Consequences of measurement error. Psychology 588: Covariance structure and factor models
|
|
- Jewel Stevens
- 5 years ago
- Views:
Transcription
1 Consequences of measurement error Psychology 588: Covariance structure and factor models
2 Scaling indeterminacy of latent variables Scale of a latent variable is arbitrary and determined by a convention for convenience Typically set to variance of one (factor analysis convention) or to be identical to an arbitrarily chosen indicator s scale By centering indicator variables, we set latent variables means to zero Consider the following transformation: x j J a b b j j j j,,...,,, 0 a j j j j b b
3 If all J indicators are considered simultaneously, vector notation is more convenient: x νλ δ, a b a ν λ λ δ b b meaning that the linear transformation of ξ can be exactly compensated in the accordingly transformed ν = ν λa/b and λ = λ/b, leaving the errors δ unchanged (i.e., same fit)
4 What s great about measurement errors in equation 4 Regression weights and correlations are interpreted, implicitly assuming that the operationally defined variables involve no measurement error --- hardly realized for theoretical constructs (e.g., self esteem, IQ, etc.) Ignoring the measurement error will lead to inconsistent estimates We will see consequences of ignoring measurement errors
5 Univariate consequences 5 Consider a mean-included equation for X (hours worked per week) to indicate ξ (achievement motivation): X, E, E 0, E 0 E X X var X var Given only one indicator per latent variable, the intercept and loading (i.e., weight) are simply scaling constants for ξ However, if the ξ scale is set comparable to the X scale (i.e., λ = ), we see that var(x) is an over-estimation of ϕ = var(ξ) if δ is not included in the equation
6 Bivariate relation and simple regression 6 True data structure: x y η: job satisfaction y: satisfaction scale xi x d gamma eta y e zeta cov(x, y) is unbiased estimate of cov(ξ, η) with λ = λ =, since no other variables (δ and ε) can explain cov(x, y) cov, cov, cov xy, cov,
7 From the previous equations, and by analogy with y = γ x + ζ if measurement errors are ignored, cov, cov xy, var x var The parenthesized ratio (reliability) becomes only with no measurement error; otherwise, γ is an attenuated estimate of γ and s s is an inconsistent estimator of γ, ˆ xy xx If the bias of regression weight has an additional factor as --- but such scaling is unusual xx when there is only one indicator per latent variable xx
8 Correlations: cov, var var xy cov xy, var x var y var x var y var var x var y var xx yy which shows an attenuation of the true correlation due to measurement error, with the familiar correction formula: xy xx yy
9 Consequences in multiple regression 9 True data structure: γξ x ξδ y with Λ x = I and λ y = d d d3 x x x3 xi xi xi3 g g g3 eta zeta y e Ignoring measurement errors: y γ x σ cov ξ, cov ξ, ξγ Φγ σ cov x, y cov ξδ, ξγ Φγ xy
10 γ Φ σ and by analogy with γ x, γ Σxxσ xy ΣxxΦγ Φ Θ Φγ y Without measurement error (Θ δ = 0), γ γ ; otherwise, γ γ Alternatively written: since --- where i Σ Σ xx x γ is the OLS estimator of B in i.e., regression weights for prediction of ξ by x Σ Σ γ xx x Σ Φ x ξ Bx e, Again, without measurement error, Σ Σ xx x I Note: in Bollen (pp ), are meant to be respectively, for the multiple regression model ΓΣ,, Σ γσ,,, xy σxy
11 As a very simplified case, suppose x is the only fallible as: x x, i,..., q i i with the true and estimated regression equations: x q q q In this special case, the regression weight matrix has a simple multiplicative form of bias (hint: use ): Σ Σ Φ Θ Φ I c 0 xx x q q c q Φ Φ Θ c 0 I,
12 Consequently, resulting bias factors are: i b x q b,,, i q i x i Bias-factor for x is less than in absolute value ( without measurement error), and so is biased toward the bias factor indicates regression weight b in ξ = b 0 + b x + b ξ + + b q ξ q ~ i b x q Consequences for x i, i =,,q are additive, depending on relationships between ξ and ξ i holding all other IVs constant, and γ
