Regression And Mediation Analysis Using Mplus
Bengt O. Muthén, Linda K. Muthén, Tihomir Asparouhov
The inspiration to write this book came from many years of teaching about
Mplus and answering questions on Mplus Discussion and Mplus support. It became clear that once
people leave school, it is difficult to keep up with the newest methodology. The purpose of this
book is to provide researchers with information that is not readily available to them and that we
believe is important for their research. Many topics such as linear regression analysis; mediation
analysis; causal inference; regression analysis with categorical, count, and censored outcomes;
Bayesian analysis; and missing data analysis have entire books devoted to them. This book does not
attempt to replace these books but rather to give useful and manageable summaries of these topics
and show how the analyses are implemented in Mplus. The technical level is kept at a minimum but
still requires an introductory statistics background and a good background in regression.
Chapter 1 covers linear regression analysis including regression with an interaction,
multiplegroup analysis, missing data on covariates, and heteroscedasticity modeling. Chapter 2
covers mediation analysis with a continuous mediator and a continuous outcome including moderated
mediation. Chapter 3 covers special topics in mediation analysis that are not normally found in
books on mediation analysis. These include Monte Carlo simulation studies of mediation and
moderated mediation, model misspecification due to omitted variables and confounders, instrumental
variable estimation, sensitivity analysis, multiple group analysis of moderated mediation,
and measurement error. Chapter 4 covers causal inference based on counterfactuals for mediation
analysis with a continuous mediator and a continuous outcome. Chapter 5 covers regression analysis
for categorical dependent variables including binary, ordinal, and nominal variables. Chapter 6
covers regression analysis for a count dependent variable including the following models: Poisson,
Poisson with a random intercept, zeroinflated Poisson, negative binomial, zeroinflated negative
binomial, twopart (hurdle) with zerotruncation, and varyingexposure. Chapter 7 covers
regression analysis for a censored dependent variable including the following models: censorednormal
(tobit), censoredinflated, sample selection (Heckman), twopart, and switching regressions. Chapter
8 covers causal inference for mediation analysis with a binary outcome and a continuous mediator, a
count outcome and a continuous mediator, a two part outcome and a continuous mediator, a binary and
an ordinal mediator, a nominal mediator, and a mediator with measurement error. Chapter 9 discusses
Bayesian analysis and uses it to estimate several mediation examples which show how it can be used
as an alternative to maximum likelihood estimation. Chapter 10 discusses several approaches to
missing data modelling including missing completely at random (MCAR), missing at random (MAR), and
not missing at random (NMAR) including selection modeling.
Product Details
Paperback: 519 pages
Publisher: Muthén & Muthén
Language: English
ISBN 9780982998311
Product dimensions: 9.5 x 7.25 x 1
To Order
The book can be ordered from the online store.
Web Page for Examples
Mplus inputs and outputs for the book examples are posted along with data sets where available. A total of 140 analyses are shown.
Errata
Errata for first and second printing.
Table of Contents
 Linear regression analysis
 Linear regression analysis assumptions
 Linear regression estimation
 Residuals
 Outliers
 Linear regression Rsquare
 Linear regression standardization
 Example: Regression with one covariate
 Individual residuals and outliers
 Reporting results
 Multiple covariates
 Linear regression with two continuous covariates
Interaction between two continuous covariates
 Linear regression with one binary and one continuous covariate
Interaction between a binary and a continuous covariate
 Example: Regression with two covariates
 Reporting results
 Example: Regression with an interaction
 Reporting results
Presenting parameter estimates
Presenting results graphically
 Special topics
 Standardized coefficients greater than one
 Standardized coefficients differing in significance from unstandardized coefficients
 Twogroup regression analysis
Example: Twogroup regression analysis of an intervention study
 Bringing covariates into the model
Missing data on x
Example: Bringing a covariate into the model for the intervention example using a twogroup analysis
 The Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC)
 Heteroscedasticity modeling
Example: Heteroscedasticity modeling of LSAY math data
 Random coefficient regression
Example: Random coefficient regression for LSAY math data
 Mediation analysis
 A prototypical mediation model
 Mediation modeling techniques
 Estimation
Indirect effect standard errors and confidence intervals
 Standardization for mediation models
 Model testing
 Example: Sex discrimination
