Chapter 4 Experiments with Blocking Factors
|
|
- Alannah Ross
- 6 years ago
- Views:
Transcription
1 Chapter 4 Experiments with Blocking Factors 許湘伶 Design and Analysis of Experiments (Douglas C. Montgomery) hsuhl (NUK) DAE Chap. 4 1 / 54
2 The Randomized Complete Block Design (RCBD; 隨機化完全集區設計 ) 1 Variability arising from a nuisance factor can affect the results. 2 A nuisance factor: has an effect on the response but we are not interested in that effect. unknown and uncontrolled: Randomization known and uncontrollable: compensate( 補償 ) in ANCOVA; (Chap. 14) known and controllable: design technique called blocking; 3 blocking: systematically eliminate its effect on the statistical comparisons among treatments hsuhl (NUK) DAE Chap. 4 2 / 54
3 The Randomized Complete Block Design the hardness testing experiment determine whether or not four different tips produce different readings on a hardness testing machine one factor: tip type completely randomized single-factor design the metal coupons differ slightly in their hardness to remove the variability between coupons from the experimental error hsuhl (NUK) DAE Chap. 4 3 / 54
4 The Randomized Complete Block Design (cont.) Complete: each block contains all the treatments improves the accuracy of the comparisons among tips by eliminating the variability among the coupons Block factor: 1 batches of raw material 2 people 3 time hsuhl (NUK) DAE Chap. 4 4 / 54
5 The Randomized Complete Block Design (cont.) a treatment; b blocks one observation per treatment in each block only randomization of treatments is within the block hsuhl (NUK) DAE Chap. 4 5 / 54
6 The Randomized Complete Block Design (cont.) Statitical model for RCBD: { i = 1, 2,..., a y ij = µ + τ i + β j + ɛ ij j = 1, 2,..., b a b ( τ i = 0 and β j = 0) i=1 j=1 = µ ij + ɛ ij (µ ij = µ + τ i + β j mean model) β j : the effect of the jth block ɛ ij NID(0, σ 2 ) hsuhl (NUK) DAE Chap. 4 6 / 54
7 Estimating parameters Two constraints: a ˆτ i = 0 i=1 b ˆβ j = 0 j=1 Estimates: ˆµ = ȳ ˆτ i = ȳ i ȳ ˆβ j = ȳ j ȳ i = 1,..., a j = 1,..., b ŷ ij = ˆµ + ˆτ i + ˆβ j = ȳ i + ȳ j ȳ hsuhl (NUK) DAE Chap. 4 7 / 54
8 Testing hypothesis for RCBD Testing the equality of the treatment means: H 0 : µ 1 = µ 2 = = µ a H 1 : at least one µ i µ j H 0 : τ 1 = τ 2 = = τ a = 0 H 1 : τ i 0at least one i the ith treatment mean: µ i = 1 b b (µ + τ i + β j ) = µ + τ i j=1 hsuhl (NUK) DAE Chap. 4 8 / 54
9 Testing hypothesis for RCBD (cont.) partitioning of total variability: d.f.: ab 1 = (a 1) + (b 1) + (a 1)(b 1) hsuhl (NUK) DAE Chap. 4 9 / 54
10 Testing hypothesis for RCBD (cont.) SS T = SS Treatments = 1 b SS Blocks = 1 a a i=1 a i=1 b j=1 b j=1 y ij y2 N y 2 i y2 N y 2 j y2 N SS E =SS T SS Treatments SS Blocks hsuhl (NUK) DAE Chap / 54
11 Testing hypothesis for RCBD (cont.) SS Treatments /σ 2, SS Blocks /σ 2, SS E /σ 2 : are independently χ 2 r.v. Expected of the mean squares: E(MS Treatments ) =σ 2 + b a i=1 τ i 2 a 1 E(MS Blocks ) =σ 2 + a a j=1 β2 j b 1 E(MS E ) =σ 2 Test statistic for testing the equality of treatment means: F 0 = MS Treatments MS E H 0 Fa 1,(a 1)(b 1) hsuhl (NUK) DAE Chap / 54
12 Testing hypothesis for RCBD (cont.) randomization has been applied only to treatments within blocks: restriction on randomization Comparing block means? F 0 = MS Blocks MS E? for H 0 : β j = 0 If ɛ ij NID(0, σ 2 ) F 0 = MS Blocks MS E can be used to compare block means. On the other hand, F ratio for comparing block means is a test for the equality of the block means + the randomization restriction. (Anderson and McLean (1974)) Because the normality assumption is often questionable, to exclude F 0 = MS Blocks MS E from ANOVA table hsuhl (NUK) DAE Chap / 54
13 Testing hypothesis for RCBD (cont.) hsuhl (NUK) DAE Chap / 54
14 Testing hypothesis for RCBD (cont.) RCBD for the Vascular Graft Experiment( 人工血管 ) Factor: extrusion pressure(psi; 擠壓力 ) Block: batch of resin( 樹脂 ) hsuhl (NUK) DAE Chap / 54
15 Testing hypothesis for RCBD (cont.) hsuhl (NUK) DAE Chap / 54
16 Testing hypothesis for RCBD (cont.) hsuhl (NUK) DAE Chap / 54