13 So far all reasoning is based on rather unrealistic assumptions: Only single indicator per latent variable, and so its loading becomes simply scaling constant Only one fallible IV Without such assumptions (e.g., all IVs fallible), consequences of measurement error become too complicated and hard to simplify algebraically --- no particular simple form of aσ Σ One clear conclusion: all estimates are inconsistent --- systematically different from what they meant to be xx x
14 Consequence in standardization: standardized ii var i i i var var Consequence in SMC is similar to the bivariate case: plim R plimr What should we do with essentially omnipresent measurement error? Use SEM which allows for measurement errors in the model --- though we are limited in certain models regarding the model identification (e.g., Table 5., p. 64)
15 Correlated errors of measurement 5 Consequence in regression weights further complicated: γ Σ Σ γ Σ σ xx x xx For simple regression: Now, is not necessarily < γ cov, xx var x If correlated measurement errors are only within IVs (i.e., σ δε = 0, Σ xx = Φ + Θ σ where Θ σ is not diagonal), γ Σ Σ γ still holds (but the bias factor will have a more complicated form, also involving off-diagonal entries of Θ σ ) xx x
16 With multi-equations 6 In path models with sequential causal paths, consequences of measurement errors very hard to simply generalize --- see the union sentiment (Fig. 5., p. 69) and SES (Fig. 5.4, p. 73) examples If reliabilities are known, the corresponding error variances can be constrained; if unknown, the error variances may be modeled as free parameters provided that they are identifiable To keep in mind: we need more than one indicator per latent variable for identifiability and statistical testing --- leading to measurement models with multiple indicators or CFA
General structural model Part 1: Covariance structure and identification. Psychology 588: Covariance structure and factor models
General structural model Part 1: Covariance structure and identification Psychology 588: Covariance structure and factor models Latent variables 2 Interchangeably used: constructs --- substantively defined
More informationProblem Set #6: OLS. Economics 835: Econometrics. Fall 2012
Problem Set #6: OLS Economics 835: Econometrics Fall 202 A preliminary result Suppose we have a random sample of size n on the scalar random variables (x, y) with finite means, variances, and covariance.
More informationSEM with observed variables: parameterization and identification. Psychology 588: Covariance structure and factor models
SEM with observed variables: parameterization and identification Psychology 588: Covariance structure and factor models Limitations of SEM as a causal modeling 2 If an SEM model reflects the reality, the
More informationSEM 2: Structural Equation Modeling
SEM 2: Structural Equation Modeling Week 1 - Causal modeling and SEM Sacha Epskamp 18-04-2017 Course Overview Mondays: Lecture Wednesdays: Unstructured practicals Three assignments First two 20% of final
More information4. Distributions of Functions of Random Variables
4. Distributions of Functions of Random Variables Setup: Consider as given the joint distribution of X 1,..., X n (i.e. consider as given f X1,...,X n and F X1,...,X n ) Consider k functions g 1 : R n
More informationSEM 2: Structural Equation Modeling
SEM 2: Structural Equation Modeling Week 2 - Causality and equivalent models Sacha Epskamp 15-05-2018 Covariance Algebra Let Var(x) indicate the variance of x and Cov(x, y) indicate the covariance between
More informationStructural Equation Modeling An Econometrician s Introduction
S Structural Equation Modeling An Econometrician s Introduction PD Dr. Stefan Klößner Winter Term 2016/17 U N I V E R S I T A S A R A V I E N I S S Overview NOT: easy to digest introduction for practitioners
More informationMeasurement Theory. Reliability. Error Sources. = XY r XX. r XY. r YY
Y -3 - -1 0 1 3 X Y -10-5 0 5 10 X Measurement Theory t & X 1 X X 3 X k Reliability e 1 e e 3 e k 1 The Big Picture Measurement error makes it difficult to identify the true patterns of relationships between
More informationECON Introductory Econometrics. Lecture 6: OLS with Multiple Regressors
