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Statistical Analysis With Latent Variables User’s Guide Linda K. Muthén Bengt O. Muthén

Following is the correct citation for this document: Muthén, L.K. and Muthén, B.O. (1998-2010). Mplus User’s Guide. Sixth Edition. Los Angeles, CA: Muthén & Muthén Copyright © 1998-2010 Muthén & Muthén Program Copyright © 1998-2010 Muthén & Muthén Version 6 April 2010 The development of this software has been funded in whole or in part with Federal funds from the National Institute on Alcohol Abuse and Alcoholism, National Institutes of Health, under Contract No. N44AA52008 and Contract No. N44AA92009. Muthén & Muthén 3463 Stoner Avenue Los Angeles, CA 90066 Tel: (310) 391-9971 Fax: (310) 391-8971 Web: www.StatModel.com Support@StatModel.com

TABLE OF CONTENTS Chapter 1: Introduction Chapter 2: Getting started with Mplus Chapter 3: Regression and path analysis Chapter 4: Exploratory factor analysis Chapter 5: Confirmatory factor analysis and structural equation modeling Chapter 6: Growth modeling and survival analysis Chapter 7: Mixture modeling with cross-sectional data Chapter 8: Mixture modeling with longitudinal data Chapter 9: Multilevel modeling with complex survey data Chapter 10: Multilevel mixture modeling Chapter 11: Missing data modeling and Bayesian analysis Chapter 12: Monte Carlo simulation studies Chapter 13: Special features Chapter 14: Special modeling issues Chapter 15: TITLE, DATA, VARIABLE, and DEFINE commands Chapter 16: ANALYSIS command Chapter 17: MODEL command Chapter 18: OUTPUT, SAVEDATA, and PLOT commands Chapter 19: MONTECARLO command Chapter 20: A summary of the Mplus language 1 13 19 41 51 97 141 197 233 289 337 357 391 407 449 519 567 633 689 711

PREFACE We started to develop Mplus fifteen years ago with the goal of providing researchers with powerful new statistical modeling techniques. We saw a wide gap between new statistical methods presented in the statistical literature and the statistical methods used by researchers in substantively-oriented papers. Our goal was to help bridge this gap with easy-to-use but powerful software. Version 1 of Mplus was released in November 1998; Version 2 was released in February 2001; Version 3 was released in March 2004; Version 4 was released in February 2006; and Version 5 was released in November 2007. We are now proud to present the new and unique features of Version 6. With Version 6, we have gone a considerable way toward accomplishing our goal, and we plan to continue to pursue it in the future. The new features that have been added between Version 5 and Version 6 would never have been accomplished without two very important team members, Tihomir Asparouhov and Thuy Nguyen. It may be hard to believe that the Mplus team has only two programmers, but these two programmers are extraordinary. Tihomir has developed and programmed sophisticated statistical algorithms to make the new modeling possible. Without his ingenuity, they would not exist. His deep insights into complex modeling issues and statistical theory are invaluable. Thuy has developed the post-processing graphics module and the Mplus editor and language generator. In addition, Thuy has programmed the Mplus language and is responsible for keeping control of the entire code which has grown enormously. Her unwavering consistency, logic, and steady and calm approach to problems keep everyone on target. We feel fortunate to work with such a talented team. Not only are they extremely bright, but they are also hard-working, loyal, and always striving for excellence. Mplus Version 6 would not have been possible without them. Another important team member is Michelle Conn. Michelle was with us at the beginning when she was instrumental in setting up the Mplus office and has been managing the office for the past six years. In addition, Michelle is responsible for creating the pictures of the models in the example chapters of the Mplus User’s Guide. She has patiently and quickly changed them time and time again as we have repeatedly changed our minds. She is also responsible for keeping the website updated and interacting with customers. Her calm under pressure is much appreciated. Jean Maninger joined the Mplus team after Version 4 was released. Jean works with Michelle and has proved to be a valuable team member.

We would also like to thank all of the people who have contributed to the development of Mplus in past years. These include Stephen Du Toit, Shyan Lam, Damir Spisic, Kerby Shedden, and John Molitor. Part of the work has been supported by SBIR contracts from NIAAA that we acknowledge gratefully. We thank Bridget Grant for her encouragement in this work. Linda K. Muthén Bengt O. Muthén Los Angeles, California April 2010

Introduction CHAPTER 1 INTRODUCTION Mplus is a statistical modeling program that provides researchers with a flexible tool to analyze their data. Mplus offers researchers a wide choice of models, estimators, and algorithms in a program that has an easy-to-use interface and graphical displays of data and analysis results. Mplus allows the analysis of both cross-sectional and longitudinal data, single-level and multilevel data, data that come from different populations with either observed or unobserved heterogeneity, and data that contain missing values. Analyses can be carried out for observed variables that are continuous, censored, binary, ordered categorical (ordinal), unordered categorical (nominal), counts, or combinations of these variable types. In addition, Mplus has extensive capabilities for Monte Carlo simulation studies, where data can be generated and analyzed according to any of the models included in the program. The Mplus modeling framework draws on the unifying theme of latent variables. The generality of the Mplus modeling framework comes from the unique use of both continuous and categorical latent variables. Continuous latent variables are used to represent factors corresponding to unobserved constructs, random effects corresponding to individual differences in development, random effects corresponding to variation in coefficients across groups in hierarchical data, frailties corresponding to unobserved heterogeneity in survival time, liabilities corresponding to genetic susceptibility to disease, and latent response variable values corresponding to missing data. Categorical latent variables are used to represent latent classes corresponding to homogeneous groups of individuals, types of development components corresponding to finite mixtures of unobserved populations, and latent response variable categories corresponding to missing data. trajectory classes corresponding in unobserved populations, mixture latent to THE Mplus MODELING FRAMEWORK The purpose of modeling data is to describe the structure of data in a simple way so that it is understandable and interpretable. Essentially, the modeling of data amounts to specifying a set of relationships 1

between variables. The figure below shows the types of relationships that can be modeled in Mplus. The rectangles represent observed variables. Observed variables can be outcome variables or background variables. Background variables are referred to as x; continuous and censored outcome variables are referred to as y; and binary, ordered categorical (ordinal), unordered categorical (nominal), and count outcome variables are referred to as u. The circles represent latent variables. Both continuous and categorical latent variables are allowed. Continuous latent variables are referred to as f. Categorical latent variables are referred to as c. The arrows in the figure represent regression relationships between variables. Regressions relationships that are allowed but not specifically shown in the figure include regressions among observed outcome variables, among continuous latent variables, and among categorical latent variables. For continuous outcome variables, linear regression models are used. For censored outcome variables, censored (tobit) regression models are used, with or without inflation at the censoring point. For binary and ordered categorical outcomes, probit or logistic regressions models are used. For unordered categorical outcomes, multinomial logistic regression models are used. For count outcomes, Poisson and negative binomial regression models are used, with or without inflation at the zero point. CHAPTER 1 2

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