17 Testing hypothesis for RCBD (cont.) > graft<-scan("graft.txt") > PSI.labels <- c(8500,8700,8900,9100) > vasc.graft <- data.frame(psi=gl(4,6,24),block=gl(6,1,24),graft) > graft.aov <- aov(graft block+psi,vasc.graft) > summary(graft.aov) Df Sum Sq Mean Sq F value Pr(>F) block ** PSI ** Residuals Signif. codes: 0 *** ** 0.01 * hsuhl (NUK) DAE Chap / 54
18 Testing hypothesis for RCBD (cont.) Residuals: e ij = y ij ŷ ij = y ij (ȳ i + ȳ j ȳ ) hsuhl (NUK) DAE Chap / 54
19 Testing hypothesis for RCBD (cont.) hsuhl (NUK) DAE Chap / 54
20 Testing hypothesis for RCBD (cont.) Mean yields for the four extrusion pressures relative to a scaled t distribution with a scale factor MS E /b = 1.10 The highest pressure (9100 psi) results in a mean yield that is much lower than all other means. hsuhl (NUK) DAE Chap / 54
21 Testing hypothesis for RCBD (cont.) > library(agricolae) > graft.lsd<-lsd.test(graft.aov,"psi",group=f) # without grouping > graft.lsd $statistics Mean CV MSerror $parameters Df ntr t.value $means graft std r LCL UCL Min Max $comparison hsuhl (NUK) DAE Chap / 54
22 Testing hypothesis for RCBD (cont.) Difference pvalue sig. LCL UCL * *** ** $groups NULL hsuhl (NUK) DAE Chap / 54
23 Testing hypothesis for RCBD (cont.) Additivity of the Randomized Block Model y ij = µ + τ i + β j + ɛ ij If interaction is present, it can seriously affect and possibly invalidate the ANOVA. Interaction: inflates MS E ; affect the comparison of treatment means factorial designs hsuhl (NUK) DAE Chap / 54
24 Random Treatments and Blocks Assume β j NID(0, σβ 2 ), j = 1, 2,..., b, the treatments are fixed E(y ij ) = µ + τ i, V(y ij ) = σ 2 β + σ 2 i = 1, 2,..., a Cov(y ij, y i j ) = 0, j j Cov(y ij, y i j) = σ 2 β, i i Covariance between any two observations in different block is zero Covariance between any two observations from the same block is σ 2 β hsuhl (NUK) DAE Chap / 54
25 Random Treatments and Blocks (cont.) Expected mean squares: E(MS Treatments ) = σ 2 + b a i=1 τ i 2 a 1 E(MS Blocks ) = σ 2 + aσβ 2 E(MS E ) = σ 2 The null hypothesis of no treatment effects: τ i = 0 F 0 = MS Treatmens MS E hsuhl (NUK) DAE Chap / 54
26 Random Treatments and Blocks (cont.) Estimator: ˆσ 2 β = MS Blocks MS E a Interaction between treatments and blocks: y ij = µ + τ i + β j + (τβ) ij + ɛ ij { i = 1, 2,..., a j = 1, 2,..., b στβ 2 : variance component for the block treatment interaction E(MS Treatments ) = σ 2 + στβ 2 + b a i=1 τ i 2 a 1 E(MS Blocks ) = σ 2 + aσβ 2 E(MS E ) = σ 2 + σ 2 τβ hsuhl (NUK) DAE Chap / 54
27 Estimating missing values Sometimes, an observation in one of the blocks is missing Suppose y ij for treatment i in block j is missing y ij, y i, y j: represent the one missing observation in the grand total, the treatment and the block, respectively x: the missing observation to be estimated; Choosing x s.t. min SS E SSE = x 2 1 b (y i + x) 2 1 a (y j + x) ab (y + x) 2 + R min x SSE x = ay i + by j y (a 1)(b 1) (1) hsuhl (NUK) DAE Chap / 54
28 Estimating missing values (cont.) If several observations are missing, they may be estimated by writing SS E as a function of the missing values, differentiating with respect to each missing value, equating the results to zero, and solving the resulting equations. Using (1), iteratively to estimate the missing values; continued until convergence is obtained. hsuhl (NUK) DAE Chap / 54
29 Latin Square Design several other types of designs that utilize the blocking principle Example: five different formulations of a rocket propellant( 火箭推進燃料 ) uses in aircrew escape systems( 彈射救生系統 ) on the observed burning rate a batch of raw material: large enough for five formulations to be tested operators: differences in the skills and experience two nuisance factors to be average out testing each formulation exactly once in each batch of raw material and for each formulation to prepared exactly once by each of five operators hsuhl (NUK) DAE Chap / 54
30 Latin Square Design (cont.) Latin square design used to eliminate two nuisance sources of variability rows, columns: two restrictions on randomization hsuhl (NUK) DAE Chap / 54