ECON4150 - Introductory Econometrics Lecture 6: OLS with Multiple Regressors Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 6 Lecture outline 2 Violation of first Least Squares assumption
More informationOnline Appendices for: Modeling Latent Growth With Multiple Indicators: A Comparison of Three Approaches
Online Appendices for: Modeling Latent Growth With Multiple Indicators: A Comparison of Three Approaches Jacob Bishop and Christian Geiser Utah State University David A. Cole Vanderbilt University Contents
More informationEstimation of Parameters
CHAPTER Probability, Statistics, and Reliability for Engineers and Scientists FUNDAMENTALS OF STATISTICAL ANALYSIS Second Edition A. J. Clark School of Engineering Department of Civil and Environmental
More informationConfirmatory Factor Analysis: Model comparison, respecification, and more. Psychology 588: Covariance structure and factor models
Confirmatory Factor Analysis: Model comparison, respecification, and more Psychology 588: Covariance structure and factor models Model comparison 2 Essentially all goodness of fit indices are descriptive,
More informationFinal Exam. Economics 835: Econometrics. Fall 2010
Final Exam Economics 835: Econometrics Fall 2010 Please answer the question I ask - no more and no less - and remember that the correct answer is often short and simple. 1 Some short questions a) For each
More informationTHE GENERAL STRUCTURAL EQUATION MODEL WITH LATENT VARIATES
THE GENERAL STRUCTURAL EQUATION MODEL WITH LATENT VARIATES I. Specification: A full structural equation model with latent variables consists of two parts: a latent variable model (which specifies the relations
More informationFactor Analysis. Qian-Li Xue
Factor Analysis Qian-Li Xue Biostatistics Program Harvard Catalyst The Harvard Clinical & Translational Science Center Short course, October 7, 06 Well-used latent variable models Latent variable scale
More informationVariance. Standard deviation VAR = = value. Unbiased SD = SD = 10/23/2011. Functional Connectivity Correlation and Regression.
10/3/011 Functional Connectivity Correlation and Regression Variance VAR = Standard deviation Standard deviation SD = Unbiased SD = 1 10/3/011 Standard error Confidence interval SE = CI = = t value for
More informationInverse of a Square Matrix. For an N N square matrix A, the inverse of A, 1
Inverse of a Square Matrix For an N N square matrix A, the inverse of A, 1 A, exists if and only if A is of full rank, i.e., if and only if no column of A is a linear combination 1 of the others. A is
More informationFirst Year Examination Department of Statistics, University of Florida
First Year Examination Department of Statistics, University of Florida August 19, 010, 8:00 am - 1:00 noon Instructions: 1. You have four hours to answer questions in this examination.. You must show your
More informationRandom Vectors, Random Matrices, and Matrix Expected Value
Random Vectors, Random Matrices, and Matrix Expected Value James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) 1 / 16 Random Vectors,
More informationRegression and correlation. Correlation & Regression, I. Regression & correlation. Regression vs. correlation. Involve bivariate, paired data, X & Y
Regression and correlation Correlation & Regression, I 9.07 4/1/004 Involve bivariate, paired data, X & Y Height & weight measured for the same individual IQ & exam scores for each individual Height of
More informationFinancial Econometrics
Financial Econometrics Estimation and Inference Gerald P. Dwyer Trinity College, Dublin January 2013 Who am I? Visiting Professor and BB&T Scholar at Clemson University Federal Reserve Bank of Atlanta
More informationEC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix)
1 EC212: Introduction to Econometrics Review Materials (Wooldridge, Appendix) Taisuke Otsu London School of Economics Summer 2018 A.1. Summation operator (Wooldridge, App. A.1) 2 3 Summation operator For
More informationIntroduction to Structural Equations
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationECON 3150/4150, Spring term Lecture 7