 Inspecting the data and reporting results
 Example: Head circumference
 Reporting results
 The saturated model
 Multiple mediators
 Example: Parallel mediators for media influence
 Example: Sequential mediators of socioeconomic status
 Moderated mediation
 Case 1 (xz): Regression of y on x, m on x, both moderated by z
 Case 2 (mz): Regression of y on m moderated by z
 Case 3 (mx): Regression of y on m moderated by x
 Combined moderation case
 Example: Case 1 moderated mediation in an intervention of aggressive behavior in the classroom
Testing significance of effects at specific moderator values
Creating a plot with bootstrap confidence intervals for the effects at a range of moderator values
Combination of significance of effect at specific moderator values and plot of confidence intervals using MODEL INDIRECT
 Example: Case 2 moderated mediation for work team behavior
 Example: Case 3 moderated mediation of simulated data
 Example: Combined moderated mediation for sex discrimination
 Special topics in mediation analysis
 Monte Carlo simulation study of mediation
 Example: Monte Carlo study of indirect effects
 Monte Carlo studies of moderation
 Example: Moderation of the regression of m on x
 Example: Moderation of the regression of y on m
 Model misspecification
 Example: Omitted moderator
 Example: Omitted mediators
 Example: Confounders
 Instrumental variable estimation
 Example: IV estimation with mediatoroutcome confounding
Bias and coverage of IV estimation of the indirect effect
Comparing IV and ML standard errors and power
IV estimation dependence on the size of the x, m correlation
Comparison of IV and maximumlikelihood estima tion when the assumptions behind both approaches are violated
 Sensitivity analysis
 Example: Sensitivity analysis for an experimental study of sex discrimination in the workplace
 Example: Sensitivity analysis in a Monte Carlo study
 Multiplegroup mediation analysis
 Relating multiplegroup parameters to interaction parameters
 Modification indices
 Example: Twogroup analysis of moderated mediation for sex discrimination
 Measurement errors and latent variables
 Measurement error in an independent variable
 Measurement error in a mediator
 Example: Monte Carlo simulation study of measurement error in the mediator
 Known reliability
 Multiple indicators
Reliability of a sum of indicators
Structural equation modeling with a factor analysis measurement model
 Causal inference for mediation
 Causal assumptions
 Potential outcomes and counterfactuals
 Example: Hypothetical potential outcome data
 Basics of counterfactuallydefined effects
 Example: Hypothetical mediation potential outcomes
 Direct and indirect effects
 Direct effects
 Indirect effects
 Total effect decomposition
 Example: Hypothetical mediation data analysis
 Causal effect formulas
 Example: Effects in the simple mediation case
 Example: Effects with moderation of Y regressed on M (case 3)
 Example: Effects in the combined moderation case
 Example: Effects combining case 1 and case 2 moderation
 Multiple mediators
 Summary
 Categorical dependent variable
 Basic concepts for categorical variables
 Binary variables
 Binary dependent variable
 Example: OLS, logistic, and probit regression of coal miner respiratory problems
 Modeling with a logistic regression function
 Modeling with a probit regression function
 Estimation of the logistic and probit regressions
 Probability curve formulation versus a latent response variable formulation
 R2 and standardization
R2 for a binary outcome
Standardization
 Example: Logistic and probit regression of British coal miner data
Logistic regression
Probit regression
Computation of estimated probabilities
Comparing logistic and probit regression coefficients
Comparing the logistic and probit regression models by BIC
 Logistic and probit regression with one binary and one continuous x
 Logistic regression and adjusted odds ratios
 Example: Adjusted odds ratios for alcohol survey data
 Example: Adjusted odds ratios for educational achievement data
 Ordinal dependent variable
 Probability curve formulation of ordinal dependent variable regression
 Latent response variable formulation of ordinal dependent variable regression
 Example: Sample probits for drinking related to age and income
 Testing the parallel probability curve assumption behind the ordinal regression model
 Example: Ordinal logistic regression of mental impairment
Estimated odds ratio
Estimated probabilities
Odds ratio with an interaction
 Nominal dependent variable
 Example: Multinomial logistic regression of antisocial behavior
 Count dependent variable
 Poisson model
 Poisson model with a random intercept
 Zeroinflated Poisson model
 Negative binomial model
 Zeroinflated negative binomial model
 Twopart (hurdle) model with zerotruncation
 Varyingexposure model
 Comparing models
 Example: Count regression of marital affairs
 Poisson, Poisson with a random intercept, and negative binomial models
Negative binomial model
 Zeroinflated Poisson and zeroinflated negative binomial models
 Twopart (hurdle) modeling