31 Latin Square Design (cont.) Each of the p 2 cells contains one of the p letters that corresponds to the treatments, and each letter occurs once and only once in each row and column. 4 4 A B D C B C A D C D B A D A C B 5 5 A D B E C D A C B E C B E D A B E A C D E C D A B hsuhl (NUK) DAE Chap / 54
32 Latin Square Design (cont.) statistical model for a Latin square: y ijk = µ + α i + τ j + β k + ɛ ijk (row) (treatment) (column) partitioning SS T : i = 1, 2,..., p j = 1, 2,..., p k = 1, 2,..., p SS T = SS Rows + SS Columns + SS Treatments + SS E (p 2 1) (p 1) (p 1) (p 1) ((p 2)(p 1)) no interaction between factors The statistical analysis (ANOVA) is much like the analysis for the RCBD. hsuhl (NUK) DAE Chap / 54
33 Latin Square Design (cont.) Testing for no differences in treatment means: Figure : Example 4.3 hsuhl (NUK) DAE Chap / 54
34 Latin Square Design (cont.) Figure : Example 4.3 hsuhl (NUK) DAE Chap / 54
35 Latin Square Design (cont.) hsuhl (NUK) DAE Chap / 54
36 Latin Square Design (cont.) > rocket.lm <- lm(y factor(op)+factor(batch)+treat,rocket) > anova(rocket.lm) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) factor(op) * factor(batch) treat ** Residuals Signif. codes: 0 *** ** 0.01 * hsuhl (NUK) DAE Chap / 54
37 Latin Square Design (cont.) Residuals: e ijk = y ijk ŷ ijk = y ijk ȳ i ȳ j ȳ k + 2ȳ hsuhl (NUK) DAE Chap / 54
38 Latin Square Design (cont.) Standard Latin Square: obtained by writing the first row in alphabetical order and then writing each successive row as the row of letters just above shifted one place to the left. hsuhl (NUK) DAE Chap / 54
39 Latin Square Design (cont.) Replication-Case 1: the same batch and operators in each replicate hsuhl (NUK) DAE Chap / 54
40 Latin Square Design (cont.) Replication-Case 2: the same batch but different operators in each replicate hsuhl (NUK) DAE Chap / 54
41 Latin Square Design (cont.) Replication-Case 3: different batched and different operators in each replicate hsuhl (NUK) DAE Chap / 54
42 Crossover Designs the time period is a factor in the experiment two replacement fluids (A,B) on dehyfration( 脫水 ) on 20 subjects 1 First period: half of the subjects-a; the other half of the subjects-b 2 measure response at the end of the period 3 after the effect of the fluids is eliminated: change the assignment half of the subjects-a B; the other half of the subjects-b A hsuhl (NUK) DAE Chap / 54
43 Graeco-Latin Square Design hsuhl (NUK) DAE Chap / 54
44 Graeco-Latin Square Design (cont.) block in three direction each at p levels in only p 2 runs statistical model: y ijkl = µ + θ i + τ j + w k + Φ l + ɛ ijkl, i, j, k, l = 1,..., p hsuhl (NUK) DAE Chap / 54
45 Balanced Incomplete Block Designs (BIBD) BIBD: 平衡不完全集區設計 any two treatments appear together an equal number of times all treatment comparisons are equally important the treatment combinations used in each block should be selected in a balanced manner hsuhl (NUK) DAE Chap / 54
46 Balanced Incomplete Block Designs (BIBD) (cont.) a treatment; each block can hold exactly k (k < a) treatments ( ) a A BIBD: constructed by blocks; assigning a different k combination of treatmens to each block hsuhl (NUK) DAE Chap / 54
47 Balanced Incomplete Block Designs (BIBD) (cont.) Assume a treatment; b blocks each block contains k treatments; each treatment occurs r times in the design N = ar = bk each pair of treaments appears in the same block: λ = r(k 1) a 1 hsuhl (NUK) DAE Chap / 54
48 Balanced Incomplete Block Designs (BIBD) (cont.) statistical model for BIBD: Patitioned total variability: y ij = µ + τ i + β j + ɛ ij SS T = SS Treatments(adjusted) + SS Blocks + SS E hsuhl (NUK) DAE Chap / 54
49 Balanced Incomplete Block Designs (BIBD) (cont.) The adjusted treatment totals will always sum to zero. SS Treatments(adjusted) = k a i=1 Q2 i λa Q = y i 1 b n ij y j, i = 1,..., a k j=1 { 1, if treatment i appears in block j n ij = 0, otherwise SS Blocks = 1 b y 2 j y2 k N j=1 hsuhl (NUK) DAE Chap / 54
50 Balanced Incomplete Block Designs (BIBD) (cont.) hsuhl (NUK) DAE Chap / 54
51 Balanced Incomplete Block Designs (BIBD) (cont.) ### Ex 4.5 > catalyst.y <- matrix(c(73,na,73,75,74,75,75,na,na,67,68,72,71,72,na,75) > catalyst <- data.frame(rep=as.vector(catalyst.y), + treat=factor(rep(1:4,4)), + block=factor(rep(1:4,each=4))) > summary(aov(rep treat+block+error(block),catalyst)) Error: block Df Sum Sq Mean Sq treat Error: Within Df Sum Sq Mean Sq F value Pr(>F) treat * Residuals Signif. codes: 0 *** ** 0.01 * > hsuhl (NUK) DAE Chap / 54