ECON 3150/4150, Spring term 2014. Lecture 7 The multivariate regression model (I) Ragnar Nymoen University of Oslo 4 February 2014 1 / 23 References to Lecture 7 and 8 SW Ch. 6 BN Kap 7.1-7.8 2 / 23 Omitted
More informationGaussian Processes. Le Song. Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012
Gaussian Processes Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 01 Pictorial view of embedding distribution Transform the entire distribution to expected features Feature space Feature
More informationDiscussion of Sensitivity and Informativeness under Local Misspecification
Discussion of Sensitivity and Informativeness under Local Misspecification Jinyong Hahn April 4, 2019 Jinyong Hahn () Discussion of Sensitivity and Informativeness under Local Misspecification April 4,
More informationBayesian Analysis of Latent Variable Models using Mplus
Bayesian Analysis of Latent Variable Models using Mplus Tihomir Asparouhov and Bengt Muthén Version 2 June 29, 2010 1 1 Introduction In this paper we describe some of the modeling possibilities that are
More informationECON The Simple Regression Model
ECON 351 - The Simple Regression Model Maggie Jones 1 / 41 The Simple Regression Model Our starting point will be the simple regression model where we look at the relationship between two variables In
More informationThe returns to schooling, ability bias, and regression
The returns to schooling, ability bias, and regression Jörn-Steffen Pischke LSE October 4, 2016 Pischke (LSE) Griliches 1977 October 4, 2016 1 / 44 Counterfactual outcomes Scholing for individual i is
More informationState-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Fin. Econometrics / 53
State-space Model Eduardo Rossi University of Pavia November 2014 Rossi State-space Model Fin. Econometrics - 2014 1 / 53 Outline 1 Motivation 2 Introduction 3 The Kalman filter 4 Forecast errors 5 State
More informationExploring Cultural Differences with Structural Equation Modelling
Exploring Cultural Differences with Structural Equation Modelling Wynne W. Chin University of Calgary and City University of Hong Kong 1996 IS Cross Cultural Workshop slide 1 The objectives for this presentation
More informationSimple Linear Regression Analysis
LINEAR REGRESSION ANALYSIS MODULE II Lecture - 6 Simple Linear Regression Analysis Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Prediction of values of study
More informationPsychology 282 Lecture #3 Outline
Psychology 8 Lecture #3 Outline Simple Linear Regression (SLR) Given variables,. Sample of n observations. In study and use of correlation coefficients, and are interchangeable. In regression analysis,
More informationEconometrics. 7) Endogeneity
30C00200 Econometrics 7) Endogeneity Timo Kuosmanen Professor, Ph.D. http://nomepre.net/index.php/timokuosmanen Today s topics Common types of endogeneity Simultaneity Omitted variables Measurement errors
More informationCopula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011
Copula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011 Outline Ordinary Least Squares (OLS) Regression Generalized Linear Models
More informationA Study of Statistical Power and Type I Errors in Testing a Factor Analytic. Model for Group Differences in Regression Intercepts
A Study of Statistical Power and Type I Errors in Testing a Factor Analytic Model for Group Differences in Regression Intercepts by Margarita Olivera Aguilar A Thesis Presented in Partial Fulfillment of
More informationLecture 9 SLR in Matrix Form
Lecture 9 SLR in Matrix Form STAT 51 Spring 011 Background Reading KNNL: Chapter 5 9-1 Topic Overview Matrix Equations for SLR Don t focus so much on the matrix arithmetic as on the form of the equations.
More informationSingle-Equation GMM: Endogeneity Bias
Single-Equation GMM: Lecture for Economics 241B Douglas G. Steigerwald UC Santa Barbara January 2012 Initial Question Initial Question How valuable is investment in college education? economics - measure
More informationEconometrics. Week 6. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 6 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 21 Recommended Reading For the today Advanced Panel Data Methods. Chapter 14 (pp.
