Split-Plot Designs. David M. Allen University of Kentucky. January 30, 2014
|
|
- Vivian Haynes
- 6 years ago
- Views:
Transcription
1 Split-Plot Designs David M. Allen University of Kentucky January 30, 2014
2 1 Introduction In this talk we introduce the split-plot design and give an overview of how SAS determines the denominator degrees of freedom for various tests. Back 2
3 2 Drug-Alcohol Study The drug-alcohol study presented here is based on an actual study. It has been scaled down to facilitate more explicit displays. The responses have be changed because the original data are proprietary. See Allen and Cady [1] for more discussion. Back 3
4 Background Tranquilizers are one of the most prescribed classes of drugs. Unfortunately, the combination of tranquilizers and alcohol can compromise a driver s ability to operate a motor vehicle. It is desirable to develop a new tranquilizer that serves its intended purpose but does not combine with alcohol to give an undesirable effect. This trial is to compare effects of drug, effects of alcohol, and the effects of their interaction. The drugs are A a new drug, B a currently popular drug, and C a placebo. The response is the subject s performance on a simulated driving test. While multiple response measurements are recorded, the mean deviation (in feet) from the center of the driving lane is used here. Back 4
5 Randomization Subjects are the whole-plot unit. The alcohol and no alcohol treatments are randomly assigned to the twelve subjects with the restriction that there is the same number of subjects in each treatment group. Separately for each subject, the order of drugs A, B, and C is randomized. There is an adequate interval of time between administration of the different drugs to insure there are no carry-over effects. Back 5
6 The data Drugs Alcohol Subject A B C Yes EAS Yes JBM Yes ARE Yes JBH Yes WJT Yes EEA No JWL No CJW No RDF No RLA No HW No AMR Back 6
7 The model is The model y jk = μ + α + s j + δ k + (αδ) k + ε jk where y jk is the observation on the response variable; μ is the over-all mean; α is the effect of the th level of alcohol; s j is the effect of the jth subject; δ k is the effect of the kth drug; (αδ) k is the effect of the interaction of the th level of alcohol and kth level of drug; and ε jk is a random error. We assume s j N(0, σ 2 s ), ε jk N(0, σ 2 ), and that these effects are mutually independent. All other effects are considered fixed parameters. We have that j = 1 6 for = 1, and j = 7 12 for = 2. Back 7
8 Symbolic data Drugs Alcohol Subject A B C Yes EAS y 1,1,1 y 1,1,2 y 1,1,3 y 1,1, Yes JBM y 1,2,1 y 1,2,2 y 1,2,3 y 1,2, Yes WJT y 1,5,1 y 1,5,2 y 1,5,3 y 1,5, Yes EEA y 1,6,1 y 1,6,2 y 1,6,3 y 1,6, y 1,,1 y 1,,2 y 1,,3 y 1,, No JWL y 2,7,1 y 2,7,2 y 2,7,3 y 2,7, No CJW y 2,8,1 y 2,8,2 y 2,8,3 y 2,8, No HW y 2,11,1 y 2,11,2 y 2,11,3 y 2,11, No AMR y 2,12,1 y 2,12,2 y 2,12,3 y 2,12, y 2,,1 y 2,,2 y 2,,3 y 2,, Back 8
9 Symbolic analysis of variance Degrees of Sum of Mean Expected Source Freedom Squares Square Mean Square Alcohol 1 SS α MS α σ 2 + 3σ 2 s + Q(α, ( Subjects 10 SS s MS s σ 2 + 3σ 2 s Drugs 2 SS δ MS δ σ 2 + Q(δ, (αδ)) Alcohol*Drug 2 SS (αδ) MS (αδ) σ 2 + Q((αδ)) Residual 20 SS ε MS ε σ 2 Back 9
10 Numeric analysis of variance Degrees of Sum of Mean F- Source Freedom Squares Square statistic Alcohol Subjects Drugs Alcohol*Drug Residual Back 10
11 3 Nested factors A factor B is said to be nested within factor A if the levels of factor B are different within each level of factor A. In this case, we say factor A contains factor B. Back 11
12 An example To facilitate explicit displays, we use a smaller version of the drug-alcohol study: Drug Alcohol Subject SubWithin A B Yes dma Yes lwh Yes rla No clw No red No bbs No kmd The levels of Subjects are completely different for the yes Back 12
13 and no levels of Alcohol. We say that Subjects are nested within Alcohol and that Alcohol contains Subjects. Back 13
14 Coding Sometimes a nested factor is coded such that the levels are unique only within levels of the containing factor. For example, the factor SubWithin in the above display is unique only within levels of Alcohol. The remainder of this section deals with building the Z matrix. We assume Alcohol, Subject, and SubWithin are classes variables. Back 14
15 Z = Building the Z matrix We can build Z by putting Subject in a random statement. We call this the direct method. Back 15
16 SAS notation We can build Z by putting either of the equivalent terms, Alcohol*SubWithin or SubWithin(Alcohol), in a random statement. We call this the product method. Back 16
17 Z = = Back 17
18 An editorial The product method has little to recommend it: A variable having a unique subject code must exist, for otherwise the randomization could not have been carried out. Why not use it? If there are unequal numbers of subjects in the alcohol groups, the second method will put one or more columns of all zeros in the design matrix. This increases computational time. Back 18
