Suppose we needed four batches of formaldehyde, and coulddoonly4runsperbatch. Thisisthena2 4 factorial in 2 2 blocks.
|
|
- Tabitha Cross
- 5 years ago
- Views:
Transcription
1 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 blocks, then so is their product, which is defined by multiplication mod 2 : 0 = 2 = = E.g. = 2 =. Pick two effects to be confounded with blocks: and. Then also = is confounded. We wouldn t pick and, since =.
2 59 For the choices and we have = = = = with () a b ab c ac bc abc Block I IV II III IV I III II d ad bd abd cd acd bcd abcd Block III II IV I II III I IV Block I : () Block II : Block III : Block IV :
3 A B C D blocks ABC ACD BD y
4 6 > g <- lm(y ~blocks + A + B + C + D + A*B + A*C + A*D + B*C + C*D + A*B*D + B*C*D + A*B*C*D) > anova(g) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) blocks A B C D A:B A:C.6.6 A:D B:C C:D A:B:D B:C:D A:B:C:D Residuals 0 0.0
5 62 Normal Q Q Plot Sample Quantiles A:C A:B C:D D B:C:D A:B:D A:B:C:D B blocks3 blocks2 A:D B:C C A blocks4 0 Theoretical Quantiles Fig Half normal plot for 2 4 factorial in 2 2 blocks. Itlookslikewecandropthemaineffect of D if we keep some of its interactions.
6 63 R will, by default, estimate a main effect if an interaction is in the model. To fit blocks, A, B, C, AB, AD, BC, CD but not D, we can add the SS and df for D to those for Error. > h <- lm(y ~blocks + A + B + C + B*C + A*B + A*D + C*D) > anova(h) Df Sum Sq Mean Sq F value Pr(>F) blocks *** A *** B *** C *** D B:C ** A:B * A:D ** C:D Residuals This would change to ( ) 5 =3 08 on 5 d.f. - not a helpful step (since was larger than ).
7 64 mean of y B mean of y D A A mean of y C mean of y D B C Fig Interaction plots. The best combination seemstobea,c,dhigh,blow.
8 Partial confounding To get an estimate of error, we have to either drop certain effects from the model, or replicate the design. If we replicate, we can either: Confoundthesameeffects with blocks in each replication - complete confounding, or Confound different effects with each replication - partial confounding. Partial confounding is often better, since we then get estimates of effects from the replications in which they are not confounded.
9 66 Example 7-3 from text. Two replicates of a 2 3 factorial are to be run, in 2 block each. Replicate : Confound ABC with blocks. So = =0for() and =for Replicate 2: Confound AB with blocks. So = + 2 =0for() and = for Rep Rep2 Block Block 2 Block 3 Block 4 () = 550 = 669 () = 604 =650 =642 = 633 = 052 = 60 =749 = 037 =635 =868 = 075 =729 =860 =063
10 67 A B C Rep Block ABC AB y I I I I I I I I II II II II II II II II When the levels of one factor (Blocks) make sense only within the levels of another factor (Replicates) we say that the first is nested within the second. A waytoindicatethisinrisas:
11 > h <- lm(y ~Rep + Block%in%Rep + A + B + C + A*B + A*C + B*C + A*B*C) > anova(h) Analysis of Variance Table 68 Response: y Df Sum Sq Mean Sq F value Pr(>F) Rep A * B C e-05 *** Rep:Block A:B A:C ** B:C A:B:C Residuals Through the partial confounding we are able to estimate all interactions. It looks like only A, C, and AC are significant.
12 69 resids resids c(a) c(c) resids resids c(b) c(block) Normal Q Q Plot resids Sample Quantiles fits Theoretical Quantiles Fig Residuals for fit to A, C, and AC only.
13 70 mean of y II I A B C Block mean of y A Factors C Fig Design and interactions. How is ( ) computed? One way is to compute in Rep I, where this effect is confounded with blocks, and similarly in
14 7 RepII,andaddthem: " # 2 ( ) +( ) = 8 = 338 = =20 25 ( ) = = in agreement with the ANOVA output. See the programme on the course web site to see how to do this calculation very easily. Another method goes back to general principles. We calculate a SS for blocks within each replicate (since blocks make sense only within the replicates): ( ) =4 X X ³ 2 = = 2 = 2 Here is the average in block of replicate, and is the overall average of that replicate, which is the only one in which that block makes sense. See the R calculation.
