# Introduction to Multilevel Modeling Using HLM 6. By ATS Statistical Consulting Group

Save this PDF as:

Size: px
Start display at page:

Download "Introduction to Multilevel Modeling Using HLM 6. By ATS Statistical Consulting Group"

## Transcription

1 Introduction to Multilevel Modeling Using HLM 6 By ATS Statistical Consulting Group

2 Multilevel data structure Students nested within schools Children nested within families Respondents nested within interviewers Repeated measures nested within individuals longitudinal data, growth curve modeling In the example of student nested within schools: Level-1 variables, such as student s gender and age Level-2 variables, such as school type and size

3

4 How would we analyze such OLS regression multilevel data? OLS regression with robust standard error Aggregation Disaggregation Ecological fallacy interpreting analyses on aggregated data at the individual level

5 Ecological Fallacy See figure 3.1, on page 14 from Multilevel Analysis by Snijders and Bosker

6 Hierarchical linear model Random Intercept model Yij = β0j + rij β0j = γ00 + u0j Written in mixed model format: Yij = γ00 + u0j + rij i is for individuals and j is for schools β0j is the mean of Yij for school j γ00 is the average of all the β0j s, therefore the grand rij and u0j are normally distributed rij and u0j are independent of each other Parameters to be estimated include regression coefficients and variance components: γ00, var(rij) and var(u0j)

7 Hierarchical linear model Random Intercept and random slope model Yij = β0j + β1jx + rij β0j = γ00 + u0j β1j = γ10 + u1j Written in mixed model format: Yij = γ00 + γ10x + u0j + u1jx+ rij β0j is the mean of Yij for school j when X is zero β1j is the slope of X for school j (or the effect of X for school j) rij, u0j and u1j are normally distributed u0j and u1j are assumed to be correlated cross-level error terms are assumed to be independent parameters: γ00, γ10, var(u0j), var(u1j), cov(u0j, u1j) and var(rij)

8 Hierarchical linear model Random Intercept and random slope model Level-2 variable(s) to predict intercept and/or slope Yij = β0j + β1jx + rij β0j = γ00 + γ01w + u0j β1j = γ10 + γ11w + u1j Written in mixed model format: Yij = γ00 + γ01w + γ10x + γ11w*x + u0j + u1j*x + rij β0j is the mean of Yij for school j when X is zero β1j is the slope of X for school j (or the effect of X for school j) γ00 is the average intercept γ11 is the coefficient for the cross-level interaction term rij, u0j and u1j are normally distributed u0j and u1j are assumed to be correlated Cross-level error terms are assumed to be independent parameters to be estimated: γ00, γ01, γ10, γ11, var(u0j), var(u1j), cov(u0j, u1j) and var(rij)

9 Comparing the assumptions for hierarchical linear models with OLS models OLS Assumptions Linearity: function form is linear Normality: residuals are normally distributed Homoscedasticity: residual variance is constant Independence: observations are independent of each other HLM assumptions Linearity: function forms are linear at each level Normality: level-1 residuals are normally distributed and level-2 random effects u s have a multivariate normal distribution Homoscedasticity: level-1 residual variance is constant Independence: level-1 residuals and level-2 residuals are uncorrelated Independence: observations at highest level are independent of each other

10 Estimation Methods: REML vs. ML Reading: Section 4.6 Parameter Estimation from Snijder and Bosker REML and ML produce similar regression coefficients REML and ML differ in terms of estimating the variance components If the number of level-2 units is small, then ML variance estimates will be smaller than REML, leading to artificially short confidence interval and biased significant tests. REML is the default estimation method for HLM Likelihood ratio test for nested models When fixed effects are the same, model has fewer random effects, then both REML or ML may be used When one model has fewer fixed effects and possibly fewer random effects, then ML may be used

11 Issues with Centering Reading: Section 5.2 The effects of centering from Kreft and De Leeuw In OLS centering is to change the interpretation of the intercept Centering in HLM is not a simple issue Grand-mean centering The raw score model and the grand mean centered model are equivalent linear models. Group-mean centering Most of the times, the group mean centered model and the raw score model are neither equivalent in the fixed part nor in the random part. Combining substantive and statistical reasons in choosing raw score group-centering with reintroducing the means group-centering without reintroducing the means