52 Balanced Incomplete Block Designs (BIBD) (cont.) > summary(aov(rep treat+block,catalyst)) Df Sum Sq Mean Sq F value Pr(>F) treat * block *** Residuals Signif. codes: 0 *** ** 0.01 * observations deleted due to missingness > summary(aov(rep block+treat,catalyst)) Df Sum Sq Mean Sq F value Pr(>F) block ** treat * Residuals Signif. codes: 0 *** ** 0.01 * observations deleted due to missingness hsuhl (NUK) DAE Chap / 54
53 Balanced Incomplete Block Designs (BIBD) (cont.) like to assess the block effects alternate partitioning of SS T SS Treatments : unadjusted SS T = SS Treatments + SS Blocks(adjusted) + SS E SS Blocks(adjusted) = r b j=1 (Q j) 2 λb Q j = y j 1 4 n n ij y i, i=1 j = 1, 2,..., b SS T SS Treatments(adjusted) + SS Blocks(adjusted) + SS E hsuhl (NUK) DAE Chap / 54
54 Balanced Incomplete Block Designs (BIBD) (cont.) hsuhl (NUK) DAE Chap / 54
Chapter 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 informationDESAIN EKSPERIMEN BLOCKING FACTORS. Semester Genap 2017/2018 Jurusan Teknik Industri Universitas Brawijaya
DESAIN EKSPERIMEN BLOCKING FACTORS Semester Genap Jurusan Teknik Industri Universitas Brawijaya Outline The Randomized Complete Block Design The Latin Square Design The Graeco-Latin Square Design Balanced
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 informationSTAT 430 (Fall 2017): Tutorial 8
STAT 430 (Fall 2017): Tutorial 8 Balanced Incomplete Block Design Luyao Lin November 7th/9th, 2017 Department Statistics and Actuarial Science, Simon Fraser University Block Design Complete Random Complete
More informationChapter 9 Other Topics on Factorial and Fractional Factorial Designs
Chapter 9 Other Topics on Factorial and Fractional Factorial Designs 許湘伶 Design and Analysis of Experiments (Douglas C. Montgomery) hsuhl (NUK) DAE Chap. 9 1 / 26 The 3 k Factorial Design 3 k factorial
More informationWeek 3 & 4 Randomized Blocks, Latin Squares, and Related Designs
Week 3 & 4Randomized Blocks, Latin Squares, and Related Designs p 1/81 Week 3 & 4 Randomized Blocks, Latin Squares, and Related Designs Joslin Goh Simon Fraser University Week 3 & 4Randomized Blocks, Latin
More informationLecture 10. Factorial experiments (2-way ANOVA etc)
Lecture 10. Factorial experiments (2-way ANOVA etc) Jesper Rydén Matematiska institutionen, Uppsala universitet jesper@math.uu.se Regression and Analysis of Variance autumn 2014 A factorial experiment
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
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 informationUnit 8: 2 k Factorial Designs, Single or Unequal Replications in Factorial Designs, and Incomplete Block Designs
Unit 8: 2 k Factorial Designs, Single or Unequal Replications in Factorial Designs, and Incomplete Block Designs STA 643: Advanced Experimental Design Derek S. Young 1 Learning Objectives Revisit your
More informationUnit 6: Orthogonal Designs Theory, Randomized Complete Block Designs, and Latin Squares
Unit 6: Orthogonal Designs Theory, Randomized Complete Block Designs, and Latin Squares STA 643: Advanced Experimental Design Derek S. Young 1 Learning Objectives Understand the basics of orthogonal designs
More informationContents. TAMS38 - Lecture 6 Factorial design, Latin Square Design. Lecturer: Zhenxia Liu. Factorial design 3. Complete three factor design 4
Contents Factorial design TAMS38 - Lecture 6 Factorial design, Latin Square Design Lecturer: Zhenxia Liu Department of Mathematics - Mathematical Statistics 28 November, 2017 Complete three factor design
More informationLecture 7: Latin Square and Related Design
Lecture 7: Latin Square and Related Design Montgomery: Section 4.2-4.3 Page 1 Automobile Emission Experiment Four cars and four drivers are employed in a study for possible differences between four gasoline
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 information22s:152 Applied Linear Regression. Take random samples from each of m populations.