More informationAN INTRODUCTION TO STRUCTURAL EQUATION MODELING
AN INTRODUCTION TO STRUCTURAL EQUATION MODELING Giorgio Russolillo CNAM, Paris giorgio.russolillo@cnam.fr Structural Equation Modeling (SEM) Structural Equation Models (SEM) are complex models allowing
More informationOmitted Variable Bias, Coefficient Stability and Other Issues. Chase Potter 6/22/16
Omitted Variable Bias, Coefficient Stability and Other Issues Chase Potter 6/22/16 Roadmap 1. Omitted Variable Bias 2. Coefficient Stability Oster 2015 Paper 3. Poorly Measured Confounders Pischke and
More informationMultiple Linear Regression
Multiple Linear Regression Asymptotics Asymptotics Multiple Linear Regression: Assumptions Assumption MLR. (Linearity in parameters) Assumption MLR. (Random Sampling from the population) We have a random
More informationConfirmatory Factor Analysis. Psych 818 DeShon
Confirmatory Factor Analysis Psych 818 DeShon Purpose Takes factor analysis a few steps further. Impose theoretically interesting constraints on the model and examine the resulting fit of the model with
More information6348 Final, Fall 14. Closed book, closed notes, no electronic devices. Points (out of 200) in parentheses.
6348 Final, Fall 14. Closed book, closed notes, no electronic devices. Points (out of 200) in parentheses. 0 11 1 1.(5) Give the result of the following matrix multiplication: 1 10 1 Solution: 0 1 1 2
More informationEconometrics Summary Algebraic and Statistical Preliminaries
Econometrics Summary Algebraic and Statistical Preliminaries Elasticity: The point elasticity of Y with respect to L is given by α = ( Y/ L)/(Y/L). The arc elasticity is given by ( Y/ L)/(Y/L), when L
More informationDependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline.
Practitioner Course: Portfolio Optimization September 10, 2008 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y ) (x,
More informationSteps in Regression Analysis
MGMG 522 : Session #2 Learning to Use Regression Analysis & The Classical Model (Ch. 3 & 4) 2-1 Steps in Regression Analysis 1. Review the literature and develop the theoretical model 2. Specify the model:
More informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 6 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 53 Outline of Lecture 6 1 Omitted variable bias (SW 6.1) 2 Multiple
More informationLecture 14 Simple Linear Regression
Lecture 4 Simple Linear Regression Ordinary Least Squares (OLS) Consider the following simple linear regression model where, for each unit i, Y i is the dependent variable (response). X i is the independent
More informationRESMA course Introduction to LISREL. Harry Ganzeboom RESMA Data Analysis & Report #4 February
RESMA course Introduction to LISREL Harry Ganzeboom RESMA Data Analysis & Report #4 February 17 2009 LISREL SEM: Simultaneous [Structural] Equations Model: A system of linear equations ( causal model )
More informationGov 2000: 9. Regression with Two Independent Variables
Gov 2000: 9. Regression with Two Independent Variables Matthew Blackwell Fall 2016 1 / 62 1. Why Add Variables to a Regression? 2. Adding a Binary Covariate 3. Adding a Continuous Covariate 4. OLS Mechanics
More informationThe Delta Method and Applications
Chapter 5 The Delta Method and Applications 5.1 Local linear approximations Suppose that a particular random sequence converges in distribution to a particular constant. The idea of using a first-order
More informationInference using structural equations with latent variables
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationECON 3150/4150, Spring term Lecture 6
