Orthogonal contrasts for a 2x2 factorial design Example p130
|
|
- Bryan Thompson
- 5 years ago
- Views:
Transcription
1 Week 9: Orthogonal comparisons for a 2x2 factorial design. The general two-factor factorial arrangement. Interaction and additivity. ANOVA summary table, tests, CIs. Planned/post-hoc comparisons for the factors or treatments. (ch13) [pp13-153] Orthogonal contrasts for a 2x2 factorial design Example p13 Tabulated statistics: Stress, Diet Rows: Stress D1 Columns: Diet D2 High Low Cell Contents: Cholesterol : Mean Cholesterol : DATA 1
2 Data Display D1_HighStress D2_HighStress D1_LowStress D2_LowStress One-way ANOVA: D1_HighStress, D2_HighStress, D1_LowStress, D2_LowStress Factor Error.8.2 Total S =.11 R-Sq = 98.55% R-Sq(adj) = 97.7% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev D1_HighStress (---*---) D2_HighStress (---*---) D1_LowStress (---*---) D2_LowStress (---*---) Interaction Plot (data means) for Cholesterol Stress High Low Mean D1 Diet D2 2
3 H : No interaction in the effects of the two factors H : µ µ = µ µ S1D1 S1D2 S2D1 S2D2 H : µ µ µ µ S1D1 S1D2 S2D1 S2D2 H : µ + µ µ + µ SD 1 1 S2D2 SD 1 2 S2D2 ψ = µ µ µ µ int S1D1 S1D2 S2D1 S2D2 Check that the three contrasts are orthogonal. SS (int contrast) =.25 SS(Stress) =.85 SS(Diet) =.5 = = µ SD 1 1 µ = simple effect of diet at low SD 1 2 stress level µ SD 1 1 µ = simple effect of diet at high SD 1 2 stress level 3
4 Note: When the interaction is significant, the tests for main effect hypotheses are of dubious value. P132 CN - illustrative plots are important - interaction between two factors A and B implies that both factors have effects, but the effect of factor A depend on the level of factor B present and vice versa. P133 CN - Three way interaction
5 The model: The General Two Factor Model Observation = fit + error y ijk = µ ij + ε ijk y ijk = µ + µ µ µ µ µ + ε ijk effect of level i of factor A = µ ( µ ) + ( µ ) + ( ij i + µ ) µ i effect of level j of factor B = µ µ j interaction effect at treatment (i,j) = µ ij µ µ + µ i j ε ijk ~ iid N (, σ ) Estimates y ijk = y + ( y i y) + ( y j y) + ( y ij y i y j + y) + ( y ijk y ij ) 5
6 SS decomposition SSTot = SSA + SSB +SS(AB)+SSE It can be shown that SS(AB) = SSTrt SS(A)-SS(B) 6
7 Standard hypotheses: H (1) : no interaction H (2) : there is no main effect of factor A i.e µ µ = for all i i H (3) : there is no main effect of factor B i.e µ µ = for all j j 7
8 Example (Example 3 CN p11) Data Display Row Yield Variety Nitrogen
9 Or I tabulated form: Tabulated statistics: Nitrogen, Variety Rows: Nitrogen Columns: Variety Cell Contents: Yield : DATA 9
10 One-way ANOVA: Yield versus Trt Trt Error Total Residual Plots for Yield 99 Normal Probability Plot of the Residuals 8 Residuals Versus the Fitted Values 9 Percent 5 1 Residual Residual Fitted Value 8 Histogram of the Residuals 8 Residuals Versus the Order of the Data Frequency Residual Residual Observation Order
11 Two-way ANOVA: Yield versus Nitrogen, Variety Nitrogen Variety Interaction Error Total S = 6.53 R-Sq = 92.9% R-Sq(adj) = 86.7% Residual Plots for Yield Normal Probability Plot of the Residuals Residuals Versus the Fitted Values Percent 5 1 Residual Residual Fitted Value 8 Histogram of the Residuals 8 Residuals Versus the Order of the Data Frequency Residual Residual Observation Order Anderson Darling test for normality Probability Plot of RESI1 Normal Percent Mean StDev.697 N 2 AD.63 P-Value RESI