12 An Example The dataset is a subsample from the 1982 High School and Beyond Survey and is used extensively in Hierarchical Linear Models by Raudenbush and Bryk. It consists of 7185 students nested in 160 schools. The outcome variable of interest is the student-level math achievement score, mathach. Predictor variables Level-1 (student level) predictor variables: ses: social-economic-status of a student female 0 = male and 1 = female Level-2 (school level) predictor variables: meanses: mean ses at school level, aggregated from student level schtype: type of school: 0 = public and 1 = private, there are 90 public and 70 private schools size: size of a school

13 Model Building Reading: Section 6.4 Model specification from Snijder and Bosker Unconditional model: mathachij = β0j + rij β0j = γ00 + u0j Random intercept model with level-2 predictor(s): mathachij = β0j + rij β0j = γ00 + γ01(meanses) + u0j Random intercept and random slope model: mathachij = β0j + β1j(ses) + rij β0j = γ00 + u0j β1j = γ10 + u1j Full model: mathachij = β0j + β1j(group_mean_centered_ses) + rij β0j = γ00 + γ01(schtype) + γ02(meanses) + u0j β1j = γ10 + γ11(schtype) + γ12(meanses) + u1j

14 Model 1: Unconditional Means Model mathachij = β0j + rij β0j = γ00 + u0j γ00 = var(rij) = var(u0j) = Rho = var(u0j)/(var(u0j) + var(rij)) = /( ) =

15 Final model mathachij = β0j + β1j(group_mean_centered_ses) + rij β0j = γ00 + γ01(schtype) + γ02(meanses) + u0j β1j = γ10 + γ11(schtype) + γ12(meanses) + u1j Tau INTRCPT1,B SES,B Final estimation of fixed effects (with robust standard errors) Standard Approx. Fixed Effect Coefficient Error T-ratio d.f. P-value For INTRCPT1, B0 INTRCPT2, G SCHTYPE, G MEANSES, G For SES slope, B1 INTRCPT2, G SCHTYPE, G MEANSES, G Final estimation of variance components: Random Effect Standard Variance df Chi-square P-value Deviation Component INTRCPT1, U SES slope, U level-1, R

16 Final Model (continued) mathachij = β0j + β1j(group_mean_centered_ses) + rij β0j = γ00 + γ01(schtype) + γ02(meanses) + u0j β1j = γ10 + γ11(schtype) + γ12(meanses) + u1j γ00 = : the intercept for public schools with meanses =0 (average ses) γ01 = 1.226: the change in intercept from a public school to a private school the intercept for private school with meanses = 0 is = γ02 = 5.333: the change in intercept for a one-unit change in meanses the intercept for public school with meanses = 1 is = γ10 = 2.94: the slope of gcses for public schools with meanses = 0. the effect of gcses for public schools with meanses = 0 is 2.94 γ11 = : the change in slope from a public school to a private school the effect of gcses for private schools with meanses = 0 is = γ12 = 1.034: the change in slope for a one-unit change in meanses the effect of gcses for public schools with meanses = 0 is 2.94 the effect of gcses for public schools with meanses = 1 is = For For INTRCPT1, B0 INTRCPT2, G SCHTYPE, G MEANSES, G SES slope, B1 INTRCPT2, G SCHTYPE, G MEANSES, G

17 The following paragraph is based on: What s new in HLM 6 HLM 6 greatly broadens the range of hierarchical models that can be estimated. It also offers greater convenience of use than previous versions. Here is a quick overview of key new features and options: All new graphical displays of data. Greater expanded graphics for fitted models. Model equations displayed in hierarchical or mixed-model format with or without subscripts - easy to save for publication. Distribution assumptions and link functions are presented in detail. Slightly different and easier way for specifying random effects. Cross-classified random effects models for linear models and non-linear link functions with convenient Windows interface. High-order Laplace approximation with EM algorithm for stable convergence and accurate estimation in two-level hierarchical generalized linear models (HGLM). Multinomial and ordinal models for three-level data. Also see the types of models. New flexible and accurate sample design weighting for two- and three-level HLMs and HGLMs. Easier automated input from a wide variety of software packages, including the current versions of SAS, SPSS, and STATA. Residual files can be saved directly as SPSS (*.sav) or STATA (*.dta) files. Analyses are based on MDM files, replacing the older less flexible SSM format.

