Stats 112/203: Sample Midterm Exam Solutions
|
|
- Lee Rose
- 7 years ago
- Views:
Transcription
1 Stats 112/203: Sample Midterm Exam Solutions 1. Consider the following linear mixed model: Y ij = β 0 + b 0i + (β 1 + b 1i )t ij + ɛ ij, where Y ij is the response measurement for the ith individual s jth occasion taken at time t ij. The coefficients β 0 and β 1 are fixed effects. Assume and ( b 0i ( b0i b 1i ) iid N N(0, σ 2 ), (( ) ( 0 σ 2, 0 σ 01 0 σ 01 σ1 2 ɛ ij iid b 1i ) and ɛij are independent. )), (a) Derive the marginal variance V ar(y ij ) and covariance Cov(Y ij, Y ik ) (j k). (b) What is the conditional variance V ar(y ij b 0i, b 1i )? (c) Suppose the observed responses for the first individual are Y 11 = 32, Y 12 = 24, Y 13 = 25, and Y 14 = 28, observed at days t 11 = 0, t 12 = 8, t 13 = 16, and t 14 = 30. Write the linear mixed model for this observation in matrix notation: Y i = X i β + Z i b i + ɛ i. (d) Suppose the true values of the fixed effects are β 0 = 20 and β 1 = 5. Draw a time plot that demonstrates Solution: (a) i the overall population mean response trajectory, ii a hypothetical random sample of three individual mean response trajectories, and iii the hypothetical observed response measurements for each of the three chosen individuals under each of the following conditions: 1. σ 2 0 = 0 and 0 < σ2 < σ σ 2 1 = 0 and 0 < σ2 < σ σ 2 = 0, σ 2 0 > 0, and σ2 1 > 0. V ar(y ij ) = Cov(β 0 + b 0i + (β 1 + b 1i )t ij + ɛ ij, β 0 + b 0i + (β 1 + b 1i )t ij + ɛ ij ) = Cov(b 0i + b 1i t ij + ɛ ij, b 0i + b 1i t ij + ɛ ij ) = V ar(b 0i ) + t 2 ijv ar(b 1i ) + V ar(ɛ ij ) + 2t ij Cov(b 0i, b 1i ) + 2Cov(b 0i, ɛ ij ) + 2t ij Cov(b 1i, ɛ ij ) = σ t 2 ijσ σ 2 + 2t ij σ = σ t 2 ijσ t ij σ 01 + σ 2 since b 0i and ɛ ij are independent, and b 1i and ɛ ij are independent. 1
2 Similarly, Cov(Y ij, Y ik ) = Cov(β 0 + b 0i + (β 1 + b 1i )t ij + ɛ ij, β 0 + b 0i + (β 1 + b 1i )t ik + ɛ ik ) = Cov(b 0i + b 1i t ij + ɛ ij, b 0i + b 1i t ik + ɛ ik ) = V ar(b 0i ) + t ij t ik V ar(b 1i ) + Cov(ɛ ij, ɛ ik ) + (t ij + t ik )Cov(b 0i, b 1i ) + Cov(b 0i, ɛ ij ) + Cov(b 0i, ɛ ik ) + t ij Cov(b 1i, ɛ ij ) + t ik Cov(b 1i, ɛ ik ) = σ0 2 + t ij t ik σ1 2 + (t ij + t ik )σ 01 since b 0i, ɛ ij and ɛ ik are independent; and b 1i, ɛ ij and ɛ ik are independent. (b) V ar(y ij b 0i, b 1i ) = V ar(ɛ ij ) = σ 2. (c) = [ β0 β 1 ] [ b01 b 11 ] ɛ 11 + ɛ 12 ɛ 13 ɛ 14 (d) σ 2 0 = 0 and 0 < σ2 < σ 2 1 : Problem 2(b)1. Y Pop. Mean Indiv. 1 Indiv. 2 Indiv Time 2
3 σ 2 1 = 0 and 0 < σ2 < σ 2 0 : Problem 2(b)2. Y Pop. Mean Indiv. 1 Indiv. 2 Indiv Time σ 2 = 0, σ 2 0 > 0, and σ2 1 > 0: Problem 2(b)3. Y Pop. Mean Indiv. 1 Indiv. 2 Indiv Time 3
4 2. Data were collected on N = 22 students by the Minneapolis Public School District in Minnesota to comply with federal accountability requirements, namely Title X of the No Child Left Behind Act of Reading achievement scores (Read) were measured in grades 5, 6, 7, and 8 for each student in the study. The predictor of interest is called Risk, which takes on the value DADV if the student was designated as disadvantaged (based on poverty or homeless measures), and ADV if the student was designated as advantaged. R code, output, and summary plots are in the Appendix. (a) Write out the equation of the mod1 fitted model for the subject-specific (conditional) mean response. Define any variables used. (b) Give an interpretation of the estimated Grade coefficient in mod1 in context of the problem. (c) Does the intercept for mod1 have a useful interpretation? If yes, give the interpretation. If no, explain why. (d) What is the estimated variance of the random slopes using mod1.reml? Provide an interpretation of the magnitude of this value in context of the problem. (e) Here are the predicted random effects (EBLUPs) for individual 1, who was classified as disadvantaged: > ranef(mod1.reml) (Intercept) Grade You may leave your answers to the following three questions unsimplified. i. What is the predicted subject-specific slope for individual 1? ii. What is the predicted mean reading score for individual 1 in grade 9? iii. What is the estimated marginal mean reading score for disadvantaged students in grade 9? (f) The R output includes a likelihood ratio test of mod2.reml vs. mod1.reml. State the null and alternative hypotheses being tested (in mathematical symbols), the p-value (report at least four digits), and state a conclusion of the test in context of the problem. (g) Write a short paragraph addressing the effect of being advantaged or disadvantaged on the changes in reading scores across grades. Include both an estimate of the effect as well as results from an appropriate test assessing statistical significance of the effect. 4