22s:152 Applied Linear Regression Chapter 8: ANOVA NOTE: We will meet in the lab on Monday October 10. One-way ANOVA Focuses on testing for differences among group means. Take random samples from each
More informationAllow the investigation of the effects of a number of variables on some response
Lecture 12 Topic 9: Factorial treatment structures (Part I) Factorial experiments Allow the investigation of the effects of a number of variables on some response in a highly efficient manner, and in a
More informationSTAT22200 Spring 2014 Chapter 13B
STAT22200 Spring 2014 Chapter 13B Yibi Huang May 27, 2014 13.3.1 Crossover Designs 13.3.4 Replicated Latin Square Designs 13.4 Graeco-Latin Squares Chapter 13B - 1 13.3.1 Crossover Design (A Special Latin-Square
More informationStat 502 Design and Analysis of Experiments General Linear Model
1 Stat 502 Design and Analysis of Experiments General Linear Model Fritz Scholz Department of Statistics, University of Washington December 6, 2013 2 General Linear Hypothesis We assume the data vector
More informationANOVA (Analysis of Variance) output RLS 11/20/2016
ANOVA (Analysis of Variance) output RLS 11/20/2016 1. Analysis of Variance (ANOVA) The goal of ANOVA is to see if the variation in the data can explain enough to see if there are differences in the means.
More informationThe Pennsylvania State University The Graduate School Eberly College of Science INTRABLOCK, INTERBLOCK AND COMBINED ESTIMATES
The Pennsylvania State University The Graduate School Eberly College of Science INTRABLOCK, INTERBLOCK AND COMBINED ESTIMATES IN INCOMPLETE BLOCK DESIGNS: A NUMERICAL STUDY A Thesis in Statistics by Yasin
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 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 informationIncomplete Block Designs
Incomplete Block Designs Recall: in randomized complete block design, each of a treatments was used once within each of b blocks. In some situations, it will not be possible to use each of a treatments
More informationCuckoo Birds. Analysis of Variance. Display of Cuckoo Bird Egg Lengths
Cuckoo Birds Analysis of Variance Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison Statistics 371 29th November 2005 Cuckoo birds have a behavior in which they lay their
More information22s:152 Applied Linear Regression. There are a couple commonly used models for a one-way ANOVA with m groups. Chapter 8: ANOVA
22s:152 Applied Linear Regression Chapter 8: ANOVA NOTE: We will meet in the lab on Monday October 10. One-way ANOVA Focuses on testing for differences among group means. Take random samples from each
More informationDesign and Analysis of Experiments Prof. Jhareswar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur
Design and Analysis of Experiments Prof. Jhareswar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur Lecture - 27 Randomized Complete Block Design (RCBD):
More informationRandomized Complete Block Designs Incomplete Block Designs. Block Designs. 1 Randomized Complete Block Designs. 2 Incomplete Block Designs
Block Designs Randomized Complete Block Designs 1 Randomized Complete Block Designs 2 0 / 18 1 Randomized Complete Block Designs 2 1 / 18 Randomized Complete Block Design RCBD is the most widely used experimental
More informationTwo-Way Factorial Designs
81-86 Two-Way Factorial Designs Yibi Huang 81-86 Two-Way Factorial Designs Chapter 8A - 1 Problem 81 Sprouting Barley (p166 in Oehlert) Brewer s malt is produced from germinating barley, so brewers like
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 informationMultiple comparisons - subsequent inferences for two-way ANOVA
1 Multiple comparisons - subsequent inferences for two-way ANOVA the kinds of inferences to be made after the F tests of a two-way ANOVA depend on the results if none of the F tests lead to rejection of
More informationIntroduction to Design and Analysis of Experiments with the SAS System (Stat 7010 Lecture Notes)
Introduction to Design and Analysis of Experiments with the SAS System (Stat 7010 Lecture Notes) Asheber Abebe Discrete and Statistical Sciences Auburn University Contents 1 Completely Randomized Design
More informationTwo-Way Analysis of Variance - no interaction
1 Two-Way Analysis of Variance - no interaction Example: Tests were conducted to assess the effects of two factors, engine type, and propellant type, on propellant burn rate in fired missiles. Three engine
More informationDesign and Analysis of Experiments Prof. Jhareswar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur
Design and Analysis of Experiments Prof. Jhareswar Maiti Department of Industrial and Systems Engineering Indian Institute of Technology, Kharagpur Lecture 26 Randomized Complete Block Design (RCBD): Estimation
More informationAnalysis of Covariance
Analysis of Covariance (ANCOVA) Bruce A Craig Department of Statistics Purdue University STAT 514 Topic 10 1 When to Use ANCOVA In experiment, there is a nuisance factor x that is 1 Correlated with y 2
More information3. Design Experiments and Variance Analysis
3. Design Experiments and Variance Analysis Isabel M. Rodrigues 1 / 46 3.1. Completely randomized experiment. Experimentation allows an investigator to find out what happens to the output variables when
More informationSTAT22200 Spring 2014 Chapter 8A
STAT22200 Spring 2014 Chapter 8A Yibi Huang May 13, 2014 81-86 Two-Way Factorial Designs Chapter 8A - 1 Problem 81 Sprouting Barley (p166 in Oehlert) Brewer s malt is produced from germinating barley,
More information1 Use of indicator random variables. (Chapter 8)
1 Use of indicator random variables. (Chapter 8) let I(A) = 1 if the event A occurs, and I(A) = 0 otherwise. I(A) is referred to as the indicator of the event A. The notation I A is often used. 1 2 Fitting
More informationBALANCED INCOMPLETE BLOCK DESIGNS
BALANCED INCOMPLETE BLOCK DESIGNS V.K. Sharma I.A.S.R.I., Library Avenue, New Delhi -110012. 1. Introduction In Incomplete block designs, as their name implies, the block size is less than the number of
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 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 informationLecture 6 Multiple Linear Regression, cont.