19 From the computational point of view, the worst possible specification is combine the two methods. For example, Subject(Alcohol) would introduce fourteen columns in the design matrix, and one-half of them would be all zeros. There is an additional consideration: SAS treats models specified by the product and direct methods differently. Back 19
20 4 Satterthwaite procedure In this section we give the simplest form of the Satterthwaite approximation [3]. This approximation may be thought of as synthesizing a mean square. Back 20
21 The setup Suppose a model depends on vector of fixed effects, β, and two variances, σ 2 1 and σ2. Our interest is in a linear 2 function of the fixed effects which we denote by δ. Assume that we have a normally distributed estimator, ˆδ, with variance c 1 σ c 2σ 2 2 where c 1 and c 2 are known constants. Available are SS 1 and SS 2 such that SS 1 σ 2 1 χ2 (ν 1 ) and SS 2 σ 2 2 χ2 (ν 2 ). You may look back to page 9 for an example of SS 1 and SS 2. SS 1, SS 2, and ˆδ are mutually independent. The test statistic for the null hypothesis that δ is equal a specified value δ 0 is t = ˆδ δ 0 c1 SS 1 /ν 1 + c 2 SS 2 /ν 2. Back 21
22 The question is: what is the distribution of t? Back 22
23 Decomposing t The approach used here is to approximate the distribution of t by a t-distribution. That reduces the problem to finding the degrees of freedom of the approximating t-distribution. Define and Z = ˆδ δ 0 c 1 σ c 2σ 2 2 U = c 1 σ 2 SS 1 1 c 2 σ 2 SS 2 2 ν 1 (c 1 σ c 2σ 2 2 ) σ ν 2 (c 1 σ c 2σ 2 2 ) σ 2 2 then t = Z/ U. Under the null hypothesis, the distribution of Z is standard normal. Back 23
24 It remains to approximate the distribution of U by a Chi-square divided by it degrees of freedom, i.e. there exist a ν such that U χ 2 (ν)/ν is approximately satisfied. Back 24
25 Degrees of freedom for approximating distribution By approximately satisfied we mean U and χ 2 (ν)/ν should have the same variance. Now V r(u) = and c 1 σ 2 2 c 1 2 σ 2 2 ν 1 (c 1 σ c 2σ 2 2 ) 2ν 1 + ν 2 (c 1 σ c 2σ 2 2 ) = 2 c2 1 σ4 1 /ν 1 + c 2 2 σ4 2 /ν 2 (c 1 σ c 2σ 2 2 )2 V r χ 2 (ν)/ν = 2 ν. 2 Back 25
26 Equating these two variances and solving for ν gives ν = (c 1σ c 2σ 2 2 )2 c 2 1 σ4 1 /ν 1 + c 2 2 σ4 2 /ν 2 Back 26
27 5 Estimation with balanced data Estimators of linear combinations of fixed effects can be categorized in three ways: 1. estimators that are orthogonal to subjects; 2. estimators that involve only subject totals; and 3. other estimators. We will illustrate a represenitive estimator from each category. The estimators discussed in this section are defined in terms of notation given on page 8. Back 27
28 Drug A versus Drug C A comparison of Drug A with Drug C, averaged over possible interaction effects, is orthogonal to subjects. This is because each drug is used on each subject. The estimator of δ 1 δ 3 is (y 1,,1 + y 2,,1 y 1,,3 y 2,,3 )/2, and its variance is σ 2 /6. The residual mean square is an estimator of σ 2 and is distributed proportional to Chi-square. The t-distribution is used in the usual way for testing or or confidence intervals. A similar result is true for all contrasts among drug effects or among interaction effects. Back 28
29 Alcohol versus no alcohol A comparison alcohol with no alcohol, averaged over any interaction effects, involves only subject totals. The estimator of α 1 α 2 is y 1,, y 2,,, and its variance is (3σ 2 s + σ2 )/9. The subject mean square is an estimator of 3σ 2 s + σ2 and is distributed proportional to Chi-square. The t-distribution is used in the usual way for testing or or confidence intervals. Back 29
30 Response with Drug A and Alcohol The estimated response for a subject on Drug A and Alcohol is y 1,,1, and its variance is σ 2 + σ 2. We estimate s σ 2 + σ 2 s by 1 3 MS s MS ε. Unfortunately, 1 3 MS s MS ε is not distributed proportional to Chi-square, so the usual confidence interval based on the t-distribution not strictly valid. Back 30
31 We use the Satterthwaite procedure to find the degrees of freedom of the approximating Chi-square distribution. The correspondence of notation is σ 2 1 = 3σ2 s + σ2 σ 2 2 = σ2 ν 1 = 10 ν 2 = 20 c 1 = 1/3 c 2 = 2/3 Since the variances are not known, substitute the corresponding mean squares. The result is ν = 15. We proceed with the inference assuming a t-distribution with Back 31
32 fifteen degrees of freedom. Back 32
33 6 SAS degrees of freedom options On the estimate statement one may use the df option to specify the denominator degrees of freedom for the approximate t-distribution. However, except for simple tests with balanced data, most people will want SAS to provide the degrees of freedom. In this section we describe five different methods for determining denominator degrees of freedom that a accessible in SAS. Back 33
34 The containment method The containment method is the default when the RANDOM statement is used. Otherwise, the containment method is invoked with the DDFM = CONTAIN option on the model statement. Denote the fixed effect in question A, and search the RANDOM effect list for the effects that syntactically contain A. Among the random effects that contain A, compute their rank contribution to the [X Z] matrix. The denominator degrees of freedom assigned to A is the smallest of these rank contributions. If A is not found on the random statement, the containment method is not invoked, and the denominator degrees of freedom are the residual degrees of freedom. Back 34