5 Solution: (a) where ˆ Y ij ˆb i = D i t ij t ij D i + ˆb 0i + ˆb 1i t ij Ŷij is the fitted mean reading achievement score of the ith individual in grade t ij, t ij is the grade level of individual i at the time of occasion j, and { 1 individual i is disadvantaged D i = 0 individual i is advantaged (b) For advantaged students, the estimated mean reading score increases by per grade level. (c) No. The intercept is the mean reading level at grade zero for advanted students, which is an extrapolation beyond the observed data (grades 5-8). (d) Vˆar(b 1i ) = (2.8595) 2 = Approximately 95% of advantaged students have an estimated mean change in reading score between a 1.15 decrease and a increase per grade level (4.570± 2(2.860)). (e) i = ii ( )(9) = iii ( )(9) = (f) We are testing the set of hypotheses H 0 : σ 2 1 = 0 H a : σ 2 1 > 0 where σ 2 = V ar(b 1i ). The p-value is (from a χ 2 (2) and χ 2 (1) mixture distribution), which is less than a significance level of 0.05, so we reject H 0. There is statistically significant evidence of individual variation in the mean change in read reading score per grade level. (g) Disadvantaged students have an estimated increase in mean reading score that is larger than the estimated increase in mean reading score for advantaged students per grade level. However, this difference between the two groups is not statistically significant (p-value = 0.710). 5
6 3. Consider a three-year longitudinal study of binge drinking among 100 college students, with a weekly indicator variable Y ij as the response variable (1 for any binge drinking episode that week; 0 otherwise). Covariates are time t ij in weeks, and membership in organized group housing X i (1 for yes, such as fraternity, sorority, or student cooperative; 0 otherwise). (Note that membership in organized group housing for an individual does not change over time.) One of the goals of the study is to assess changes in binge drinking behavior over time, and particularly whether changes over time are influenced by membership in group housing. (a) For the goal of this study, would a model for population-averaged effects or subject-specific effects be more appropriate? Explain. (b) Write out an appropriate generalized linear mixed model for this study, including any model assumptions. (c) Using your model from part (a), write out the null and alternative hypotheses that match the study goal of determining whether changes over time are influenced by membership in group housing in terms of the model parameters. Solution: (a) To assess within-subject changes in binge drinking behavior over time, a subjectspecific model would be more appropriate. For the question of whether changes over time are influenced by membership in group housing, a population-averaged model would be more appropriate since group membership does not change over time within an individual in this study. (b) Assume Y ij b i Bin(1, π ij ) where we model the conditional probability of any binge drinking episode in week j for individual i, π ij = E(Y ij b i ), as ( ) πij log = β 0 + b i + β 1 X i + β 2 t ij + β 3 X i t ij, 1 π ij where we assume b i iid N(0, σ 2 b ). (Note that the model must include the interaction term between X i and t ij in order to assess whether changes over time are influenced by membership in group housing.) (c) H 0 : β 3 = 0 vs. H a : β