Lecture 6 Multiple Linear Regression, cont. BIOST 515 January 22, 2004 BIOST 515, Lecture 6 Testing general linear hypotheses Suppose we are interested in testing linear combinations of the regression
More informationLecture 7: Latin Squares and Related Designs
Lecture 7: Latin Squares and Related Designs Montgomery: Section 4.2 and 4.3 1 Lecture 7 Page 1 Automobile Emission Experiment Four cars and four drivers are employed in a study of four gasoline additives(a,b,c,
More informationSTATISTICS 368 AN EXPERIMENT IN AIRCRAFT PRODUCTION Christopher Wiens & Douglas Wiens
STATISTICS 368 AN EXPERIMENT IN AIRCRAFT PRODUCTION Christopher Wiens & Douglas Wiens April 21, 2005 The progress of science. 1. Preliminary description of experiment We set out to determine the factors
More informationSTAT 135 Lab 10 Two-Way ANOVA, Randomized Block Design and Friedman s Test
STAT 135 Lab 10 Two-Way ANOVA, Randomized Block Design and Friedman s Test Rebecca Barter April 13, 2015 Let s now imagine a dataset for which our response variable, Y, may be influenced by two factors,
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 informationLecture 11: Nested and Split-Plot Designs
Lecture 11: Nested and Split-Plot Designs Montgomery, Chapter 14 1 Lecture 11 Page 1 Crossed vs Nested Factors Factors A (a levels)and B (b levels) are considered crossed if Every combinations of A and
More informationTopic 9: Factorial treatment structures. Introduction. Terminology. Example of a 2x2 factorial
Topic 9: Factorial treatment structures Introduction A common objective in research is to investigate the effect of each of a number of variables, or factors, on some response variable. In earlier times,
More informationLecture 1: Linear Models and Applications
Lecture 1: Linear Models and Applications Claudia Czado TU München c (Claudia Czado, TU Munich) ZFS/IMS Göttingen 2004 0 Overview Introduction to linear models Exploratory data analysis (EDA) Estimation
More informationResidual Analysis for two-way ANOVA The twoway model with K replicates, including interaction,
Residual Analysis for two-way ANOVA The twoway model with K replicates, including interaction, is Y ijk = µ ij + ɛ ijk = µ + α i + β j + γ ij + ɛ ijk with i = 1,..., I, j = 1,..., J, k = 1,..., K. In carrying
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 informationAnalysis of Variance and Design of Experiments-I
Analysis of Variance and Design of Experiments-I MODULE VIII LECTURE - 35 ANALYSIS OF VARIANCE IN RANDOM-EFFECTS MODEL AND MIXED-EFFECTS MODEL Dr. Shalabh Department of Mathematics and Statistics Indian
More informationUnit 7: Random Effects, Subsampling, Nested and Crossed Factor Designs
Unit 7: Random Effects, Subsampling, Nested and Crossed Factor Designs STA 643: Advanced Experimental Design Derek S. Young 1 Learning Objectives Understand how to interpret a random effect Know the different
More informationChapter 2 Inferences in Regression and Correlation Analysis
Chapter 2 Inferences in Regression and Correlation Analysis 許湘伶 Applied Linear Regression Models (Kutner, Nachtsheim, Neter, Li) hsuhl (NUK) LR Chap 2 1 / 102 Inferences concerning the regression parameters
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 informationTopic 13. Analysis of Covariance (ANCOVA) - Part II [ST&D Ch. 17]
Topic 13. Analysis of Covariance (ANCOVA) - Part II [ST&D Ch. 17] 13.5 Assumptions of ANCOVA The assumptions of analysis of covariance are: 1. The X s are fixed, measured without error, and independent
More informationCh 3: Multiple Linear Regression
Ch 3: Multiple Linear Regression 1. Multiple Linear Regression Model Multiple regression model has more than one regressor. For example, we have one response variable and two regressor variables: 1. delivery
More informationLecture 15. Hypothesis testing in the linear model
14. Lecture 15. Hypothesis testing in the linear model Lecture 15. Hypothesis testing in the linear model 1 (1 1) Preliminary lemma 15. Hypothesis testing in the linear model 15.1. Preliminary lemma Lemma
More informationHomework 3 - Solution
STAT 526 - Spring 2011 Homework 3 - Solution Olga Vitek Each part of the problems 5 points 1. KNNL 25.17 (Note: you can choose either the restricted or the unrestricted version of the model. Please state
More informationSummary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)
Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ
More informationChapter 4 Randomized Blocks, Latin Squares, and Related Designs Solutions
Solutions from Montgomery, D. C. (008) Design and Analysis of Experiments, Wiley, NY Chapter 4 Randomized Blocks, Latin Squares, and Related Designs Solutions 4.. The ANOVA from a randomized complete block