19. 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationAdvanced Digital Design with the Verilog HDL, Second Edition Michael D. Ciletti Prentice Hall, Pearson Education, 2011
Problem 2-1 Recall that a minterm is a cube in which every variable appears. A Boolean expression in SOP form is canonical if every cube in the expression has a unique representation in which all of the
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 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 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 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 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 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 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 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 informationWritten Exam (2 hours)
M. Müller Applied Analysis of Variance and Experimental Design Summer 2015 Written Exam (2 hours) General remarks: Open book exam. Switch off your mobile phone! Do not stay too long on a part where you
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 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 informationMath Treibergs. Peanut Oil Data: 2 5 Factorial design with 1/2 Replication. Name: Example April 22, Data File Used in this Analysis:
Math 3080 1. Treibergs Peanut Oil Data: 2 5 Factorial design with 1/2 Replication. Name: Example April 22, 2010 Data File Used in this Analysis: # Math 3080-1 Peanut Oil Data April 22, 2010 # Treibergs
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 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 informationFractional Replications
Chapter 11 Fractional Replications Consider the set up of complete factorial experiment, say k. If there are four factors, then the total number of plots needed to conduct the experiment is 4 = 1. When
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 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 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 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 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 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 informationThese are multifactor experiments that have
Design of Engineering Experiments Nested Designs Text reference, Chapter 14, Pg. 525 These are multifactor experiments that have some important industrial applications Nested and split-plot designs frequently
More informationStat 217 Final Exam. Name: May 1, 2002
Stat 217 Final Exam Name: May 1, 2002 Problem 1. Three brands of batteries are under study. It is suspected that the lives (in weeks) of the three brands are different. Five batteries of each brand are
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 informationMODELS WITHOUT AN INTERCEPT
Consider the balanced two factor design MODELS WITHOUT AN INTERCEPT Factor A 3 levels, indexed j 0, 1, 2; Factor B 5 levels, indexed l 0, 1, 2, 3, 4; n jl 4 replicate observations for each factor level
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 informationFractional designs and blocking.
Fractional designs and blocking Petter Mostad mostad@chalmers.se Review of two-level factorial designs Goal of experiment: To find the effect on the response(s) of a set of factors each factor can be set
More informationCHAPTER 3 BOOLEAN ALGEBRA
CHAPTER 3 BOOLEAN ALGEBRA (continued) This chapter in the book includes: Objectives Study Guide 3.1 Multiplying Out and Factoring Expressions 3.2 Exclusive-OR and Equivalence Operations 3.3 The Consensus
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 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 informationCHAPTER 5 KARNAUGH MAPS
CHAPTER 5 1/36 KARNAUGH MAPS This chapter in the book includes: Objectives Study Guide 5.1 Minimum Forms of Switching Functions 5.2 Two- and Three-Variable Karnaugh Maps 5.3 Four-Variable Karnaugh Maps
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 informationLecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps
EE210: Switching Systems Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps Prof. YingLi Tian Feb. 21/26, 2019 Department of Electrical Engineering The City College of New York
More informationKarnaugh Maps Objectives
Karnaugh Maps Objectives For Karnaugh Maps of up to 5 variables Plot a function from algebraic, minterm or maxterm form Obtain minimum Sum of Products and Product of Sums Understand the relationship between
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 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 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 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 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 informationSolution to Final Exam
Stat 660 Solution to Final Exam. (5 points) A large pharmaceutical company is interested in testing the uniformity (a continuous measurement that can be taken by a measurement instrument) of their film-coated
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 informationDesign theory for relational databases
Design theory for relational databases 1. Consider a relation with schema R(A,B,C,D) and FD s AB C, C D and D A. a. What are all the nontrivial FD s that follow from the given FD s? You should restrict
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 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 informationSTAT22200 Spring 2014 Chapter 14
STAT22200 Spring 2014 Chapter 14 Yibi Huang May 27, 2014 Chapter 14 Incomplete Block Designs 14.1 Balanced Incomplete Block Designs (BIBD) Chapter 14-1 Incomplete Block Designs A Brief Introduction to
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 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 informationMapping QTL to a phylogenetic tree
Mapping QTL to a phylogenetic tree Karl W Broman Department of Biostatistics & Medical Informatics University of Wisconsin Madison www.biostat.wisc.edu/~kbroman Human vs mouse www.daviddeen.com 3 Intercross
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 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 informationNesting and Mixed Effects: Part I. Lukas Meier, Seminar für Statistik
Nesting and Mixed Effects: Part I Lukas Meier, Seminar für Statistik Where do we stand? So far: Fixed effects Random effects Both in the factorial context Now: Nested factor structure Mixed models: a combination
More informationMATH 556 Homework 13 Due: Nov 21, Wednesday
MATH 556 Homework 13 Due: Nov 21, Wednesday Ex. A. Based on the model concluded in 6.9, carry out a simulation study (In particular, use the LSE from the data in 6.9, generate new data for A.1, A.2, A.3,
More informationUNIT 3 BOOLEAN ALGEBRA (CONT D)
UNIT 3 BOOLEAN ALGEBRA (CONT D) Spring 2011 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions Exclusive-OR and Exclusive-NOR operations The consensus theorem Summary of algebraic
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 informationStatistics GIDP Ph.D. Qualifying Exam Methodology May 26 9:00am-1:00pm
Statistics GIDP Ph.D. Qualifying Exam Methodology May 26 9:00am-1:00pm Instructions: Put your ID (not name) on each sheet. Complete exactly 5 of 6 problems; turn in only those sheets you wish to have graded.