35 Note that for a nested model, specified by the direct method, the containment method will not be invoked. Back 35
36 The between-within method The DDFM = BETWITHIN option is the default for REPEATED statement specifications (with no RANDOM statements). It is computed by dividing the residual degrees of freedom into between-subject and within-subject portions. PROC MIXED then checks whether a fixed effect changes within any subject. If so, it assigns within-subject degrees of freedom to the effect; otherwise, it assigns the between-subject degrees of freedom to the effect. If there are multiple within-subject effects containing classification variables, the within-subject degrees of freedom is partitioned into components corresponding to the subject-by-effect interactions. Back 36
37 The residual degrees of freedom The denominator degrees of freedom are the residual degrees of freedom. This will give exact test for all effects that are orthogonal to the Z matrix; i.e. split-plot treatment and interaction with whole-plot treatment. Back 37
38 The Satterthwaite method The Satterthwaite method is a generalization of the Satterthwaite method described in Section 4. The generalization is discussed in considerable detail in another lecture. Back 38
39 The Kenward-Roger method The Kenward-Roger method implements the method described in [2]. This method is in SAS starting with Version 8. The Kenward-Roger method uses the Satterthwaite method for determining the denominator degrees of freedom, but it modifies the estimator as well. Calling the Kenward-Roger method a denominator degrees of freedom method is a misnomer. Back 39
40 7 Comparison of degrees of freedom In section 5 we looked at three different estimators using traditional methods and taking advantage of the balanced data. In this section, we look at how SAS computes the denominator degrees of freedom for these estimates. We then remove some of the data and repeat the exercise. Back 40
41 Drug-Alcohol data with missing values Drugs Alcohol Subject A B C Yes HW Yes JBM Yes JWL Yes JBH Yes ARE Yes EEA No DCJ No CJW No RDF No RLA No EAS No AMR Back 41
42 We have removed seven observations or 19.4%. Four are from the alcohol group, and three are from the no alcohol group. Three observations are removed from both the Drug A and Drug B groups, and one observation is removed from Drug C. Back 42
43 The SAS code The SAS code used for this demonstration is proc mixed data = balanced; classes Alcohol Subject SubWithin Drug; model y = Alcohol Drug Alcohol*Drug / ddfm = conta random Subject; estimate 1 intercept 1 Alcohol 1 0 Drug 1 Alcoho estimate 2 Alcohol -1 1 ; estimate 3 Drug ; run; The high lighted parts of the code are changed from run to run. We use the balanced data and the data with missing observations. We use all five methods of Back 43
44 computing the denominator degrees of freedom. We use both the direct and product method of specifying the random effect. Back 44
45 Estimate 1 Drug A with no alcohol Denominator degrees of freedom Method Balanced Missing Containment Between-within Residual Satterthwaite Kenward-Roger Back 45
46 Estimate 2 Alcohol versus no alcohol Denominator degrees of freedom Method Balanced Missing Containment 20(10) 13(10) Between-within Residual Satterthwaite Kenward-Roger For the containment method, the first number is for direct specification, and the number in parentheses is for product specification. Back 46
47 Estimate 3 Drug A versus drug C Denominator degrees of freedom Method Balanced Missing Containment Between-within Residual Satterthwaite Kenward-Roger Back 47
48 References [1] David M. Allen and Foster B. Cady. Analyzing Experimental Data by Regression. VanNostrand-Reinhold, Belmont, California, [2] M. G. Kenward and J. H. Roger. Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53: , [3] F. E. Satterthwaite. An approximate distribution of estimates of variance components. Biometrics Bulletin, 2: , Back 48
Randomized Complete Block Designs
Randomized Complete Block Designs David Allen University of Kentucky February 23, 2016 1 Randomized Complete Block Design There are many situations where it is impossible to use a completely randomized
More informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Today s Topics: What happens to missing predictors Effects of time-invariant predictors Fixed vs. systematically varying vs. random effects Model building
More informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Today s Class (or 3): Summary of steps in building unconditional models for time What happens to missing predictors Effects of time-invariant predictors
More informationSTA441: Spring Multiple Regression. This slide show is a free open source document. See the last slide for copyright information.