More informationFigure 1: The fitted line using the shipment route-number of ampules data. STAT5044: Regression and ANOVA The Solution of Homework #2 Inyoung Kim
0.0 1.0 1.5 2.0 2.5 3.0 8 10 12 14 16 18 20 22 y x Figure 1: The fitted line using the shipment route-number of ampules data STAT5044: Regression and ANOVA The Solution of Homework #2 Inyoung Kim Problem#
More informationPower & Sample Size Calculation
Chapter 7 Power & Sample Size Calculation Yibi Huang Chapter 7 Section 10.3 Power & Sample Size Calculation for CRDs Power & Sample Size for Factorial Designs Chapter 7-1 Power & Sample Size Calculation
More informationInference for Regression
Inference for Regression Section 9.4 Cathy Poliak, Ph.D. cathy@math.uh.edu Office in Fleming 11c Department of Mathematics University of Houston Lecture 13b - 3339 Cathy Poliak, Ph.D. cathy@math.uh.edu
More informationChapter 14 Logistic Regression, Poisson Regression, and Generalized Linear Models
Chapter 14 Logistic Regression, Poisson Regression, and Generalized Linear Models 許湘伶 Applied Linear Regression Models (Kutner, Nachtsheim, Neter, Li) hsuhl (NUK) LR Chap 10 1 / 29 14.1 Regression Models
More informationAnalysis of Variance
Analysis of Variance Blood coagulation time T avg A 62 60 63 59 61 B 63 67 71 64 65 66 66 C 68 66 71 67 68 68 68 D 56 62 60 61 63 64 63 59 61 64 Blood coagulation time A B C D Combined 56 57 58 59 60 61
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 informationProcess Capability Analysis Using Experiments
Process Capability Analysis Using Experiments A designed experiment can aid in separating sources of variability in a quality characteristic. Example: bottling soft drinks Suppose the measured syrup content
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 informationChapter 9 Time-Weighted Control Charts. Statistical Quality Control (D. C. Montgomery)
Chapter 9 Time-Weighted Control Charts 許湘伶 Statistical Quality Control (D. C. Montgomery) Introduction I Shewhart control chart: Chap. 5 7: basic SPC methods Useful in phase I implementation( 完成 ) of SPC
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 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 informationFactorial Treatment Structure: Part I. Lukas Meier, Seminar für Statistik
Factorial Treatment Structure: Part I Lukas Meier, Seminar für Statistik Factorial Treatment Structure So far (in CRD), the treatments had no structure. So called factorial treatment structure exists if
More information3. Factorial Experiments (Ch.5. Factorial Experiments)
3. Factorial Experiments (Ch.5. Factorial Experiments) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University DOE and Optimization 1 Introduction to Factorials Most experiments for process
More informationReference: Chapter 13 of Montgomery (8e)
Reference: Chapter 1 of Montgomery (8e) Maghsoodloo 89 Factorial Experiments with Random Factors So far emphasis has been placed on factorial experiments where all factors are at a, b, c,... fixed levels
More informationSTAT22200 Chapter 14
STAT00 Chapter 4 Yibi Huang Chapter 4 Incomplete Block Designs 4. Balanced Incomplete Block Designs (BIBD) Chapter 4 - Incomplete Block Designs A Brief Introduction to a Class of Most Useful Designs in
More informationSTAT 8200 Design and Analysis of Experiments for Research Workers Lecture Notes
STAT 8200 Design and Analysis of Experiments for Research Workers Lecture Notes Basics of Experimental Design Terminology Response (Outcome, Dependent) Variable: (y) The variable who s distribution is
More informationRegression Analysis. Regression: Methodology for studying the relationship among two or more variables
Regression Analysis Regression: Methodology for studying the relationship among two or more variables Two major aims: Determine an appropriate model for the relationship between the variables Predict the
More informationChapter 12. Analysis of variance
Serik Sagitov, Chalmers and GU, January 9, 016 Chapter 1. Analysis of variance Chapter 11: I = samples independent samples paired samples Chapter 1: I 3 samples of equal size J one-way layout two-way layout
More informationOutline. Statistical inference for linear mixed models. One-way ANOVA in matrix-vector form
Outline Statistical inference for linear mixed models Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark general form of linear mixed models examples of analyses using linear mixed
More informationStat 6640 Solution to Midterm #2
Stat 6640 Solution to Midterm #2 1. A study was conducted to examine how three statistical software packages used in a statistical course affect the statistical competence a student achieves. At the end