More informationConstruction of Mixed-Level Orthogonal Arrays for Testing in Digital Marketing
Construction of Mixed-Level Orthogonal Arrays for Testing in Digital Marketing Vladimir Brayman Webtrends October 19, 2012 Advantages of Conducting Designed Experiments in Digital Marketing Availability
More informationMultiple Predictor Variables: ANOVA
Multiple Predictor Variables: ANOVA 1/32 Linear Models with Many Predictors Multiple regression has many predictors BUT - so did 1-way ANOVA if treatments had 2 levels What if there are multiple treatment
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 informationChapter 2. Digital Logic Basics
Chapter 2 Digital Logic Basics 1 2 Chapter 2 2 1 Implementation using NND gates: We can write the XOR logical expression B + B using double negation as B+ B = B+B = B B From this logical expression, we
More informationStatistics GIDP Ph.D. Qualifying Exam Methodology May 26 9:00am-1:00pm
Statistics GIDP Ph.D. Qualifying Exam Methodology May 26 9:00am-1:00pm Instructions: Put your ID (not name) on each sheet. Complete exactly 5 of 6 problems; turn in only those sheets you wish to have graded.
More informationLecture 7: Karnaugh Map, Don t Cares
EE210: Switching Systems Lecture 7: Karnaugh Map, Don t Cares Prof. YingLi Tian Feb. 28, 2019 Department of Electrical Engineering The City College of New York The City University of New York (CUNY) 1
More informationStatistics for EES Factorial analysis of variance
Statistics for EES Factorial analysis of variance Dirk Metzler June 12, 2015 Contents 1 ANOVA and F -Test 1 2 Pairwise comparisons and multiple testing 6 3 Non-parametric: The Kruskal-Wallis Test 9 1 ANOVA
More informationComputer Organization I. Lecture 13: Design of Combinational Logic Circuits
Computer Organization I Lecture 13: Design of Combinational Logic Circuits Overview The optimization of multiple-level circuits Mapping Technology Verification Objectives To know how to optimize the multiple-level
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 informationA Study on Factorial Designs with Blocks Influence and Inspection Plan for Radiated Emission Testing of Information Technology Equipment
A Study on Factorial Designs with Blocks Influence and Inspection Plan for Radiated Emission Testing of Information Technology Equipment By Kam-Fai Wong Department of Applied Mathematics National Sun Yat-sen
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 informationLecture 12: Feb 16, 2017
CS 6170 Computational Topology: Topological Data Analysis Spring 2017 Lecture 12: Feb 16, 2017 Lecturer: Prof Bei Wang University of Utah School of Computing Scribe: Waiming Tai This
More informationMultiple Regression: Example
Multiple Regression: Example Cobb-Douglas Production Function The Cobb-Douglas production function for observed economic data i = 1,..., n may be expressed as where O i is output l i is labour input c
More informationR version ( ) Copyright (C) 2009 The R Foundation for Statistical Computing ISBN
Math 3080 1. Treibergs Spray Data: 2 3 Factorial with Multiple Replicates Per ell ANOVA Name: Example April 7, 2010 Data File Used in this Analysis: # Math 3081-1 Spray Data April 17,2010 # Treibergs #
More informationChapter 10 Exercise 10.1
Chapter 0 Exercise 0. Q.. A(, ), B(,), C(, ), D(, ), E(0,), F(,), G(,0), H(, ) Q.. (i) nd (vi) st (ii) th (iii) nd (iv) rd (v) st (vii) th (viii) rd (ix) st (viii) rd Q.. (i) Y (v) X (ii) Y (vi) X (iii)
More informationProbability Distribution
Probability Distribution 1. In scenario 2, the particle size distribution from the mill is: Counts 81
More informationAmount of Weight Gained. Regained 5 lb or Less Regained More Than 5 lb Total. In Person Online Newsletter
This is an open book test. You are allowed your textbook, notes and a calculator. Other books, laptops, or messaging devices are not permitted. Give complete solutions. Be clear about the order of logic
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 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 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 & 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 informationStatistical Design and Analysis of Experiments Part Two
0.1 Statistical Design and Analysis of Experiments Part Two Lecture notes Fall semester 2007 Henrik Spliid nformatics and Mathematical Modelling Technical University of Denmark List of contents, cont.
More informationStatistics 512: Solution to Homework#11. Problems 1-3 refer to the soybean sausage dataset of Problem 20.8 (ch21pr08.dat).
Statistics 512: Solution to Homework#11 Problems 1-3 refer to the soybean sausage dataset of Problem 20.8 (ch21pr08.dat). 1. Perform the two-way ANOVA without interaction for this model. Use the results
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 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 information