ECON 3150/4150, Spring term 2013. Lecture 6 Review of theoretical statistics for econometric modelling (II) Ragnar Nymoen University of Oslo 31 January 2013 1 / 25 References to Lecture 3 and 6 Lecture
More informationChapter 8. Models with Structural and Measurement Components. Overview. Characteristics of SR models. Analysis of SR models. Estimation of SR models
Chapter 8 Models with Structural and Measurement Components Good people are good because they've come to wisdom through failure. Overview William Saroyan Characteristics of SR models Estimation of SR models
More informationWhat is in the Book: Outline
Estimating and Testing Latent Interactions: Advancements in Theories and Practical Applications Herbert W Marsh Oford University Zhonglin Wen South China Normal University Hong Kong Eaminations Authority
More informationBivariate distributions
Bivariate distributions 3 th October 017 lecture based on Hogg Tanis Zimmerman: Probability and Statistical Inference (9th ed.) Bivariate Distributions of the Discrete Type The Correlation Coefficient
More informationReliability-Constrained Latent Structure Models
Reliability-Constrained Latent Structure Models Peter Westfall and Kevin Henning Texas Tech University Latent Structure Models (LSMs) AKA "LISREL," or "structural equation models" Common in social science,
More informationStatistics 910, #5 1. Regression Methods
Statistics 910, #5 1 Overview Regression Methods 1. Idea: effects of dependence 2. Examples of estimation (in R) 3. Review of regression 4. Comparisons and relative efficiencies Idea Decomposition Well-known
More informationTwo-Variable Regression Model: The Problem of Estimation
Two-Variable Regression Model: The Problem of Estimation Introducing the Ordinary Least Squares Estimator Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Two-Variable
More informationAGEC 661 Note Fourteen
AGEC 661 Note Fourteen Ximing Wu 1 Selection bias 1.1 Heckman s two-step model Consider the model in Heckman (1979) Y i = X iβ + ε i, D i = I {Z iγ + η i > 0}. For a random sample from the population,
More informationUsing Structural Equation Modeling to Conduct Confirmatory Factor Analysis
Using Structural Equation Modeling to Conduct Confirmatory Factor Analysis Advanced Statistics for Researchers Session 3 Dr. Chris Rakes Website: http://csrakes.yolasite.com Email: Rakes@umbc.edu Twitter:
More informationVariance reduction. Michel Bierlaire. Transport and Mobility Laboratory. Variance reduction p. 1/18
Variance reduction p. 1/18 Variance reduction Michel Bierlaire michel.bierlaire@epfl.ch Transport and Mobility Laboratory Variance reduction p. 2/18 Example Use simulation to compute I = 1 0 e x dx We
More informationstatistical sense, from the distributions of the xs. The model may now be generalized to the case of k regressors:
Wooldridge, Introductory Econometrics, d ed. Chapter 3: Multiple regression analysis: Estimation In multiple regression analysis, we extend the simple (two-variable) regression model to consider the possibility
More informationAn Efficient State Space Approach to Estimate Univariate and Multivariate Multilevel Regression Models
An Efficient State Space Approach to Estimate Univariate and Multivariate Multilevel Regression Models Fei Gu Kristopher J. Preacher Wei Wu 05/21/2013 Overview Introduction: estimate MLM as SEM (Bauer,
More information4. Path Analysis. In the diagram: The technique of path analysis is originated by (American) geneticist Sewell Wright in early 1920.