12 Interaction Plot (data means) for Yield 9 8 Variety Mean Nitrogen 25 12
13 Compare the two ANOVA tables above One-way ANOVA: Yield versus Trt Trt Error Total S = 6.53 R-Sq = 92.9% R-Sq(adj) = 86.7% Two-way ANOVA: Yield versus Nitrogen, Variety Nitrogen Variety Interaction Error Total S = 6.53 R-Sq = 92.9% R-Sq(adj) = 86.7% 13
14 Elements of the ANOVA table Tabulated statistics: Nitrogen, Variety Rows: Nitrogen Columns: Variety All All Cell Contents: Yield : Mean Yield : Sum Yield : Standard deviation Count Yield : DATA 1
15 Further analysis Two-way ANOVA: Yield versus Nitrogen, Variety Nitrogen Linear sig Quadratic n.s. Variety vs n.s 2 vs 1 n.s (1,3) vs (2,) sig Interaction Error Total Main Effects Plot (data means) for Yield 75 7 Mean of Yield Nitrogen 25 15
16 CI for the difference between means at two levels of variety (or nitrogen) y y ± t 1 1 i j /2 s n + α n i j Ex Find a 95% CI for the difference between the means of variety 1 and variety 2 Ex Find a 95% CI for the difference between the means of the treatments (V1, 15) and (V2 and 15). 16
17 Fits and residuals from the interaction model ANOVA: Yield versus Nitrogen, Variety Factor Type Levels Values Nitrogen fixed 3 15, 2, 25 Variety fixed 1, 2, 3, Analysis of Variance for Yield Nitrogen Variety Nitrogen*Variety Error Total S = R-Sq = 92.9% R-Sq(adj) = 86.7% Data Display Row Yield Variety Nitrogen RESI1 FITS
18 Additive model ANOVA: Yield versus Nitrogen, Variety Factor Type Levels Values Nitrogen fixed 3 15, 2, 25 Variety fixed 1, 2, 3, Analysis of Variance for Yield Nitrogen Variety Error Total S = R-Sq = 86.58% R-Sq(adj) = 82.86% - Compare with the interaction model ANOVA: Yield versus Nitrogen, Variety Factor Type Levels Values Nitrogen fixed 3 15, 2, 25 Variety fixed 1, 2, 3, Analysis of Variance for Yield Nitrogen Variety Nitrogen*Variety Error Total S = R-Sq = 92.9% R-Sq(adj) = 86.7% 18
19 Residuals from the additive model Data Display Row Yield Variety Nitrogen RESI1 FITS
20 GLM approach for unbalanced designs (eg if some observations are missing) -Use indicator variables for qualitative factors and use GLM approach using regression procedure Regression Analysis: Yield versus Nitrogen, nsq,... The regression equation is Yield = Nitrogen -.31 nsq + 25 v v v v1n v2n v3n v1nsq +.63 v2nsq +.53 v3nsq S = R-Sq = 92.9% R-Sq(adj) = 86.5% Analysis of Variance Regression Residual Error Total Source DF Seq SS Nitrogen nsq v v v v1n v2n 1 5. v3n v1nsq v2nsq v3nsq Note: we should include nitrogen_sq (denoted n2 above ) when there are three levels for that factor 2
21 - Compare with ANOVA approach ANOVA: Yield versus Nitrogen, Variety Factor Type Levels Values Nitrogen fixed 3 15, 2, 25 Variety fixed 1, 2, 3, Analysis of Variance for Yield Nitrogen Variety Nitrogen*Variety Error Total
Confidence Interval for the mean response
Week 3: Prediction and Confidence Intervals at specified x. Testing lack of fit with replicates at some x's. Inference for the correlation. Introduction to regression with several explanatory variables.
