 III III
Response Surface Methodology, Stat 579 Fall 2014 Homework 04 Name: Answer Key Prof. Erik B. Erhardt Part I. (130 points) I recommend reading through all the parts of the HW (with my adjustments) before
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 informationMidterm: CS 6375 Spring 2015 Solutions
Midterm: CS 6375 Spring 2015 Solutions The exam is closed book. You are allowed a one-page cheat sheet. Answer the questions in the spaces provided on the question sheets. If you run out of room for an
More informationMath 147 Lecture Notes: Lecture 12
Math 147 Lecture Notes: Lecture 12 Walter Carlip February, 2018 All generalizations are false, including this one.. Samuel Clemens (aka Mark Twain) (1835-1910) Figures don t lie, but liars do figure. Samuel
More informationMidterm. Introduction to Machine Learning. CS 189 Spring Please do not open the exam before you are instructed to do so.
CS 89 Spring 07 Introduction to Machine Learning Midterm Please do not open the exam before you are instructed to do so. The exam is closed book, closed notes except your one-page cheat sheet. Electronic
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 informationRegression analysis is a tool for building mathematical and statistical models that characterize relationships between variables Finds a linear
Regression analysis is a tool for building mathematical and statistical models that characterize relationships between variables Finds a linear relationship between: - one independent variable X and -
More informationTackling Statistical Uncertainty in Method Validation
Tackling Statistical Uncertainty in Method Validation Steven Walfish President, Statistical Outsourcing Services steven@statisticaloutsourcingservices.com 301-325 325-31293129 About the Speaker Mr. Steven
More informationChapter 1 Statistical Inference
Chapter 1 Statistical Inference causal inference To infer causality, you need a randomized experiment (or a huge observational study and lots of outside information). inference to populations Generalizations
More informationDesign & Analysis of Experiments 7E 2009 Montgomery
1 What If There Are More Than Two Factor Levels? The t-test does not directly apply ppy There are lots of practical situations where there are either more than two levels of interest, or there are several
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 informationIE 316 Exam 1 Fall 2011
IE 316 Exam 1 Fall 2011 I have neither given nor received unauthorized assistance on this exam. Name Signed Date Name Printed 1 1. Suppose the actual diameters x in a batch of steel cylinders are normally
More information