7 Appendix: R code, output, and plots for Problem 2 Individual Reading Score Trajectories Mean Reading Score Trajectories by Group Reading Score Advantaged Disadvantaged Mean Reading Score Advantaged Disadvantaged Grade Grade > mod1 = lme(read ~ Risk*Grade, random = ~ 1+Grade ID, method="ml",...) > summary(mod1) Linear mixed-effects model fit by maximum likelihood... Fixed effects: Read ~ Risk * Grade Value Std.Error DF t-value p-value (Intercept) RiskDADV Grade RiskDADV:Grade > mod1.reml = lme(read ~ Risk*Grade, random = ~ 1+Grade ID, method="reml",...) > summary(mod1.reml) Linear mixed-effects model fit by REML... Random effects: Formula: ~1 + Grade ID Structure: General positive-definite, Log-Cholesky parametrization StdDev Corr (Intercept) (Intr) Grade Residual > mod2.reml = lme(read ~ Risk*Grade, random = ~ 1 ID, data=dat.long, method="reml", na.action=na.omit) > anova(mod2.reml,mod1.reml) Model df AIC BIC loglik Test L.Ratio p-value mod2.reml mod1.reml vs > pchibarsq( , 2, lower.tail=false) [1] > pchibarsq( , 1, lower.tail=false) [1]
E(y i ) = x T i β. yield of the refined product as a percentage of crude specific gravity vapour pressure ASTM 10% point ASTM end point in degrees F
Random and Mixed Effects Models (Ch. 10) Random effects models are very useful when the observations are sampled in a highly structured way. The basic idea is that the error associated with any linear,
More informationIntroducing the Multilevel Model for Change
Department of Psychology and Human Development Vanderbilt University GCM, 2010 1 Multilevel Modeling - A Brief Introduction 2 3 4 5 Introduction In this lecture, we introduce the multilevel model for change.
More informationApplied Statistics. J. Blanchet and J. Wadsworth. Institute of Mathematics, Analysis, and Applications EPF Lausanne
Applied Statistics J. Blanchet and J. Wadsworth Institute of Mathematics, Analysis, and Applications EPF Lausanne An MSc Course for Applied Mathematicians, Fall 2012 Outline 1 Model Comparison 2 Model
More informationANOVA. February 12, 2015
ANOVA February 12, 2015 1 ANOVA models Last time, we discussed the use of categorical variables in multivariate regression. Often, these are encoded as indicator columns in the design matrix. In [1]: %%R
More informationUse of deviance statistics for comparing models
A likelihood-ratio test can be used under full ML. The use of such a test is a quite general principle for statistical testing. In hierarchical linear models, the deviance test is mostly used for multiparameter
More informationMultilevel Modeling in R, Using the nlme Package
Multilevel Modeling in R, Using the nlme Package William T. Hoyt (University of Wisconsin-Madison) David A. Kenny (University of Connecticut) March 21, 2013 Supplement to Kenny, D. A., & Hoyt, W. (2009)
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby
More informationDEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9
DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationChapter 7: Simple linear regression Learning Objectives
Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -
More informationHighlights the connections between different class of widely used models in psychological and biomedical studies. Multiple Regression
GLMM tutor Outline 1 Highlights the connections between different class of widely used models in psychological and biomedical studies. ANOVA Multiple Regression LM Logistic Regression GLM Correlated data
More informationTime-Series Regression and Generalized Least Squares in R
Time-Series Regression and Generalized Least Squares in R An Appendix to An R Companion to Applied Regression, Second Edition John Fox & Sanford Weisberg last revision: 11 November 2010 Abstract Generalized
More informationModels for Longitudinal and Clustered Data
Models for Longitudinal and Clustered Data Germán Rodríguez December 9, 2008, revised December 6, 2012 1 Introduction The most important assumption we have made in this course is that the observations
More informationA Primer on Forecasting Business Performance
A Primer on Forecasting Business Performance There are two common approaches to forecasting: qualitative and quantitative. Qualitative forecasting methods are important when historical data is not available.