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 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 informationModel Building Chap 5 p251
Model Building Chap 5 p251 Models with one qualitative variable, 5.7 p277 Example 4 Colours : Blue, Green, Lemon Yellow and white Row Blue Green Lemon Insects trapped 1 0 0 1 45 2 0 0 1 59 3 0 0 1 48 4
More informationModels with qualitative explanatory variables p216
Models with qualitative explanatory variables p216 Example gen = 1 for female Row gpa hsm gen 1 3.32 10 0 2 2.26 6 0 3 2.35 8 0 4 2.08 9 0 5 3.38 8 0 6 3.29 10 0 7 3.21 8 0 8 2.00 3 0 9 3.18 9 0 10 2.34
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 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 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 informationUnbalanced Data in Factorials Types I, II, III SS Part 1
Unbalanced Data in Factorials Types I, II, III SS Part 1 Chapter 10 in Oehlert STAT:5201 Week 9 - Lecture 2 1 / 14 When we perform an ANOVA, we try to quantify the amount of variability in the data accounted
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 informationANOVA Situation The F Statistic Multiple Comparisons. 1-Way ANOVA MATH 143. Department of Mathematics and Statistics Calvin College
1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College An example ANOVA situation Example (Treating Blisters) Subjects: 25 patients with blisters Treatments: Treatment A, Treatment
More information1-Way ANOVA MATH 143. Spring Department of Mathematics and Statistics Calvin College
1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Spring 2010 The basic ANOVA situation Two variables: 1 Categorical, 1 Quantitative Main Question: Do the (means of) the quantitative
More informationIncreasing precision by partitioning the error sum of squares: Blocking: SSE (CRD) à SSB + SSE (RCBD) Contrasts: SST à (t 1) orthogonal contrasts
Lecture 13 Topic 9: Factorial treatment structures (Part II) Increasing precision by partitioning the error sum of squares: s MST F = = MSE 2 among = s 2 within SST df trt SSE df e Blocking: SSE (CRD)
More informationANOVA: Analysis of Variation
ANOVA: Analysis of Variation The basic ANOVA situation Two variables: 1 Categorical, 1 Quantitative Main Question: Do the (means of) the quantitative variables depend on which group (given by categorical
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 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 informationDifference in two or more average scores in different groups
ANOVAs Analysis of Variance (ANOVA) Difference in two or more average scores in different groups Each participant tested once Same outcome tested in each group Simplest is one-way ANOVA (one variable as
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 informationSteps for Regression. Simple Linear Regression. Data. Example. Residuals vs. X. Scatterplot. Make a Scatter plot Does it make sense to plot a line?
Steps for Regression Simple Linear Regression Make a Scatter plot Does it make sense to plot a line? Check Residual Plot (Residuals vs. X) Are there any patterns? Check Histogram of Residuals Is it Normal?
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 informationOne-Way Analysis of Variance. With regression, we related two quantitative, typically continuous variables.
One-Way Analysis of Variance With regression, we related two quantitative, typically continuous variables. Often we wish to relate a quantitative response variable with a qualitative (or simply discrete)
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 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 informationDisadvantages of using many pooled t procedures. The sampling distribution of the sample means. The variability between the sample means
Stat 529 (Winter 2011) Analysis of Variance (ANOVA) Reading: Sections 5.1 5.3. Introduction and notation Birthweight example Disadvantages of using many pooled t procedures The analysis of variance procedure
More informationAnalysis of Variance Bios 662
Analysis of Variance Bios 662 Michael G. Hudgens, Ph.D. mhudgens@bios.unc.edu http://www.bios.unc.edu/ mhudgens 2008-10-21 13:34 BIOS 662 1 ANOVA Outline Introduction Alternative models SS decomposition
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 informationMultiple Regression Examples
Multiple Regression Examples Example: Tree data. we have seen that a simple linear regression of usable volume on diameter at chest height is not suitable, but that a quadratic model y = β 0 + β 1 x +
More informationHistogram of Residuals. Residual Normal Probability Plot. Reg. Analysis Check Model Utility. (con t) Check Model Utility. Inference.