4. Path Analysis The technique of path analysis is originated by (American) geneticist Sewell Wright in early 1920. The relationships between variables are presented in a path diagram. The system of relationships
More informationBasic econometrics. Tutorial 3. Dipl.Kfm. Johannes Metzler
Basic econometrics Tutorial 3 Dipl.Kfm. Introduction Some of you were asking about material to revise/prepare econometrics fundamentals. First of all, be aware that I will not be too technical, only as
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationAdvanced Econometrics
Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 16, 2013 Outline Univariate
More informationA note on structured means analysis for a single group. André Beauducel 1. October 3 rd, 2015
Structured means analysis for a single group 1 A note on structured means analysis for a single group André Beauducel 1 October 3 rd, 2015 Abstract The calculation of common factor means in structured
More informationNesting and Equivalence Testing
Nesting and Equivalence Testing Tihomir Asparouhov and Bengt Muthén August 13, 2018 Abstract In this note, we discuss the nesting and equivalence testing (NET) methodology developed in Bentler and Satorra
More information1 of 7 7/16/2009 6:12 AM Virtual Laboratories > 7. Point Estimation > 1 2 3 4 5 6 1. Estimators The Basic Statistical Model As usual, our starting point is a random experiment with an underlying sample
More informationNotes, March 4, 2013, R. Dudley Maximum likelihood estimation: actual or supposed
18.466 Notes, March 4, 2013, R. Dudley Maximum likelihood estimation: actual or supposed 1. MLEs in exponential families Let f(x,θ) for x X and θ Θ be a likelihood function, that is, for present purposes,
More informationWeek 9 The Central Limit Theorem and Estimation Concepts
Week 9 and Estimation Concepts Week 9 and Estimation Concepts Week 9 Objectives 1 The Law of Large Numbers and the concept of consistency of averages are introduced. The condition of existence of the population
More informationStat 206: Sampling theory, sample moments, mahalanobis
Stat 206: Sampling theory, sample moments, mahalanobis topology James Johndrow (adapted from Iain Johnstone s notes) 2016-11-02 Notation My notation is different from the book s. This is partly because
More information2 Introduction to Response Surface Methodology
2 Introduction to Response Surface Methodology 2.1 Goals of Response Surface Methods The experimenter is often interested in 1. Finding a suitable approximating function for the purpose of predicting a
More informationEconomics 240A, Section 3: Short and Long Regression (Ch. 17) and the Multivariate Normal Distribution (Ch. 18)
Economics 240A, Section 3: Short and Long Regression (Ch. 17) and the Multivariate Normal Distribution (Ch. 18) MichaelR.Roberts Department of Economics and Department of Statistics University of California
More informationLinear Regression with Multiple Regressors
Linear Regression with Multiple Regressors (SW Chapter 6) Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4. Measures of fit 5. Sampling distribution
More informationReview of Econometrics
Review of Econometrics Zheng Tian June 5th, 2017 1 The Essence of the OLS Estimation Multiple regression model involves the models as follows Y i = β 0 + β 1 X 1i + β 2 X 2i + + β k X ki + u i, i = 1,...,
More informationA Probability Review
A Probability Review Outline: A probability review Shorthand notation: RV stands for random variable EE 527, Detection and Estimation Theory, # 0b 1 A Probability Review Reading: Go over handouts 2 5 in
More informationOct Simple linear regression. Minimum mean square error prediction. Univariate. regression. Calculating intercept and slope
Oct 2017 1 / 28 Minimum MSE Y is the response variable, X the predictor variable, E(X) = E(Y) = 0. BLUP of Y minimizes average discrepancy var (Y ux) = C YY 2u C XY + u 2 C XX This is minimized when u
More informationLecture # 31. Questions of Marks 3. Question: Solution:
Lecture # 31 Given XY = 400, X = 5, Y = 4, S = 4, S = 3, n = 15. Compute the coefficient of correlation between XX and YY. r =0.55 X Y Determine whether two variables XX and YY are correlated or uncorrelated
More informationState-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Financial Econometrics / 49
State-space Model Eduardo Rossi University of Pavia November 2013 Rossi State-space Model Financial Econometrics - 2013 1 / 49 Outline 1 Introduction 2 The Kalman filter 3 Forecast errors 4 State smoothing