More informationIntroduction to Hierarchical Linear Modeling with R
Introduction to Hierarchical Linear Modeling with R 5 10 15 20 25 5 10 15 20 25 13 14 15 16 40 30 20 10 0 40 30 20 10 9 10 11 12-10 SCIENCE 0-10 5 6 7 8 40 30 20 10 0-10 40 1 2 3 4 30 20 10 0-10 5 10 15
More informationSAS Syntax and Output for Data Manipulation:
Psyc 944 Example 5 page 1 Practice with Fixed and Random Effects of Time in Modeling Within-Person Change The models for this example come from Hoffman (in preparation) chapter 5. We will be examining
More informationRegression Analysis: A Complete Example
Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty
More informationLinear Mixed-Effects Modeling in SPSS: An Introduction to the MIXED Procedure
Technical report Linear Mixed-Effects Modeling in SPSS: An Introduction to the MIXED Procedure Table of contents Introduction................................................................ 1 Data preparation
More informationAuxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus
Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 15 Version 8, August 5, 2014 1 Abstract This paper discusses alternatives
More informationPremaster Statistics Tutorial 4 Full solutions
Premaster Statistics Tutorial 4 Full solutions Regression analysis Q1 (based on Doane & Seward, 4/E, 12.7) a. Interpret the slope of the fitted regression = 125,000 + 150. b. What is the prediction for
More informationIntroduction to Longitudinal Data Analysis
Introduction to Longitudinal Data Analysis Longitudinal Data Analysis Workshop Section 1 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Section 1: Introduction
More informationChapter 13 Introduction to Linear Regression and Correlation Analysis
Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing
More informationWe extended the additive model in two variables to the interaction model by adding a third term to the equation.
Quadratic Models We extended the additive model in two variables to the interaction model by adding a third term to the equation. Similarly, we can extend the linear model in one variable to the quadratic
More informationSTATISTICA Formula Guide: Logistic Regression. Table of Contents
: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary
More informationRegression III: Advanced Methods
Lecture 16: Generalized Additive Models Regression III: Advanced Methods Bill Jacoby Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Goals of the Lecture Introduce Additive Models
More information5. Multiple regression
5. Multiple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/5 QBUS6840 Predictive Analytics 5. Multiple regression 2/39 Outline Introduction to multiple linear regression Some useful
More informationStatistics courses often teach the two-sample t-test, linear regression, and analysis of variance
2 Making Connections: The Two-Sample t-test, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the two-sample
More informationdata visualization and regression
data visualization and regression Sepal.Length 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 I. setosa I. versicolor I. virginica I. setosa I. versicolor I. virginica Species Species
More informationFamily economics data: total family income, expenditures, debt status for 50 families in two cohorts (A and B), annual records from 1990 1995.
Lecture 18 1. Random intercepts and slopes 2. Notation for mixed effects models 3. Comparing nested models 4. Multilevel/Hierarchical models 5. SAS versions of R models in Gelman and Hill, chapter 12 1
More information17. SIMPLE LINEAR REGRESSION II
17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.