Steps for Regression Simple Linear Regression Make a Scatter plot Does it make sense to plot a line? Check Residual Plot (Residuals vs. X) Are there any patterns? Check Histogram of Residuals Is it Normal?
More informationSimple Linear Regression. Steps for Regression. Example. Make a Scatter plot. Check Residual Plot (Residuals vs. X)
Simple Linear Regression 1 Steps for Regression Make a Scatter plot Does it make sense to plot a line? Check Residual Plot (Residuals vs. X) Are there any patterns? Check Histogram of Residuals Is it Normal?
More informationCS 147: Computer Systems Performance Analysis
CS 147: Computer Systems Performance Analysis CS 147: Computer Systems Performance Analysis 1 / 34 Overview Overview Overview Adding Replications Adding Replications 2 / 34 Two-Factor Design Without Replications
More informationunadjusted model for baseline cholesterol 22:31 Monday, April 19,
unadjusted model for baseline cholesterol 22:31 Monday, April 19, 2004 1 Class Level Information Class Levels Values TRETGRP 3 3 4 5 SEX 2 0 1 Number of observations 916 unadjusted model for baseline cholesterol
More informationEX1. One way ANOVA: miles versus Plug. a) What are the hypotheses to be tested? b) What are df 1 and df 2? Verify by hand. , y 3
EX. Chapter 8 Examples In an experiment to investigate the performance of four different brands of spark plugs intended for the use on a motorcycle, plugs of each brand were tested and the number of miles
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 informationKey Features: More than one type of experimental unit and more than one randomization.
1 SPLIT PLOT DESIGNS Key Features: More than one type of experimental unit and more than one randomization. Typical Use: When one factor is difficult to change. Example (and terminology): An agricultural
More informationStat 501, F. Chiaromonte. Lecture #8
Stat 501, F. Chiaromonte Lecture #8 Data set: BEARS.MTW In the minitab example data sets (for description, get into the help option and search for "Data Set Description"). Wild bears were anesthetized,
More informationSMAM 314 Exam 42 Name
SMAM 314 Exam 42 Name Mark the following statements True (T) or False (F) (10 points) 1. F A. The line that best fits points whose X and Y values are negatively correlated should have a positive slope.
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 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 informationAnalysing qpcr outcomes. Lecture Analysis of Variance by Dr Maartje Klapwijk
Analysing qpcr outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk 22 October 2014 Personal Background Since 2009 Insect Ecologist at SLU Climate Change and other anthropogenic effects on interaction
More informationFactorial and Unbalanced Analysis of Variance
Factorial and Unbalanced Analysis of Variance Nathaniel E. Helwig Assistant Professor of Psychology and Statistics University of Minnesota (Twin Cities) Updated 04-Jan-2017 Nathaniel E. Helwig (U of Minnesota)
More informationChapter 6 Randomized Block Design Two Factor ANOVA Interaction in ANOVA
Chapter 6 Randomized Block Design Two Factor ANOVA Interaction in ANOVA Two factor (two way) ANOVA Two factor ANOVA is used when: Y is a quantitative response variable There are two categorical explanatory
More informationUnbalanced Designs & Quasi F-Ratios
Unbalanced Designs & Quasi F-Ratios ANOVA for unequal n s, pooled variances, & other useful tools Unequal nʼs Focus (so far) on Balanced Designs Equal n s in groups (CR-p and CRF-pq) Observation in every
More information2-way analysis of variance
2-way analysis of variance We may be considering the effect of two factors (A and B) on our response variable, for instance fertilizer and variety on maize yield; or therapy and sex on cholesterol level.