 Conclusion for marital affairs analyses
 Example: Poisson with varying exposure
 Censored dependent variable
 Basic concepts for a censored variable
 Censorednormal (tobit) regression
 Censoredinflated regression
 Sample selection (Heckman) regression
 Example: Simulated sample selection data
 Twopart regression
 Example: Methods comparison on alcohol data
 Analysis results for the four models
Loglikelihood and BIC comparisons of the four models
Comparing the results for the censorednormal (tobit) and censoredinflated models
Comparing the results for the sample selection (Heckman) and twopart models
Comparing the results for the censoredinflated and twopart models
Comparing the fit for estimated probabilities and means for the censoredinflated and twopart models
 Switching regressions
 Example: Monte Carlo simulation of switching regressions
 Mediation noncontinuous variables
 Binary outcome, continuous mediator
 A simple hypothetical mediation model
 Total, indirect, and direct effects in terms of differences in probabilities
 Causal effect formulas for a continuous M and a binary Y
 Causal effect formulas applied to a simple mediation model with a binary outcome and a continuous mediator
 Causal effect formulas defined on the odds ratio scale
Odds ratio effects assuming a rare outcome
 Example: Intention to use cigarettes
Probit regression
Logistic regression
 Example: HPV vaccination trial
No interventionmediator interaction
Interventionmediator interaction
Analysis results
 Count outcome, continuous mediator
 Causal effect formulas for a count outcome
 Example: Aggressive behavior and school removal
Estimated count probabilities
 Twopart outcome, continuous mediator
 Causal effect formulas for a twopart outcome
 Example: Twopart mediation analysis of economic stress data
Causal effects for twopart modeling
Causal effects for regular modeling with log y
Causal effects for regular modeling without log y
 Binary and ordinal mediator
 Causal effect formulas for a binary mediator
 Ordinal mediator
 Latent response variable mediator
 Estimation
 Example: Hypothetical data from the potential outcome example with a binary mediator and a continuous outcome
 Example: Ordinal mediator for intention to use cigarettes
 Example: Pearl’s artificial 2 x 2 x 2 example
 Nominal mediator
 Causal effect formulas for a nominal mediator
 Estimation
 Example: Hypothetical data with a nominal mediator and a binary outcome
 Mediator with measurement error
 Example: A Monte Carlo simulation study for a mediator measured with error
 Example: An intervention study of aggressive behavior in the classroom and juvenile court record
 Bayesian analysis
 Prior, likelihood, and posterior
 Posterior distribution for the mean of a normal distribution
 Types of priors
 Nonnormality of parameter distributions
 Markov Chain Monte Carlo (MCMC)
 Example: Bayesian estimation of a mean with missing data for LSAY math
Input for Bayesian analysis
 Plots
Trace plot
Autocorrelation plot
Posterior distribution plot
 Convergence checking
 Model fit
 Bayes versus ML intervals
 Example: Mediation model for media influence
The Potential Scale Reduction (PSR) convergence criterion
Trace plot
Autocorrelation plot
Inspecting model fit and parameter estimates
 Example: Mediation model for firefighter data
 Noninformative priors
 Informative priors
 Example: Model testing of direct effects
 Example: High school dropout and missing data
 Missing data on the mediator (n = 2, 213)
Bayesian analysis
Maximumlikelihood analysis
 Missing data on the control variables (n = 2, 898)
Assuming normality for all covariates
Acknowledging that some control variables are binary
 Missing data
 Example: Missing data information
 MCAR, MAR, and NMAR
 MCAR: Missing completely at random
 MAR: Missing at random
 NMAR: Not missing at random
 MAR for bivariate normal variables (H1 case)
 Listwise versus ML
 Maximumlikelihood estimation in the bivariate case with missing on one variable
 The EM algorithm
 Multiple imputation
 Example: Estimating sample statistics for intervention data
 MAR for regression (H0 case)
 Missing data and selection on x or y
 Regression analysis with missing data
 Technical aspects of ML assuming MAR
 Example: MAR simulated data analysis
 Missing data correlates
Example: Simulation study with missing data correlate of missing on y
 NMAR
 Example: Simulated NMAR data with missing influenced by the latent outcome
 Example: Selection modeling versus ML assuming MAR when MAR holds
 Example: Comparing missing data methods
 Missing data on covariates
 Example: Simulation study of missing on binary covariates
Inputs for generating and analyzing data assuming MAR
Inputs for generating and analyzing data under NMAR
Simulation results
Appendices
 A Covariance algebra
 A.1 Definition of expectation
 A.1.1 Rules for expectation
 A.2 Definition of covariance and variance
 A.3 Functions of random variables
 A.4 Example: Covariance algebra rules applied to linear regression
 A.5 Example: Derivation of the slope attenuation
 A.6 Example: Variance of a variable measured with error