More informationStatistical Models in R
Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Structure of models in R Model Assessment (Part IA) Anova
More informationInteraction between quantitative predictors
Interaction between quantitative predictors In a first-order model like the ones we have discussed, the association between E(y) and a predictor x j does not depend on the value of the other predictors
More informationWeek TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500 6 8480
1) The S & P/TSX Composite Index is based on common stock prices of a group of Canadian stocks. The weekly close level of the TSX for 6 weeks are shown: Week TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500
More informationAnalyzing Intervention Effects: Multilevel & Other Approaches. Simplest Intervention Design. Better Design: Have Pretest
Analyzing Intervention Effects: Multilevel & Other Approaches Joop Hox Methodology & Statistics, Utrecht Simplest Intervention Design R X Y E Random assignment Experimental + Control group Analysis: t
More informationHLM software has been one of the leading statistical packages for hierarchical
Introductory Guide to HLM With HLM 7 Software 3 G. David Garson HLM software has been one of the leading statistical packages for hierarchical linear modeling due to the pioneering work of Stephen Raudenbush
More informationIntroduction to Multilevel Modeling Using HLM 6. By ATS Statistical Consulting Group
Introduction to Multilevel Modeling Using HLM 6 By ATS Statistical Consulting Group Multilevel data structure Students nested within schools Children nested within families Respondents nested within interviewers
More informationLinear Models for Continuous Data
Chapter 2 Linear Models for Continuous Data The starting point in our exploration of statistical models in social research will be the classical linear model. Stops along the way include multiple linear
More information" Y. Notation and Equations for Regression Lecture 11/4. Notation:
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
More informationGeneral Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.
General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n
More informationNotating the Multilevel Longitudinal Model. Multilevel Modeling of Longitudinal Data. Notating (cont.) Notating (cont.)
Notating the Multilevel Modeling of Longitudinal Data Recall the typical -level model Y ij = γ 00 + (γ 0 + u j )X ij + u 0j + e ij Dr. J. Kyle Roberts Southern Methodist University Simmons School of Education
More informationPoisson Models for Count Data
Chapter 4 Poisson Models for Count Data In this chapter we study log-linear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the
More informationSimple linear regression
Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between
More informationSAS Software to Fit the Generalized Linear Model
SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling
More informationProfile analysis is the multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two cases:
Profile Analysis Introduction Profile analysis is the multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two cases: ) Comparing the same dependent variables
More informationTechnical report. in SPSS AN INTRODUCTION TO THE MIXED PROCEDURE
Linear mixedeffects modeling in SPSS AN INTRODUCTION TO THE MIXED PROCEDURE Table of contents Introduction................................................................3 Data preparation for MIXED...................................................3
More informationLongitudinal Data Analysis
Longitudinal Data Analysis Acknowledge: Professor Garrett Fitzmaurice INSTRUCTOR: Rino Bellocco Department of Statistics & Quantitative Methods University of Milano-Bicocca Department of Medical Epidemiology
More information5. Linear Regression
5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4
More informationStudy Guide for the Final Exam
Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make
More informationUsing SAS Proc Mixed for the Analysis of Clustered Longitudinal Data
Using SAS Proc Mixed for the Analysis of Clustered Longitudinal Data Kathy Welch Center for Statistical Consultation and Research The University of Michigan 1 Background ProcMixed can be used to fit Linear
More information2. Simple Linear Regression
Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according
More informationChapter 4 Models for Longitudinal Data
Chapter 4 Models for Longitudinal Data Longitudinal data consist of repeated measurements on the same subject (or some other experimental unit ) taken over time. Generally we wish to characterize the time
More informationLogit Models for Binary Data
Chapter 3 Logit Models for Binary Data We now turn our attention to regression models for dichotomous data, including logistic regression and probit analysis. These models are appropriate when the response
More informationChapter 15. Mixed Models. 15.1 Overview. A flexible approach to correlated data.