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 informationAddition of Center Points to a 2 k Designs Section 6-6 page 271
to a 2 k Designs Section 6-6 page 271 Based on the idea of replicating some of the runs in a factorial design 2 level designs assume linearity. If interaction terms are added to model some curvature results
More informationANOVA Analysis of Variance
ANOVA Analysis of Variance ANOVA Analysis of Variance Extends independent samples t test ANOVA Analysis of Variance Extends independent samples t test Compares the means of groups of independent observations
More informationTheorem A: Expectations of Sums of Squares Under the two-way ANOVA model, E(X i X) 2 = (µ i µ) 2 + n 1 n σ2
identity Y ijk Ȳ = (Y ijk Ȳij ) + (Ȳi Ȳ ) + (Ȳ j Ȳ ) + (Ȳij Ȳi Ȳ j + Ȳ ) Theorem A: Expectations of Sums of Squares Under the two-way ANOVA model, (1) E(MSE) = E(SSE/[IJ(K 1)]) = (2) E(MSA) = E(SSA/(I
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 informationTwo-factor studies. STAT 525 Chapter 19 and 20. Professor Olga Vitek
Two-factor studies STAT 525 Chapter 19 and 20 Professor Olga Vitek December 2, 2010 19 Overview Now have two factors (A and B) Suppose each factor has two levels Could analyze as one factor with 4 levels
More informationDesign of Engineering Experiments Chapter 5 Introduction to Factorials
Design of Engineering Experiments Chapter 5 Introduction to Factorials Text reference, Chapter 5 page 170 General principles of factorial experiments The two-factor factorial with fixed effects The ANOVA
More information(1) The explanatory or predictor variables may be qualitative. (We ll focus on examples where this is the case.)
Introduction to Analysis of Variance Analysis of variance models are similar to regression models, in that we re interested in learning about the relationship between a dependent variable (a response)
More informationSTAT 705 Chapter 19: Two-way ANOVA
STAT 705 Chapter 19: Two-way ANOVA Adapted from Timothy Hanson Department of Statistics, University of South Carolina Stat 705: Data Analysis II 1 / 41 Two-way ANOVA This material is covered in Sections
More informationAnalysis of Bivariate Data
Analysis of Bivariate Data Data Two Quantitative variables GPA and GAES Interest rates and indices Tax and fund allocation Population size and prison population Bivariate data (x,y) Case corr® 2 Independent
More informationIn a one-way ANOVA, the total sums of squares among observations is partitioned into two components: Sums of squares represent:
Activity #10: AxS ANOVA (Repeated subjects design) Resources: optimism.sav So far in MATH 300 and 301, we have studied the following hypothesis testing procedures: 1) Binomial test, sign-test, Fisher s
More informationSTAT Final Practice Problems
STAT 48 -- Final Practice Problems.Out of 5 women who had uterine cancer, 0 claimed to have used estrogens. Out of 30 women without uterine cancer 5 claimed to have used estrogens. Exposure Outcome (Cancer)
More informationChapter 5 Introduction to Factorial Designs Solutions
Solutions from Montgomery, D. C. (1) Design and Analysis of Experiments, Wiley, NY Chapter 5 Introduction to Factorial Designs Solutions 5.1. The following output was obtained from a computer program that
More informationMore about Single Factor Experiments
More about Single Factor Experiments 1 2 3 0 / 23 1 2 3 1 / 23 Parameter estimation Effect Model (1): Y ij = µ + A i + ɛ ij, Ji A i = 0 Estimation: µ + A i = y i. ˆµ = y..  i = y i. y.. Effect Modell
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 informationIf we have many sets of populations, we may compare the means of populations in each set with one experiment.