STA441: Spring 2018 Multiple Regression This slide show is a free open source document. See the last slide for copyright information. 1 Least Squares Plane 2 Statistical MODEL There are p-1 explanatory
More informationA Likelihood Ratio Test
A Likelihood Ratio Test David Allen University of Kentucky February 23, 2012 1 Introduction Earlier presentations gave a procedure for finding an estimate and its standard error of a single linear combination
More informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Topics: What happens to missing predictors Effects of time-invariant predictors Fixed vs. systematically varying vs. random effects Model building strategies
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 informationA discussion on multiple regression models
A discussion on multiple regression models In our previous discussion of simple linear regression, we focused on a model in which one independent or explanatory variable X was used to predict the value
More informationLOOKING FOR RELATIONSHIPS
LOOKING FOR RELATIONSHIPS One of most common types of investigation we do is to look for relationships between variables. Variables may be nominal (categorical), for example looking at the effect of an
More informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Topics: Summary of building unconditional models for time Missing predictors in MLM Effects of time-invariant predictors Fixed, systematically varying,
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 information16.400/453J Human Factors Engineering. Design of Experiments II
J Human Factors Engineering Design of Experiments II Review Experiment Design and Descriptive Statistics Research question, independent and dependent variables, histograms, box plots, etc. Inferential
More information" M A #M B. Standard deviation of the population (Greek lowercase letter sigma) σ 2
Notation and Equations for Final Exam Symbol Definition X The variable we measure in a scientific study n The size of the sample N The size of the population M The mean of the sample µ The mean of the
More informationTest 3 Practice Test A. NOTE: Ignore Q10 (not covered)
Test 3 Practice Test A NOTE: Ignore Q10 (not covered) MA 180/418 Midterm Test 3, Version A Fall 2010 Student Name (PRINT):............................................. Student Signature:...................................................
More informationOHSU OGI Class ECE-580-DOE :Design of Experiments Steve Brainerd
Why We Use Analysis of Variance to Compare Group Means and How it Works The question of how to compare the population means of more than two groups is an important one to researchers. Let us suppose that
More informationTopic 21 Goodness of Fit
Topic 21 Goodness of Fit Contingency Tables 1 / 11 Introduction Two-way Table Smoking Habits The Hypothesis The Test Statistic Degrees of Freedom Outline 2 / 11 Introduction Contingency tables, also known
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 informationChapter 7 Student Lecture Notes 7-1
Chapter 7 Student Lecture Notes 7- Chapter Goals QM353: Business Statistics Chapter 7 Multiple Regression Analysis and Model Building After completing this chapter, you should be able to: Explain model
More informationThe legacy of Sir Ronald A. Fisher. Fisher s three fundamental principles: local control, replication, and randomization.
1 Chapter 1: Research Design Principles The legacy of Sir Ronald A. Fisher. Fisher s three fundamental principles: local control, replication, and randomization. 2 Chapter 2: Completely Randomized Design
More informationAnalysis of Variance
Statistical Techniques II EXST7015 Analysis of Variance 15a_ANOVA_Introduction 1 Design The simplest model for Analysis of Variance (ANOVA) is the CRD, the Completely Randomized Design This model is also
More informationSimple logistic regression
Simple logistic regression Biometry 755 Spring 2009 Simple logistic regression p. 1/47 Model assumptions 1. The observed data are independent realizations of a binary response variable Y that follows a
More informationTopic 22 Analysis of Variance
Topic 22 Analysis of Variance Comparing Multiple Populations 1 / 14 Outline Overview One Way Analysis of Variance Sample Means Sums of Squares The F Statistic Confidence Intervals 2 / 14 Overview Two-sample
More informationChapter 3 Multiple Regression Complete Example
Department of Quantitative Methods & Information Systems ECON 504 Chapter 3 Multiple Regression Complete Example Spring 2013 Dr. Mohammad Zainal Review Goals After completing this lecture, you should be
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 informationTime Invariant Predictors in Longitudinal Models
Time Invariant Predictors in Longitudinal Models Longitudinal Data Analysis Workshop Section 9 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Section
More informationCOMPARING SEVERAL MEANS: ANOVA
LAST UPDATED: November 15, 2012 COMPARING SEVERAL MEANS: ANOVA Objectives 2 Basic principles of ANOVA Equations underlying one-way ANOVA Doing a one-way ANOVA in R Following up an ANOVA: Planned contrasts/comparisons
More informationMcGill University. Faculty of Science MATH 204 PRINCIPLES OF STATISTICS II. Final Examination
McGill University Faculty of Science MATH 204 PRINCIPLES OF STATISTICS II Final Examination Date: 20th April 2009 Time: 9am-2pm Examiner: Dr David A Stephens Associate Examiner: Dr Russell Steele Please