18 Getting ready for using HLM software for multilevel data analysis Creating MDM file separate level-1 and level2 files for HLM2, or a single file original file can be in different format, such as SPSS, Stata and SAS linking variable can be either numeric or character variables in the analyses have to be numeric mdm file: binary file used for analyses and graphics mdmt file: template file in text format for creating mdm file hlm2mdm.sts: text file containing the summary statistics Data management HLM does not have data management capability One has to use other stat package(s) to clean the data and to create variables, such as dummy variables and within-level interaction terms HLM handles cross-level interactions nicely

19 Choosing preferences and other settings

20 Demo on using HLM Input Data and Creating the "MDM" file from a single SPSS file data-based graphs box-plot scatter plot Model Building unconditional means model regression with means-as-outcomes random-coefficient model intercepts and slopes-as-outcomes model Hypothesis Testing, Model Fit Multivariate hypothesis tests on fixed effects Multivariate Tests of variance-covariance components specification Model-based graphs Other Issues Modeling Heterogeneity of Level-1 Variances Models Without a Level-1 Intercept Constraints on Fixed Effects

21 References Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling by Tom Snijders and Roel Bosker Introduction to Multilevel Modeling by Ita Kreft and Jan de Leeuw Multilevel Analysis: Techniques and Applications by Joop Hox Hierarchical Linear Models, Second Edition by Stephen Raudenbush and Anthony Bryk HLM 6 - Hierarchical Linear and Nonlinear Modeling by Raudenbush et al.

### HLM 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

### Getting Started with HLM 5. For Windows

For Windows August 2012 Table of Contents Section 1: Overview... 3 1.1 About this Document... 3 1.2 Introduction to HLM... 3 1.3 Accessing HLM... 3 1.4 Getting Help with HLM... 4 Section 2: Accessing Data

### Perform hypothesis testing

Multivariate hypothesis tests for fixed effects Testing homogeneity of level-1 variances In the following sections, we use the model displayed in the figure below to illustrate the hypothesis tests. Partial

### Εισαγωγή στην πολυεπίπεδη μοντελοποίηση δεδομένων με το HLM. Βασίλης Παυλόπουλος Τμήμα Ψυχολογίας, Πανεπιστήμιο Αθηνών

Εισαγωγή στην πολυεπίπεδη μοντελοποίηση δεδομένων με το HLM Βασίλης Παυλόπουλος Τμήμα Ψυχολογίας, Πανεπιστήμιο Αθηνών Το υλικό αυτό προέρχεται από workshop που οργανώθηκε σε θερινό σχολείο της Ευρωπαϊκής

### Use 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

### Introduction 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

### CHAPTER 9 EXAMPLES: MULTILEVEL MODELING WITH COMPLEX SURVEY DATA

Examples: Multilevel Modeling With Complex Survey Data CHAPTER 9 EXAMPLES: MULTILEVEL MODELING WITH COMPLEX SURVEY DATA Complex survey data refers to data obtained by stratification, cluster sampling and/or

### Specifications for this HLM2 run

One way ANOVA model 1. How much do U.S. high schools vary in their mean mathematics achievement? 2. What is the reliability of each school s sample mean as an estimate of its true population mean? 3. Do

### An Introduction to Hierarchical Linear Modeling for Marketing Researchers

An Introduction to Hierarchical Linear Modeling for Marketing Researchers Barbara A. Wech and Anita L. Heck Organizations are hierarchical in nature. Specifically, individuals in the workplace are entrenched

### The Basic Two-Level Regression Model

2 The Basic Two-Level Regression Model The multilevel regression model has become known in the research literature under a variety of names, such as random coefficient model (de Leeuw & Kreft, 1986; Longford,

### Illustration (and the use of HLM)

Illustration (and the use of HLM) Chapter 4 1 Measurement Incorporated HLM Workshop The Illustration Data Now we cover the example. In doing so we does the use of the software HLM. In addition, we will

### HLM: A Gentle Introduction

HLM: A Gentle Introduction D. Betsy McCoach, Ph.D. Associate Professor, Educational Psychology Neag School of Education University of Connecticut Other names for HLMs Multilevel Linear Models incorporate