Statistical Methods in Business Lecture 3. Factorial Design: If we have many sets of populations we may compare the means of populations in each set with one experiment. Assume we have two factors with
More informationOutline Topic 21 - Two Factor ANOVA
Outline Topic 21 - Two Factor ANOVA Data Model Parameter Estimates - Fall 2013 Equal Sample Size One replicate per cell Unequal Sample size Topic 21 2 Overview Now have two factors (A and B) Suppose each
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 Fall, 2013 Page 1 Tensile Strength Experiment Investigate the tensile strength of a new synthetic fiber. The factor is the
More informationExamination paper for TMA4255 Applied statistics
Department of Mathematical Sciences Examination paper for TMA4255 Applied statistics Academic contact during examination: Anna Marie Holand Phone: 951 38 038 Examination date: 16 May 2015 Examination time
More informationPLS205 Lab 2 January 15, Laboratory Topic 3
PLS205 Lab 2 January 15, 2015 Laboratory Topic 3 General format of ANOVA in SAS Testing the assumption of homogeneity of variances by "/hovtest" by ANOVA of squared residuals Proc Power for ANOVA One-way
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 informationChap The McGraw-Hill Companies, Inc. All rights reserved.
11 pter11 Chap Analysis of Variance Overview of ANOVA Multiple Comparisons Tests for Homogeneity of Variances Two-Factor ANOVA Without Replication General Linear Model Experimental Design: An Overview
More informationSMAM 314 Computer Assignment 5 due Nov 8,2012 Data Set 1. For each of the following data sets use Minitab to 1. Make a scatterplot.
SMAM 314 Computer Assignment 5 due Nov 8,2012 Data Set 1. For each of the following data sets use Minitab to 1. Make a scatterplot. 2. Fit the linear regression line. Regression Analysis: y versus x y
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 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 informationStat 511 HW#10 Spring 2003 (corrected)
Stat 511 HW#10 Spring 003 (corrected) 1. Below is a small set of fake unbalanced -way factorial (in factors A and B) data from (unbalanced) blocks. Level of A Level of B Block Response 1 1 1 9.5 1 1 11.3
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 informationMultiple Regression: Chapter 13. July 24, 2015
Multiple Regression: Chapter 13 July 24, 2015 Multiple Regression (MR) Response Variable: Y - only one response variable (quantitative) Several Predictor Variables: X 1, X 2, X 3,..., X p (p = # predictors)
More informationSTA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #6
STA 8 Applied Linear Models: Regression Analysis Spring 011 Solution for Homework #6 6. a) = 11 1 31 41 51 1 3 4 5 11 1 31 41 51 β = β1 β β 3 b) = 1 1 1 1 1 11 1 31 41 51 1 3 4 5 β = β 0 β1 β 6.15 a) Stem-and-leaf
More informationIntroduction to Regression
Introduction to Regression Using Mult Lin Regression Derived variables Many alternative models Which model to choose? Model Criticism Modelling Objective Model Details Data and Residuals Assumptions 1
More informationSchool of Mathematical Sciences. Question 1
School of Mathematical Sciences MTH5120 Statistical Modelling I Practical 8 and Assignment 7 Solutions Question 1 Figure 1: The residual plots do not contradict the model assumptions of normality, constant
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 informationAnalysis of variance
Analysis of variance 1 Method If the null hypothesis is true, then the populations are the same: they are normal, and they have the same mean and the same variance. We will estimate the numerical value
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 informationStatistics For Economics & Business
Statistics For Economics & Business Analysis of Variance In this chapter, you learn: Learning Objectives The basic concepts of experimental design How to use one-way analysis of variance to test for differences
More informationExtensions of One-Way ANOVA.