More informationSTAT 501 EXAM I NAME Spring 1999
STAT 501 EXAM I NAME Spring 1999 Instructions: You may use only your calculator and the attached tables and formula sheet. You can detach the tables and formula sheet from the rest of this exam. Show your
More informationName: Biostatistics 1 st year Comprehensive Examination: Applied in-class exam. June 8 th, 2016: 9am to 1pm
Name: Biostatistics 1 st year Comprehensive Examination: Applied in-class exam June 8 th, 2016: 9am to 1pm Instructions: 1. This is exam is to be completed independently. Do not discuss your work with
More informationSummary of Chapters 7-9
Summary of Chapters 7-9 Chapter 7. Interval Estimation 7.2. Confidence Intervals for Difference of Two Means Let X 1,, X n and Y 1, Y 2,, Y m be two independent random samples of sizes n and m from two
More informationMath 423/533: The Main Theoretical Topics
Math 423/533: The Main Theoretical Topics Notation sample size n, data index i number of predictors, p (p = 2 for simple linear regression) y i : response for individual i x i = (x i1,..., x ip ) (1 p)
More informationInferences for Regression
Inferences for Regression An Example: Body Fat and Waist Size Looking at the relationship between % body fat and waist size (in inches). Here is a scatterplot of our data set: Remembering Regression In
More informationLecture 3: Inference in SLR
Lecture 3: Inference in SLR STAT 51 Spring 011 Background Reading KNNL:.1.6 3-1 Topic Overview This topic will cover: Review of hypothesis testing Inference about 1 Inference about 0 Confidence Intervals
More informationTutorial 4: Power and Sample Size for the Two-sample t-test with Unequal Variances
Tutorial 4: Power and Sample Size for the Two-sample t-test with Unequal Variances Preface Power is the probability that a study will reject the null hypothesis. The estimated probability is a function
More informationStatistical Distribution Assumptions of General Linear Models
Statistical Distribution Assumptions of General Linear Models Applied Multilevel Models for Cross Sectional Data Lecture 4 ICPSR Summer Workshop University of Colorado Boulder Lecture 4: Statistical Distributions
More informationHypothesis Testing hypothesis testing approach
Hypothesis Testing In this case, we d be trying to form an inference about that neighborhood: Do people there shop more often those people who are members of the larger population To ascertain this, we
More informationLogistic Regression Analysis
Logistic Regression Analysis Predicting whether an event will or will not occur, as well as identifying the variables useful in making the prediction, is important in most academic disciplines as well
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 informationBasic Business Statistics, 10/e
Chapter 4 4- Basic Business Statistics th Edition Chapter 4 Introduction to Multiple Regression Basic Business Statistics, e 9 Prentice-Hall, Inc. Chap 4- Learning Objectives In this chapter, you learn:
More informationDr. Junchao Xia Center of Biophysics and Computational Biology. Fall /1/2016 1/46
BIO5312 Biostatistics Lecture 10:Regression and Correlation Methods Dr. Junchao Xia Center of Biophysics and Computational Biology Fall 2016 11/1/2016 1/46 Outline In this lecture, we will discuss topics
More informationA Re-Introduction to General Linear Models (GLM)
A Re-Introduction to General Linear Models (GLM) Today s Class: You do know the GLM Estimation (where the numbers in the output come from): From least squares to restricted maximum likelihood (REML) Reviewing
More informationLecture 21: October 19
36-705: Intermediate Statistics Fall 2017 Lecturer: Siva Balakrishnan Lecture 21: October 19 21.1 Likelihood Ratio Test (LRT) To test composite versus composite hypotheses the general method is to use
More informationMathematical statistics
November 15 th, 2018 Lecture 21: The two-sample t-test Overview Week 1 Week 2 Week 4 Week 7 Week 10 Week 14 Probability reviews Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation
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 informationChapter 10: Inferences based on two samples
November 16 th, 2017 Overview Week 1 Week 2 Week 4 Week 7 Week 10 Week 12 Chapter 1: Descriptive statistics Chapter 6: Statistics and Sampling Distributions Chapter 7: Point Estimation Chapter 8: Confidence
More informationChapter 14 Student Lecture Notes 14-1
Chapter 14 Student Lecture Notes 14-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 14 Multiple Regression Analysis and Model Building Chap 14-1 Chapter Goals After completing this
More informationLecture 2 Simple Linear Regression STAT 512 Spring 2011 Background Reading KNNL: Chapter 1
Lecture Simple Linear Regression STAT 51 Spring 011 Background Reading KNNL: Chapter 1-1 Topic Overview This topic we will cover: Regression Terminology Simple Linear Regression with a single predictor
More informationSTK4900/ Lecture 3. Program
STK4900/9900 - Lecture 3 Program 1. Multiple regression: Data structure and basic questions 2. The multiple linear regression model 3. Categorical predictors 4. Planned experiments and observational studies
More informationRegression With a Categorical Independent Variable: Mean Comparisons
Regression With a Categorical Independent Variable: Mean Lecture 16 March 29, 2005 Applied Regression Analysis Lecture #16-3/29/2005 Slide 1 of 43 Today s Lecture comparisons among means. Today s Lecture