More informationRegression Models - Introduction
Regression Models - Introduction In regression models there are two types of variables that are studied: A dependent variable, Y, also called response variable. It is modeled as random. An independent
More informationIntroduction to Confirmatory Factor Analysis
Introduction to Confirmatory Factor Analysis Multivariate Methods in Education ERSH 8350 Lecture #12 November 16, 2011 ERSH 8350: Lecture 12 Today s Class An Introduction to: Confirmatory Factor Analysis
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 89 Part II
More informationModeration 調節 = 交互作用
Moderation 調節 = 交互作用 Kit-Tai Hau 侯傑泰 JianFang Chang 常建芳 The Chinese University of Hong Kong Based on Marsh, H. W., Hau, K. T., Wen, Z., Nagengast, B., & Morin, A. J. S. (in press). Moderation. In Little,
More informationAn Introduction to Bayesian Linear Regression
An Introduction to Bayesian Linear Regression APPM 5720: Bayesian Computation Fall 2018 A SIMPLE LINEAR MODEL Suppose that we observe explanatory variables x 1, x 2,..., x n and dependent variables y 1,
More informationLatent variable interactions
Latent variable interactions Bengt Muthén & Tihomir Asparouhov Mplus www.statmodel.com November 2, 2015 1 1 Latent variable interactions Structural equation modeling with latent variable interactions has
More informationEstimating Three-way Latent Interaction Effects in Structural Equation Modeling
International Journal of Statistics and Probability; Vol. 5, No. 6; November 2016 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education Estimating Three-way Latent Interaction
More informationIntroduction to Maximum Likelihood Estimation
Introduction to Maximum Likelihood Estimation Eric Zivot July 26, 2012 The Likelihood Function Let 1 be an iid sample with pdf ( ; ) where is a ( 1) vector of parameters that characterize ( ; ) Example:
More informationApplied Econometrics (QEM)
Applied Econometrics (QEM) The Simple Linear Regression Model based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #2 The Simple
More information4.8 Instrumental Variables
4.8. INSTRUMENTAL VARIABLES 35 4.8 Instrumental Variables A major complication that is emphasized in microeconometrics is the possibility of inconsistent parameter estimation due to endogenous regressors.
More informationECON2228 Notes 2. Christopher F Baum. Boston College Economics. cfb (BC Econ) ECON2228 Notes / 47
ECON2228 Notes 2 Christopher F Baum Boston College Economics 2014 2015 cfb (BC Econ) ECON2228 Notes 2 2014 2015 1 / 47 Chapter 2: The simple regression model Most of this course will be concerned with
More informationABSTRACT. Chair, Dr. Gregory R. Hancock, Department of. interactions as a function of the size of the interaction effect, sample size, the loadings of
ABSTRACT Title of Document: A COMPARISON OF METHODS FOR TESTING FOR INTERACTION EFFECTS IN STRUCTURAL EQUATION MODELING Brandi A. Weiss, Doctor of Philosophy, 00 Directed By: Chair, Dr. Gregory R. Hancock,
More informationLongitudinal Data Analysis Using Stata Paul D. Allison, Ph.D. Upcoming Seminar: May 18-19, 2017, Chicago, Illinois
Longitudinal Data Analysis Using Stata Paul D. Allison, Ph.D. Upcoming Seminar: May 18-19, 217, Chicago, Illinois Outline 1. Opportunities and challenges of panel data. a. Data requirements b. Control
More informationMAT2377. Rafa l Kulik. Version 2015/November/26. Rafa l Kulik
MAT2377 Rafa l Kulik Version 2015/November/26 Rafa l Kulik Bivariate data and scatterplot Data: Hydrocarbon level (x) and Oxygen level (y): x: 0.99, 1.02, 1.15, 1.29, 1.46, 1.36, 0.87, 1.23, 1.55, 1.40,
More informationLecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16)
Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) 1 2 Model Consider a system of two regressions y 1 = β 1 y 2 + u 1 (1) y 2 = β 2 y 1 + u 2 (2) This is a simultaneous equation model
More informationModule 3. Latent Variable Statistical Models. y 1 y2
Module 3 Latent Variable Statistical Models As explained in Module 2, measurement error in a predictor variable will result in misleading slope coefficients, and measurement error in the response variable
More informationGLS and FGLS. Econ 671. Purdue University. Justin L. Tobias (Purdue) GLS and FGLS 1 / 22
GLS and FGLS Econ 671 Purdue University Justin L. Tobias (Purdue) GLS and FGLS 1 / 22 In this lecture we continue to discuss properties associated with the GLS estimator. In addition we discuss the practical
More information