Extensions of One-Way ANOVA http://www.pelagicos.net/classes_biometry_fa17.htm What do I want You to Know What are two main limitations of ANOVA? What two approaches can follow a significant ANOVA? How
More information1 Introduction to Minitab
1 Introduction to Minitab Minitab is a statistical analysis software package. The software is freely available to all students and is downloadable through the Technology Tab at my.calpoly.edu. When you
More informationContents. 2 2 factorial design 4
Contents TAMS38 - Lecture 10 Response surface methodology Lecturer: Zhenxia Liu Department of Mathematics - Mathematical Statistics 12 December, 2017 2 2 factorial design Polynomial Regression model First
More informationInference for Regression Inference about the Regression Model and Using the Regression Line, with Details. Section 10.1, 2, 3
Inference for Regression Inference about the Regression Model and Using the Regression Line, with Details Section 10.1, 2, 3 Basic components of regression setup Target of inference: linear dependency
More informationSleep data, two drugs Ch13.xls
Model Based Statistics in Biology. Part IV. The General Linear Mixed Model.. Chapter 13.3 Fixed*Random Effects (Paired t-test) ReCap. Part I (Chapters 1,2,3,4), Part II (Ch 5, 6, 7) ReCap Part III (Ch
More informationAnalysis of Variance. Read Chapter 14 and Sections to review one-way ANOVA.
Analysis of Variance Read Chapter 14 and Sections 15.1-15.2 to review one-way ANOVA. Design of an experiment the process of planning an experiment to insure that an appropriate analysis is possible. Some
More informationST 512-Practice Exam I - Osborne Directions: Answer questions as directed. For true/false questions, circle either true or false.
ST 512-Practice Exam I - Osborne Directions: Answer questions as directed. For true/false questions, circle either true or false. 1. A study was carried out to examine the relationship between the number
More informationWhat If There Are More Than. Two Factor Levels?
What If There Are More Than Chapter 3 Two Factor Levels? Comparing more that two factor levels the analysis of variance ANOVA decomposition of total variability Statistical testing & analysis Checking
More 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 informationSTAT 705 Chapter 19: Two-way ANOVA
STAT 705 Chapter 19: Two-way ANOVA Timothy Hanson Department of Statistics, University of South Carolina Stat 705: Data Analysis II 1 / 38 Two-way ANOVA Material covered in Sections 19.2 19.4, but a bit
More information23. Inference for regression
23. Inference for regression The Practice of Statistics in the Life Sciences Third Edition 2014 W. H. Freeman and Company Objectives (PSLS Chapter 23) Inference for regression The regression model Confidence
More informationChapter 8: Hypothesis Testing Lecture 9: Likelihood ratio tests
Chapter 8: Hypothesis Testing Lecture 9: Likelihood ratio tests Throughout this chapter we consider a sample X taken from a population indexed by θ Θ R k. Instead of estimating the unknown parameter, we
More informationField Work and Latin Square Design
Field Work and Latin Square Design Chapter 12 - Factorial Designs (covered by Jason) Interactive effects between multiple independent variables Chapter 13 - Field Research Quasi-Experimental Designs Program
More information. Example: For 3 factors, sse = (y ijkt. " y ijk
ANALYSIS OF BALANCED FACTORIAL DESIGNS Estimates of model parameters and contrasts can be obtained by the method of Least Squares. Additional constraints must be added to estimate non-estimable parameters.
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 informationMultiple Predictor Variables: ANOVA
What if you manipulate two factors? Multiple Predictor Variables: ANOVA Block 1 Block 2 Block 3 Block 4 A B C D B C D A C D A B D A B C Randomized Controlled Blocked Design: Design where each treatment
More informationOne-Way Analysis of Variance (ANOVA)
1 One-Way Analysis of Variance (ANOVA) One-Way Analysis of Variance (ANOVA) is a method for comparing the means of a populations. This kind of problem arises in two different settings 1. When a independent
More informationDesign of Experiments. Factorial experiments require a lot of resources
Design of Experiments Factorial experiments require a lot of resources Sometimes real-world practical considerations require us to design experiments in specialized ways. The design of an experiment is
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 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 informationSociology 6Z03 Review II
Sociology 6Z03 Review II John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review II Fall 2016 1 / 35 Outline: Review II Probability Part I Sampling Distributions Probability
More informationChapter 4: Randomized Blocks and Latin Squares
Chapter 4: Randomized Blocks and Latin Squares 1 Design of Engineering Experiments The Blocking Principle Blocking and nuisance factors The randomized complete block design or the RCBD Extension of the
More information