More informationChapter 11 Building the Regression Model II:
Chapter 11 Building the Regression Model II: Remedial Measures( 補救措施 ) 許湘伶 Applied Linear Regression Models (Kutner, Nachtsheim, Neter, Li) hsuhl (NUK) LR Chap 11 1 / 48 11.1 WLS remedial measures may
More informationOne-way ANOVA (Single-Factor CRD)
One-way ANOVA (Single-Factor CRD) STAT:5201 Week 3: Lecture 3 1 / 23 One-way ANOVA We have already described a completed randomized design (CRD) where treatments are randomly assigned to EUs. There is
More informationStat 705: Completely randomized and complete block designs
Stat 705: Completely randomized and complete block designs Timothy Hanson Department of Statistics, University of South Carolina Stat 705: Data Analysis II 1 / 16 Experimental design Our department offers
More informationLecture 10: Experiments with Random Effects
Lecture 10: Experiments with Random Effects Montgomery, Chapter 13 1 Lecture 10 Page 1 Example 1 A textile company weaves a fabric on a large number of looms. It would like the looms to be homogeneous
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 informationSection 4.6 Simple Linear Regression
Section 4.6 Simple Linear Regression Objectives ˆ Basic philosophy of SLR and the regression assumptions ˆ Point & interval estimation of the model parameters, and how to make predictions ˆ Point and interval
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 informationSTAT763: Applied Regression Analysis. Multiple linear regression. 4.4 Hypothesis testing
STAT763: Applied Regression Analysis Multiple linear regression 4.4 Hypothesis testing Chunsheng Ma E-mail: cma@math.wichita.edu 4.4.1 Significance of regression Null hypothesis (Test whether all β j =
More informationR 2 and F -Tests and ANOVA
R 2 and F -Tests and ANOVA December 6, 2018 1 Partition of Sums of Squares The distance from any point y i in a collection of data, to the mean of the data ȳ, is the deviation, written as y i ȳ. Definition.
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 informationPh.D. Qualifying Exam Friday Saturday, January 3 4, 2014
Ph.D. Qualifying Exam Friday Saturday, January 3 4, 2014 Put your solution to each problem on a separate sheet of paper. Problem 1. (5166) Assume that two random samples {x i } and {y i } are independently
More informationIX. Complete Block Designs (CBD s)
IX. Complete Block Designs (CBD s) A.Background Noise Factors nuisance factors whose values can be controlled within the context of the experiment but not outside the context of the experiment Covariates
More informationI i=1 1 I(J 1) j=1 (Y ij Ȳi ) 2. j=1 (Y j Ȳ )2 ] = 2n( is the two-sample t-test statistic.
Serik Sagitov, Chalmers and GU, February, 08 Solutions chapter Matlab commands: x = data matrix boxplot(x) anova(x) anova(x) Problem.3 Consider one-way ANOVA test statistic For I = and = n, put F = MS
More informationTWO OR MORE RANDOM EFFECTS. The two-way complete model for two random effects:
TWO OR MORE RANDOM EFFECTS Example: The factors that influence the breaking strength of a synthetic fiber are being studied. Four production machines and three operators are randomly selected. A two-way
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 informationANOVA Randomized Block Design
Biostatistics 301 ANOVA Randomized Block Design 1 ORIGIN 1 Data Structure: Let index i,j indicate the ith column (treatment class) and jth row (block). For each i,j combination, there are n replicates.
More information36-707: Regression Analysis Homework Solutions. Homework 3
36-707: Regression Analysis Homework Solutions Homework 3 Fall 2012 Problem 1 Y i = βx i + ɛ i, i {1, 2,..., n}. (a) Find the LS estimator of β: RSS = Σ n i=1(y i βx i ) 2 RSS β = Σ n i=1( 2X i )(Y i βx
More informationLecture 3. Experiments with a Single Factor: ANOVA Montgomery 3-1 through 3-3
Lecture 3. Experiments with a Single Factor: ANOVA Montgomery 3-1 through 3-3 Page 1 Tensile Strength Experiment Investigate the tensile strength of a new synthetic fiber. The factor is the weight percent
More informationStat/F&W Ecol/Hort 572 Review Points Ané, Spring 2010
1 Linear models Y = Xβ + ɛ with ɛ N (0, σ 2 e) or Y N (Xβ, σ 2 e) where the model matrix X contains the information on predictors and β includes all coefficients (intercept, slope(s) etc.). 1. Number of
More informationST430 Exam 2 Solutions
ST430 Exam 2 Solutions Date: November 9, 2015 Name: Guideline: You may use one-page (front and back of a standard A4 paper) of notes. No laptop or textbook are permitted but you may use a calculator. Giving
More information