### Introducing 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.

### An introduction to hierarchical linear modeling

Tutorials in Quantitative Methods for Psychology 2012, Vol. 8(1), p. 52-69. An introduction to hierarchical linear modeling Heather Woltman, Andrea Feldstain, J. Christine MacKay, Meredith Rocchi University

### Two-Level Hierarchical Linear Models. Using SAS, Stata, HLM, R, SPSS, and Mplus

Using SAS, Stata, HLM, R, SPSS, and Mplus September 2012 Table of Contents Introduction... 3 Model Considerations... 3 Intraclass Correlation Coefficient... 4 Example Dataset... 4 Intercept-only Model

### Introduction 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

### Multilevel Modeling Tutorial. Using SAS, Stata, HLM, R, SPSS, and Mplus

Using SAS, Stata, HLM, R, SPSS, and Mplus Updated: March 2015 Table of Contents Introduction... 3 Model Considerations... 3 Intraclass Correlation Coefficient... 4 Example Dataset... 4 Intercept-only Model

### Electronic 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

### Introduction to Regression and Data Analysis

Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

### Hierarchical Linear Models

Hierarchical Linear Models Joseph Stevens, Ph.D., University of Oregon (541) 346-2445, stevensj@uoregon.edu Stevens, 2007 1 Overview and resources Overview Web site and links: www.uoregon.edu/~stevensj/hlm

### Mihaela Ene, Elizabeth A. Leighton, Genine L. Blue, Bethany A. Bell University of South Carolina

Paper 134-2014 Multilevel Models for Categorical Data using SAS PROC GLIMMIX: The Basics Mihaela Ene, Elizabeth A. Leighton, Genine L. Blue, Bethany A. Bell University of South Carolina ABSTRACT Multilevel

### Using Minitab for Regression Analysis: An extended example

Using Minitab for Regression Analysis: An extended example The following example uses data from another text on fertilizer application and crop yield, and is intended to show how Minitab can be used to

### , then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (

Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we

### Qualitative vs Quantitative research & Multilevel methods

Qualitative vs Quantitative research & Multilevel methods How to include context in your research April 2005 Marjolein Deunk Content What is qualitative analysis and how does it differ from quantitative

### Chapter 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

### Analyzing 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

### I 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

### Longitudinal Meta-analysis

Quality & Quantity 38: 381 389, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands. 381 Longitudinal Meta-analysis CORA J. M. MAAS, JOOP J. HOX and GERTY J. L. M. LENSVELT-MULDERS Department

### INTRODUCTORY STATISTICS

INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

### E205 Final: Version B

Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random

### Introduction 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

### When to Use Which Statistical Test

When to Use Which Statistical Test Rachel Lovell, Ph.D., Senior Research Associate Begun Center for Violence Prevention Research and Education Jack, Joseph, and Morton Mandel School of Applied Social Sciences

### Building Path Diagrams for Multilevel Models

Psychological Methods 2007, Vol. 12, No. 3, 283 297 Copyright 2007 by the American Psychological Association 1082-989X/07/\$12.00 DOI: 10.1037/1082-989X.12.3.283 Building Path Diagrams for Multilevel Models

### Data analysis process

Data analysis process Data collection and preparation Collect data Prepare codebook Set up structure of data Enter data Screen data for errors Exploration of data Descriptive Statistics Graphs Analysis

### Simple Linear Regression in SPSS STAT 314

Simple Linear Regression in SPSS STAT 314 1. Ten Corvettes between 1 and 6 years old were randomly selected from last year s sales records in Virginia Beach, Virginia. The following data were obtained,

### Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

### MISSING DATA TECHNIQUES WITH SAS. IDRE Statistical Consulting Group

MISSING DATA TECHNIQUES WITH SAS IDRE Statistical Consulting Group ROAD MAP FOR TODAY To discuss: 1. Commonly used techniques for handling missing data, focusing on multiple imputation 2. Issues that could

### Multiple Regression in SPSS STAT 314

Multiple Regression in SPSS STAT 314 I. The accompanying data is on y = profit margin of savings and loan companies in a given year, x 1 = net revenues in that year, and x 2 = number of savings and loan

### Techniques of Statistical Analysis II second group

Techniques of Statistical Analysis II second group Bruno Arpino Office: 20.182; Building: Jaume I bruno.arpino@upf.edu 1. Overview The course is aimed to provide advanced statistical knowledge for the