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 informationSMA 6304 / MIT / MIT Manufacturing Systems. Lecture 10: Data and Regression Analysis. Lecturer: Prof. Duane S. Boning
SMA 6304 / MIT 2.853 / MIT 2.854 Manufacturing Systems Lecture 10: Data and Regression Analysis Lecturer: Prof. Duane S. Boning 1 Agenda 1. Comparison of Treatments (One Variable) Analysis of Variance
More informationReview. One-way ANOVA, I. What s coming up. Multiple comparisons
Review One-way ANOVA, I 9.07 /15/00 Earlier in this class, we talked about twosample z- and t-tests for the difference between two conditions of an independent variable Does a trial drug work better than
More informationLinear Mixed Models: Methodology and Algorithms
Linear Mixed Models: Methodology and Algorithms David M. Allen University of Kentucky January 8, 2018 1 The Linear Mixed Model This Chapter introduces some terminology and definitions relating to the main
More informationdf=degrees of freedom = n - 1
One sample t-test test of the mean Assumptions: Independent, random samples Approximately normal distribution (from intro class: σ is unknown, need to calculate and use s (sample standard deviation)) Hypotheses:
More informationChapter 13. Multiple Regression and Model Building
Chapter 13 Multiple Regression and Model Building Multiple Regression Models The General Multiple Regression Model y x x x 0 1 1 2 2... k k y is the dependent variable x, x,..., x 1 2 k the model are the
More informationQuestion. Hypothesis testing. Example. Answer: hypothesis. Test: true or not? Question. Average is not the mean! μ average. Random deviation or not?
Hypothesis testing Question Very frequently: what is the possible value of μ? Sample: we know only the average! μ average. Random deviation or not? Standard error: the measure of the random deviation.
More informationSimple linear regression
Simple linear regression Biometry 755 Spring 2008 Simple linear regression p. 1/40 Overview of regression analysis Evaluate relationship between one or more independent variables (X 1,...,X k ) and a single
More informationThe t-distribution. Patrick Breheny. October 13. z tests The χ 2 -distribution The t-distribution Summary
Patrick Breheny October 13 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction Introduction What s wrong with z-tests? So far we ve (thoroughly!) discussed how to carry out hypothesis
More informationhttp://www.statsoft.it/out.php?loc=http://www.statsoft.com/textbook/ Group comparison test for independent samples The purpose of the Analysis of Variance (ANOVA) is to test for significant differences
More informationAnalysis of Variance (ANOVA)
Analysis of Variance ANOVA) Compare several means Radu Trîmbiţaş 1 Analysis of Variance for a One-Way Layout 1.1 One-way ANOVA Analysis of Variance for a One-Way Layout procedure for one-way layout Suppose
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the
More informationEpidemiology Principles of Biostatistics Chapter 10 - Inferences about two populations. John Koval
Epidemiology 9509 Principles of Biostatistics Chapter 10 - Inferences about John Koval Department of Epidemiology and Biostatistics University of Western Ontario What is being covered 1. differences in
More informationExam details. Final Review Session. Things to Review
Exam details Final Review Session Short answer, similar to book problems Formulae and tables will be given You CAN use a calculator Date and Time: Dec. 7, 006, 1-1:30 pm Location: Osborne Centre, Unit
More informationIntroduction to SAS proc mixed
Faculty of Health Sciences Introduction to SAS proc mixed Analysis of repeated measurements, 2017 Julie Forman Department of Biostatistics, University of Copenhagen 2 / 28 Preparing data for analysis The
More informationConfidence Intervals, Testing and ANOVA Summary
Confidence Intervals, Testing and ANOVA Summary 1 One Sample Tests 1.1 One Sample z test: Mean (σ known) Let X 1,, X n a r.s. from N(µ, σ) or n > 30. Let The test statistic is H 0 : µ = µ 0. z = x µ 0
More informationBlack White Total Observed Expected χ 2 = (f observed f expected ) 2 f expected (83 126) 2 ( )2 126
Psychology 60 Fall 2013 Practice Final Actual Exam: This Wednesday. Good luck! Name: To view the solutions, check the link at the end of the document. This practice final should supplement your studying;
More informationPLSC PRACTICE TEST ONE
PLSC 724 - PRACTICE TEST ONE 1. Discuss briefly the relationship between the shape of the normal curve and the variance. 2. What is the relationship between a statistic and a parameter? 3. How is the α
More informationFigure 9.1: A Latin square of order 4, used to construct four types of design
152 Chapter 9 More about Latin Squares 9.1 Uses of Latin squares Let S be an n n Latin square. Figure 9.1 shows a possible square S when n = 4, using the symbols 1, 2, 3, 4 for the letters. Such a Latin
More informationBIOL Biometry LAB 6 - SINGLE FACTOR ANOVA and MULTIPLE COMPARISON PROCEDURES
BIOL 458 - Biometry LAB 6 - SINGLE FACTOR ANOVA and MULTIPLE COMPARISON PROCEDURES PART 1: INTRODUCTION TO ANOVA Purpose of ANOVA Analysis of Variance (ANOVA) is an extremely useful statistical method
More informationWeek 14 Comparing k(> 2) Populations
Week 14 Comparing k(> 2) Populations Week 14 Objectives Methods associated with testing for the equality of k(> 2) means or proportions are presented. Post-testing concepts and analysis are introduced.