Chapter 15 Mixed Models A flexible approach to correlated data. 15.1 Overview Correlated data arise frequently in statistical analyses. This may be due to grouping of subjects, e.g., students within classrooms,
More informationChapter 5 Analysis of variance SPSS Analysis of variance
Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,
More information1. The parameters to be estimated in the simple linear regression model Y=α+βx+ε ε~n(0,σ) are: a) α, β, σ b) α, β, ε c) a, b, s d) ε, 0, σ
STA 3024 Practice Problems Exam 2 NOTE: These are just Practice Problems. This is NOT meant to look just like the test, and it is NOT the only thing that you should study. Make sure you know all the material
More information11. Analysis of Case-control Studies Logistic Regression
Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:
More informationHomework 11. Part 1. Name: Score: / null
Name: Score: / Homework 11 Part 1 null 1 For which of the following correlations would the data points be clustered most closely around a straight line? A. r = 0.50 B. r = -0.80 C. r = 0.10 D. There is
More informationResearch Methods & Experimental Design
Research Methods & Experimental Design 16.422 Human Supervisory Control April 2004 Research Methods Qualitative vs. quantitative Understanding the relationship between objectives (research question) and
More informationDescriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
More informationNotes on Applied Linear Regression
Notes on Applied Linear Regression Jamie DeCoster Department of Social Psychology Free University Amsterdam Van der Boechorststraat 1 1081 BT Amsterdam The Netherlands phone: +31 (0)20 444-8935 email:
More informationCorrelation and Simple Linear Regression
Correlation and Simple Linear Regression We are often interested in studying the relationship among variables to determine whether they are associated with one another. When we think that changes in a
More informationI L L I N O I S UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Beckman HLM Reading Group: Questions, Answers and Examples Carolyn J. Anderson Department of Educational Psychology I L L I N O I S UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Linear Algebra Slide 1 of
More informationNCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )
Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates
More informationStatistiek II. John Nerbonne. October 1, 2010. Dept of Information Science j.nerbonne@rug.nl
Dept of Information Science j.nerbonne@rug.nl October 1, 2010 Course outline 1 One-way ANOVA. 2 Factorial ANOVA. 3 Repeated measures ANOVA. 4 Correlation and regression. 5 Multiple regression. 6 Logistic
More informationModule 5: Multiple Regression Analysis
Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College
More informationGLM I An Introduction to Generalized Linear Models
GLM I An Introduction to Generalized Linear Models CAS Ratemaking and Product Management Seminar March 2009 Presented by: Tanya D. Havlicek, Actuarial Assistant 0 ANTITRUST Notice The Casualty Actuarial
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More informationSPSS Guide: Regression Analysis
SPSS Guide: Regression Analysis I put this together to give you a step-by-step guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar
More informationJoint models for classification and comparison of mortality in different countries.
Joint models for classification and comparison of mortality in different countries. Viani D. Biatat 1 and Iain D. Currie 1 1 Department of Actuarial Mathematics and Statistics, and the Maxwell Institute
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More informationMixed-effects regression and eye-tracking data
Mixed-effects regression and eye-tracking data Lecture 2 of advanced regression methods for linguists Martijn Wieling and Jacolien van Rij Seminar für Sprachwissenschaft University of Tübingen LOT Summer
More informationOverview. Longitudinal Data Variation and Correlation Different Approaches. Linear Mixed Models Generalized Linear Mixed Models
Overview 1 Introduction Longitudinal Data Variation and Correlation Different Approaches 2 Mixed Models Linear Mixed Models Generalized Linear Mixed Models 3 Marginal Models Linear Models Generalized Linear
More informationThis can dilute the significance of a departure from the null hypothesis. We can focus the test on departures of a particular form.
One-Degree-of-Freedom Tests Test for group occasion interactions has (number of groups 1) number of occasions 1) degrees of freedom. This can dilute the significance of a departure from the null hypothesis.
More informationIntroduction to Fixed Effects Methods
Introduction to Fixed Effects Methods 1 1.1 The Promise of Fixed Effects for Nonexperimental Research... 1 1.2 The Paired-Comparisons t-test as a Fixed Effects Method... 2 1.3 Costs and Benefits of Fixed
More informationChicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011
Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this
More informationSample Size Calculation for Longitudinal Studies
Sample Size Calculation for Longitudinal Studies Phil Schumm Department of Health Studies University of Chicago August 23, 2004 (Supported by National Institute on Aging grant P01 AG18911-01A1) Introduction
More informationData Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression
Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction
More informationNonlinear Regression Functions. SW Ch 8 1/54/
Nonlinear Regression Functions SW Ch 8 1/54/ The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/ But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/ Nonlinear Regression General
More informationSection Format Day Begin End Building Rm# Instructor. 001 Lecture Tue 6:45 PM 8:40 PM Silver 401 Ballerini
NEW YORK UNIVERSITY ROBERT F. WAGNER GRADUATE SCHOOL OF PUBLIC SERVICE Course Syllabus Spring 2016 Statistical Methods for Public, Nonprofit, and Health Management Section Format Day Begin End Building
More informationChapter 6: Multivariate Cointegration Analysis
Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration
More informationSection 13, Part 1 ANOVA. Analysis Of Variance
Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability
More information1.5 Oneway Analysis of Variance
Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments
More informationFinal Exam Practice Problem Answers
Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal
More informationANALYSIS OF TREND CHAPTER 5
ANALYSIS OF TREND CHAPTER 5 ERSH 8310 Lecture 7 September 13, 2007 Today s Class Analysis of trends Using contrasts to do something a bit more practical. Linear trends. Quadratic trends. Trends in SPSS.