### Introduction to Structural Equation Modeling (SEM) Day 4: November 29, 2012

Introduction to Structural Equation Modeling (SEM) Day 4: November 29, 202 ROB CRIBBIE QUANTITATIVE METHODS PROGRAM DEPARTMENT OF PSYCHOLOGY COORDINATOR - STATISTICAL CONSULTING SERVICE COURSE MATERIALS

### Lecture 18 Linear Regression

Lecture 18 Statistics Unit Andrew Nunekpeku / Charles Jackson Fall 2011 Outline 1 1 Situation - used to model quantitative dependent variable using linear function of quantitative predictor(s). Situation

### Section 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

### NCSS 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

### The 3-Level HLM Model

James H. Steiger Department of Psychology and Human Development Vanderbilt University Regression Modeling, 2009 1 2 Basic Characteristics of the 3-level Model Level-1 Model Level-2 Model Level-3 Model

### Construct the MDM file for a HLM2 model using SPSS file input

Construct the MDM file for a HLM2 model using SPSS file input Level-1 file: For HS&B example data (distributed with the program), the level-1 file (hsb1.sav) has 7,185 cases and four variables (not including

### SPSS ADVANCED ANALYSIS WENDIANN SETHI SPRING 2011

SPSS ADVANCED ANALYSIS WENDIANN SETHI SPRING 2011 Statistical techniques to be covered Explore relationships among variables Correlation Regression/Multiple regression Logistic regression Factor analysis

### Power and sample size in multilevel modeling

Snijders, Tom A.B. Power and Sample Size in Multilevel Linear Models. In: B.S. Everitt and D.C. Howell (eds.), Encyclopedia of Statistics in Behavioral Science. Volume 3, 1570 1573. Chicester (etc.): Wiley,

### AN ILLUSTRATION OF MULTILEVEL MODELS FOR ORDINAL RESPONSE DATA

AN ILLUSTRATION OF MULTILEVEL MODELS FOR ORDINAL RESPONSE DATA Ann A. The Ohio State University, United States of America aoconnell@ehe.osu.edu Variables measured on an ordinal scale may be meaningful

### ICOTS6, 2002: O Connell HIERARCHICAL LINEAR MODELS FOR THE ANALYSIS OF LONGITUDINAL DATA WITH APPLICATIONS FROM HIV/AIDS PROGRAM EVALUATION

HIERARCHICAL LINEAR MODELS FOR THE ANALYSIS OF LONGITUDINAL DATA WITH APPLICATIONS FROM HIV/AIDS PROGRAM EVALUATION Ann A. O Connell University of Connecticut USA In this paper, two examples of multilevel

### Elements of statistics (MATH0487-1)

Elements of statistics (MATH0487-1) Prof. Dr. Dr. K. Van Steen University of Liège, Belgium December 10, 2012 Introduction to Statistics Basic Probability Revisited Sampling Exploratory Data Analysis -

### 11. 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:

### Using Hierarchical Linear Models to Measure Growth. Measurement Incorporated Hierarchical Linear Models Workshop

Using Hierarchical Linear Models to Measure Growth Measurement Incorporated Hierarchical Linear Models Workshop Chapter 6 Motivation Linear Growth Curves Quadratic Growth Curves Some other Growth Curves

### Silvermine House Steenberg Office Park, Tokai 7945 Cape Town, South Africa Telephone: +27 21 702 4666 www.spss-sa.com

SPSS-SA Silvermine House Steenberg Office Park, Tokai 7945 Cape Town, South Africa Telephone: +27 21 702 4666 www.spss-sa.com SPSS-SA Training Brochure 2009 TABLE OF CONTENTS 1 SPSS TRAINING COURSES FOCUSING

### Module 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

### The Latent Variable Growth Model In Practice. Individual Development Over Time

The Latent Variable Growth Model In Practice 37 Individual Development Over Time y i = 1 i = 2 i = 3 t = 1 t = 2 t = 3 t = 4 ε 1 ε 2 ε 3 ε 4 y 1 y 2 y 3 y 4 x η 0 η 1 (1) y ti = η 0i + η 1i x t + ε ti