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7
MA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7 1 Random Vectors Let a 0 and y be n 1 vectors, and let A be an n n matrix. Here, a 0 and A are non-random, whereas y is
More informationStatistical Inference: The Marginal Model
Statistical Inference: The Marginal Model Edps/Psych/Stat 587 Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Fall 2017 Outline Inference for fixed
More informationDESIGNING EXPERIMENTS AND ANALYZING DATA A Model Comparison Perspective
DESIGNING EXPERIMENTS AND ANALYZING DATA A Model Comparison Perspective Second Edition Scott E. Maxwell Uniuersity of Notre Dame Harold D. Delaney Uniuersity of New Mexico J,t{,.?; LAWRENCE ERLBAUM ASSOCIATES,
More informationAnalysis of Variance and Co-variance. By Manza Ramesh
Analysis of Variance and Co-variance By Manza Ramesh Contents Analysis of Variance (ANOVA) What is ANOVA? The Basic Principle of ANOVA ANOVA Technique Setting up Analysis of Variance Table Short-cut Method
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science
UNIVERSITY OF TORONTO Faculty of Arts and Science December 2013 Final Examination STA442H1F/2101HF Methods of Applied Statistics Jerry Brunner Duration - 3 hours Aids: Calculator Model(s): Any calculator
More informationIntroduction to SAS proc mixed
Faculty of Health Sciences Introduction to SAS proc mixed Analysis of repeated measurements, 2017 Julie Forman Department of Biostatistics, University of Copenhagen Outline Data in wide and long format
More informationIntroduction to Crossover Trials
Introduction to Crossover Trials Stat 6500 Tutorial Project Isaac Blackhurst A crossover trial is a type of randomized control trial. It has advantages over other designed experiments because, under certain
More informationMixed Designs: Between and Within. Psy 420 Ainsworth
Mixed Designs: Between and Within Psy 420 Ainsworth Mixed Between and Within Designs Conceptualizing the Design Types of Mixed Designs Assumptions Analysis Deviation Computation Higher order mixed designs
More informationLogistic Regression. Interpretation of linear regression. Other types of outcomes. 0-1 response variable: Wound infection. Usual linear regression
Logistic Regression Usual linear regression (repetition) y i = b 0 + b 1 x 1i + b 2 x 2i + e i, e i N(0,σ 2 ) or: y i N(b 0 + b 1 x 1i + b 2 x 2i,σ 2 ) Example (DGA, p. 336): E(PEmax) = 47.355 + 1.024
More informationST505/S697R: Fall Homework 2 Solution.
ST505/S69R: Fall 2012. Homework 2 Solution. 1. 1a; problem 1.22 Below is the summary information (edited) from the regression (using R output); code at end of solution as is code and output for SAS. a)
More informationAnswer to exercise: Blood pressure lowering drugs
Answer to exercise: Blood pressure lowering drugs The data set bloodpressure.txt contains data from a cross-over trial, involving three different formulations of a drug for lowering of blood pressure:
More informationApproximations to Distributions of Test Statistics in Complex Mixed Linear Models Using SAS Proc MIXED
Paper 6-6 Approximations to Distributions of Test Statistics in Complex Mixed Linear Models Using SAS Proc MIXED G. Bruce Schaalje, Department of Statistics, Brigham Young University, Provo, UT Justin
More informationLECTURE 5 HYPOTHESIS TESTING
October 25, 2016 LECTURE 5 HYPOTHESIS TESTING Basic concepts In this lecture we continue to discuss the normal classical linear regression defined by Assumptions A1-A5. Let θ Θ R d be a parameter of interest.