More informationModule 5: Introduction to Multilevel Modelling SPSS Practicals Chris Charlton 1 Centre for Multilevel Modelling
Module 5: Introduction to Multilevel Modelling SPSS Practicals Chris Charlton 1 Centre for Multilevel Modelling Pre-requisites Modules 1-4 Contents P5.1 Comparing Groups using Multilevel Modelling... 4
More informationInternational Statistical Institute, 56th Session, 2007: Phil Everson
Teaching Regression using American Football Scores Everson, Phil Swarthmore College Department of Mathematics and Statistics 5 College Avenue Swarthmore, PA198, USA E-mail: peverso1@swarthmore.edu 1. Introduction
More informationOne-Way Analysis of Variance
One-Way Analysis of Variance Note: Much of the math here is tedious but straightforward. We ll skim over it in class but you should be sure to ask questions if you don t understand it. I. Overview A. We
More informationPart 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
More informationWeek 5: Multiple Linear Regression
BUS41100 Applied Regression Analysis Week 5: Multiple Linear Regression Parameter estimation and inference, forecasting, diagnostics, dummy variables Robert B. Gramacy The University of Chicago Booth School
More informationBasic Statistics and Data Analysis for Health Researchers from Foreign Countries
Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Volkert Siersma siersma@sund.ku.dk The Research Unit for General Practice in Copenhagen Dias 1 Content Quantifying association
More informationPart II. Multiple Linear Regression
Part II Multiple Linear Regression 86 Chapter 7 Multiple Regression A multiple linear regression model is a linear model that describes how a y-variable relates to two or more xvariables (or transformations
More informationIntroduction to Data Analysis in Hierarchical Linear Models
Introduction to Data Analysis in Hierarchical Linear Models April 20, 2007 Noah Shamosh & Frank Farach Social Sciences StatLab Yale University Scope & Prerequisites Strong applied emphasis Focus on HLM
More informationLab 5 Linear Regression with Within-subject Correlation. Goals: Data: Use the pig data which is in wide format:
Lab 5 Linear Regression with Within-subject Correlation Goals: Data: Fit linear regression models that account for within-subject correlation using Stata. Compare weighted least square, GEE, and random
More informationLecture Notes Module 1
Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific
More informationTesting Group Differences using T-tests, ANOVA, and Nonparametric Measures
Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:
More informationBiostatistics Short Course Introduction to Longitudinal Studies
Biostatistics Short Course Introduction to Longitudinal Studies Zhangsheng Yu Division of Biostatistics Department of Medicine Indiana University School of Medicine Zhangsheng Yu (Indiana University) Longitudinal
More informationElectronic Thesis and Dissertations UCLA
Electronic Thesis and Dissertations UCLA Peer Reviewed Title: A Multilevel Longitudinal Analysis of Teaching Effectiveness Across Five Years Author: Wang, Kairong Acceptance Date: 2013 Series: UCLA Electronic
More informationCORRELATIONAL ANALYSIS: PEARSON S r Purpose of correlational analysis The purpose of performing a correlational analysis: To discover whether there
CORRELATIONAL ANALYSIS: PEARSON S r Purpose of correlational analysis The purpose of performing a correlational analysis: To discover whether there is a relationship between variables, To find out the
More information7 Time series analysis
7 Time series analysis In Chapters 16, 17, 33 36 in Zuur, Ieno and Smith (2007), various time series techniques are discussed. Applying these methods in Brodgar is straightforward, and most choices are
More informationLeast Squares Estimation
Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David
More information