### Structural Equation Models for Comparing Dependent Means and Proportions. Jason T. Newsom

Structural Equation Models for Comparing Dependent Means and Proportions Jason T. Newsom How to Do a Paired t-test with Structural Equation Modeling Jason T. Newsom Overview Rationale Structural equation

### Oklahoma School Grades: Hiding Poor Achievement

Oklahoma School Grades: Hiding Poor Achievement Technical Addendum The Oklahoma Center for Education Policy (University of Oklahoma) and The Center for Educational Research and Evaluation (Oklahoma State

### Analyzing Data from Small N Designs Using Multilevel Models: A Procedural Handbook

Analyzing Data from Small N Designs Using Multilevel Models: A Procedural Handbook Eden Nagler, M.Phil The Graduate Center, CUNY David Rindskopf, PhD Co-Principal Investigator The Graduate Center, CUNY

### Regression III: Dummy Variable Regression

Regression III: Dummy Variable Regression Tom Ilvento FREC 408 Linear Regression Assumptions about the error term Mean of Probability Distribution of the Error term is zero Probability Distribution of

### REGRESSION LINES IN STATA

REGRESSION LINES IN STATA THOMAS ELLIOTT 1. Introduction to Regression Regression analysis is about eploring linear relationships between a dependent variable and one or more independent variables. Regression

### CHAPTER 3 EXAMPLES: REGRESSION AND PATH ANALYSIS

Examples: Regression And Path Analysis CHAPTER 3 EXAMPLES: REGRESSION AND PATH ANALYSIS Regression analysis with univariate or multivariate dependent variables is a standard procedure for modeling relationships

### CHAPTER 6 EXAMPLES: GROWTH MODELING AND SURVIVAL ANALYSIS

Examples: Growth Modeling And Survival Analysis CHAPTER 6 EXAMPLES: GROWTH MODELING AND SURVIVAL ANALYSIS Growth models examine the development of individuals on one or more outcome variables over time.

### Moderation. Moderation

Stats - Moderation Moderation A moderator is a variable that specifies conditions under which a given predictor is related to an outcome. The moderator explains when a DV and IV are related. Moderation

### Assignments Analysis of Longitudinal data: a multilevel approach

Assignments Analysis of Longitudinal data: a multilevel approach Frans E.S. Tan Department of Methodology and Statistics University of Maastricht The Netherlands Maastricht, Jan 2007 Correspondence: Frans

### Simple Predictive Analytics Curtis Seare

Using Excel to Solve Business Problems: Simple Predictive Analytics Curtis Seare Copyright: Vault Analytics July 2010 Contents Section I: Background Information Why use Predictive Analytics? How to use

: 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

### Regression 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

### Binary Logistic Regression

Binary Logistic Regression Main Effects Model Logistic regression will accept quantitative, binary or categorical predictors and will code the latter two in various ways. Here s a simple model including

### Using multilevel models to analyze couple and family treatment data: Basic and advanced issues. David C. Atkins. Travis Research Institute

Multilevel analyses 1 RUNNING HEAD: Multilevel analyses Using multilevel models to analyze couple and family treatment data: Basic and advanced issues David C. Atkins Travis Research Institute Fuller Graduate

### Semester 1 Statistics Short courses

Semester 1 Statistics Short courses Course: STAA0001 Basic Statistics Blackboard Site: STAA0001 Dates: Sat. March 12 th and Sat. April 30 th (9 am 5 pm) Assumed Knowledge: None Course Description Statistical

### Estimating Multilevel Models using SPSS, Stata, SAS, and R. JeremyJ.Albright and Dani M. Marinova. July 14, 2010

Estimating Multilevel Models using SPSS, Stata, SAS, and R JeremyJ.Albright and Dani M. Marinova July 14, 2010 1 Multilevel data are pervasive in the social sciences. 1 Students may be nested within schools,

### Simple Linear Regression

Inference for Regression Simple Linear Regression IPS Chapter 10.1 2009 W.H. Freeman and Company Objectives (IPS Chapter 10.1) Simple linear regression Statistical model for linear regression Estimating

### Multivariate Normal Distribution

Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

### Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

### Regression analysis in practice with GRETL

Regression analysis in practice with GRETL Prerequisites You will need the GNU econometrics software GRETL installed on your computer (http://gretl.sourceforge.net/), together with the sample files that

### Directions for using SPSS

Directions for using SPSS Table of Contents Connecting and Working with Files 1. Accessing SPSS... 2 2. Transferring Files to N:\drive or your computer... 3 3. Importing Data from Another File Format...