More informationGeneral Linear Model (Chapter 4)
General Linear Model (Chapter 4) Outcome variable is considered continuous Simple linear regression Scatterplots OLS is BLUE under basic assumptions MSE estimates residual variance testing regression coefficients
More information13 Simple Linear Regression
B.Sc./Cert./M.Sc. Qualif. - Statistics: Theory and Practice 3 Simple Linear Regression 3. An industrial example A study was undertaken to determine the effect of stirring rate on the amount of impurity
More informationThe One-Way Repeated-Measures ANOVA. (For Within-Subjects Designs)
The One-Way Repeated-Measures ANOVA (For Within-Subjects Designs) Logic of the Repeated-Measures ANOVA The repeated-measures ANOVA extends the analysis of variance to research situations using repeated-measures
More informationTHE PEARSON CORRELATION COEFFICIENT
CORRELATION Two variables are said to have a relation if knowing the value of one variable gives you information about the likely value of the second variable this is known as a bivariate relation There
More informationCentral Limit Theorem ( 5.3)
Central Limit Theorem ( 5.3) Let X 1, X 2,... be a sequence of independent random variables, each having n mean µ and variance σ 2. Then the distribution of the partial sum S n = X i i=1 becomes approximately
More information:the actual population proportion are equal to the hypothesized sample proportions 2. H a
AP Statistics Chapter 14 Chi- Square Distribution Procedures I. Chi- Square Distribution ( χ 2 ) The chi- square test is used when comparing categorical data or multiple proportions. a. Family of only
More informationMultiple Linear Regression
Multiple Linear Regression University of California, San Diego Instructor: Ery Arias-Castro http://math.ucsd.edu/~eariasca/teaching.html 1 / 42 Passenger car mileage Consider the carmpg dataset taken from
More informationBios 6649: Clinical Trials - Statistical Design and Monitoring
Bios 6649: Clinical Trials - Statistical Design and Monitoring Spring Semester 2015 John M. Kittelson Department of Biostatistics & Informatics Colorado School of Public Health University of Colorado Denver
More informationOne-Way ANOVA. Some examples of when ANOVA would be appropriate include:
One-Way ANOVA 1. Purpose Analysis of variance (ANOVA) is used when one wishes to determine whether two or more groups (e.g., classes A, B, and C) differ on some outcome of interest (e.g., an achievement
More informationLecture 2: Linear Models. Bruce Walsh lecture notes Seattle SISG -Mixed Model Course version 23 June 2011
Lecture 2: Linear Models Bruce Walsh lecture notes Seattle SISG -Mixed Model Course version 23 June 2011 1 Quick Review of the Major Points The general linear model can be written as y = X! + e y = vector
More information10.2: The Chi Square Test for Goodness of Fit
10.2: The Chi Square Test for Goodness of Fit We can perform a hypothesis test to determine whether the distribution of a single categorical variable is following a proposed distribution. We call this
More information2 Hand-out 2. Dr. M. P. M. M. M c Loughlin Revised 2018
Math 403 - P. & S. III - Dr. McLoughlin - 1 2018 2 Hand-out 2 Dr. M. P. M. M. M c Loughlin Revised 2018 3. Fundamentals 3.1. Preliminaries. Suppose we can produce a random sample of weights of 10 year-olds
More informationGROUPED DATA E.G. FOR SAMPLE OF RAW DATA (E.G. 4, 12, 7, 5, MEAN G x / n STANDARD DEVIATION MEDIAN AND QUARTILES STANDARD DEVIATION
FOR SAMPLE OF RAW DATA (E.G. 4, 1, 7, 5, 11, 6, 9, 7, 11, 5, 4, 7) BE ABLE TO COMPUTE MEAN G / STANDARD DEVIATION MEDIAN AND QUARTILES Σ ( Σ) / 1 GROUPED DATA E.G. AGE FREQ. 0-9 53 10-19 4...... 80-89
More informationLogistic Regression. Continued Psy 524 Ainsworth
Logistic Regression Continued Psy 524 Ainsworth Equations Regression Equation Y e = 1 + A+ B X + B X + B X 1 1 2 2 3 3 i A+ B X + B X + B X e 1 1 2 2 3 3 Equations The linear part of the logistic regression
More informationLecture 3: Linear Models. Bruce Walsh lecture notes Uppsala EQG course version 28 Jan 2012
Lecture 3: Linear Models Bruce Walsh lecture notes Uppsala EQG course version 28 Jan 2012 1 Quick Review of the Major Points The general linear model can be written as y = X! + e y = vector of observed
More informationNotes for Week 13 Analysis of Variance (ANOVA) continued WEEK 13 page 1
Notes for Wee 13 Analysis of Variance (ANOVA) continued WEEK 13 page 1 Exam 3 is on Friday May 1. A part of one of the exam problems is on Predictiontervals : When randomly sampling from a normal population
More informationLecture 18 Miscellaneous Topics in Multiple Regression
Lecture 18 Miscellaneous Topics in Multiple Regression STAT 512 Spring 2011 Background Reading KNNL: 8.1-8.5,10.1, 11, 12 18-1 Topic Overview Polynomial Models (8.1) Interaction Models (8.2) Qualitative
More information