### Module 3: Multiple Regression Concepts

Contents Module 3: Multiple Regression Concepts Fiona Steele 1 Centre for Multilevel Modelling...4 What is Multiple Regression?... 4 Motivation... 4 Conditioning... 4 Data for multiple regression analysis...

### Class 6: Chapter 12. Key Ideas. Explanatory Design. Correlational Designs

Class 6: Chapter 12 Correlational Designs l 1 Key Ideas Explanatory and predictor designs Characteristics of correlational research Scatterplots and calculating associations Steps in conducting a correlational

### Lecture 16: Logistic regression diagnostics, splines and interactions. Sandy Eckel 19 May 2007

Lecture 16: Logistic regression diagnostics, splines and interactions Sandy Eckel seckel@jhsph.edu 19 May 2007 1 Logistic Regression Diagnostics Graphs to check assumptions Recall: Graphing was used to

### HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

### Individual Growth Analysis Using PROC MIXED Maribeth Johnson, Medical College of Georgia, Augusta, GA

Paper P-702 Individual Growth Analysis Using PROC MIXED Maribeth Johnson, Medical College of Georgia, Augusta, GA ABSTRACT Individual growth models are designed for exploring longitudinal data on individuals

### Modern Methods for Missing Data

Modern Methods for Missing Data Paul D. Allison, Ph.D. Statistical Horizons LLC www.statisticalhorizons.com 1 Introduction Missing data problems are nearly universal in statistical practice. Last 25 years

### Simple Linear Regression One Binary Categorical Independent Variable

Simple Linear Regression Does sex influence mean GCSE score? In order to answer the question posed above, we want to run a linear regression of sgcseptsnew against sgender, which is a binary categorical

### Lecture - 32 Regression Modelling Using SPSS

Applied Multivariate Statistical Modelling Prof. J. Maiti Department of Industrial Engineering and Management Indian Institute of Technology, Kharagpur Lecture - 32 Regression Modelling Using SPSS (Refer

### Multilevel models: Basics and extensions

Chapter 2 Multilevel models: Basics and extensions 21 Introduction: Hierarchically structured data This thesis focuses on regional labour markets within Europe Since regions within countries are likely

### Practice 3 SPSS. Partially based on Notes from the University of Reading:

Practice 3 SPSS Partially based on Notes from the University of Reading: http://www.reading.ac.uk Simple Linear Regression A simple linear regression model is fitted when you want to investigate whether

### Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C

### We 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

### SELF-TEST: SIMPLE REGRESSION

ECO 22000 McRAE SELF-TEST: SIMPLE REGRESSION Note: Those questions indicated with an (N) are unlikely to appear in this form on an in-class examination, but you should be able to describe the procedures

### Handling attrition and non-response in longitudinal data

Longitudinal and Life Course Studies 2009 Volume 1 Issue 1 Pp 63-72 Handling attrition and non-response in longitudinal data Harvey Goldstein University of Bristol Correspondence. Professor H. Goldstein

### Linear 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

### e = random error, assumed to be normally distributed with mean 0 and standard deviation σ

1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.

### What is correlational research?

Key Ideas Purpose and use of correlational designs How correlational research developed Types of correlational designs Key characteristics of correlational designs Procedures used in correlational studies

### Linear Models and Conjoint Analysis with Nonlinear Spline Transformations

Linear Models and Conjoint Analysis with Nonlinear Spline Transformations Warren F. Kuhfeld Mark Garratt Abstract Many common data analysis models are based on the general linear univariate model, including

### Statistics in Retail Finance. Chapter 2: Statistical models of default

Statistics in Retail Finance 1 Overview > We consider how to build statistical models of default, or delinquency, and how such models are traditionally used for credit application scoring and decision

### Chapter 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) -

### Models 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

### Residuals. Residuals = ª Department of ISM, University of Alabama, ST 260, M23 Residuals & Minitab. ^ e i = y i - y i

A continuation of regression analysis Lesson Objectives Continue to build on regression analysis. Learn how residual plots help identify problems with the analysis. M23-1 M23-2 Example 1: continued Case