Harvard Catalyst Biostatistical Seminar

Neuropsychological Pro�les in Alzheimer's Disease and

Cerebral Infarction: A Longitudinal MIMIC Model

An Overview of Structural EquationModeling using Mplus

Richard N. Jones, Sc.D.

jones@hsl.harvard.edu

Institute for Aging Research, Hebrew SeniorLife

Beth Israel Deaconess Medical Center, Harvard Medical School

HSPH Kresge G2 October 5, 2011

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Objective Overview of SEM in Cognitive Aging & Health Research

Objective

IntroduceI the concepts and terminology relevant to structural equation modeling

(SEM) as applied to health research

Speci�c ExampleI Cognitive EpidemiologyI Mplus software 1

EmphasisI on a broad survey of applications, results, challenges, and opportunities

1www.statmodel.com

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Introduction Use of Mplus and SEM Applications in Epidemiology

Mplus and SEM Trends in Epidemiology

Relative to the frequency

with which Cox Regression

and Epidemiology appear

in Google Scholar...

Citations matching Mplus

and Epidemiology are

increasing

Although speci�c

applications are decreasing

Mplus use is increasing

and applications are

becoming more diverse

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Introduction What is Mplus, Anyways?

Mplus: General statistical analysis software good for...

Analysis with latent variables

Clustered and correlated dataI Complex sampling, weightingI Repeated measuresI Multicomponent variables (i.e., scales, composite outcomes)I Correlated observations (e.g., twins, families)I Multilevel contexts

Particular strengthsI Missing data modelingI Bayesian data analysisI Complex models

F Joint models of change and event occurrenceF Mixture models (population heterogeneity)F Longitudinal factor analysis

Where Mplus is not strongI Data managementI Graphics

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Introduction SEM in Epidemiologic Research

Structural Equation Modeling in Epidemiologic Research

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Introduction Everything You Need to Know about SEM in 1 Slide

Structural Equation Modeling in Epidemiologic Research

DeStavola et al (2005), bottom panel Figure 3

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Introduction What is SEM?

Structural Equation Modeling (SEM) is

A general multivariate regression modeling frameworkI General - �exible model typesI Multivariate - multiple dependent variablesI Regression - it's just regression. Regression can be viewed as a special

case of SEM

SEMs often include latent variablesI Continuous latent variables (i.e., factors)I Categorical latent variables (i.e., classes, mixtures)

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Introduction Varieties of Covariance Structure Modeling

Continuous Latent Variables

No Yes

Regressions among

No

Regression Factor

Dependent or Latent (Multivari-able\-ate) Analysis

Variables y = ν + ΓX+ ε y = ν + Λη + ε

Yes

Path Structural Equa-

Analysis tion Modeling

y = ν + BY + ΓX+ ε y = ν + Λη +KX+ εη = α+ Bη + ΓX+ ζ

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Introduction Get yourself started

SEM Prerequisites

General linear modelI Linear, logistic, & probit regressionI Multivariable regressionI Mixed e�ect models for longitudinal dataI Survival and event occurrence (Cox, parametric survival)

Missing data theoryI Little and Rubin, Statistical Analysis with Missing Data. 1987

Factor AnalysisI Brown, Con�rmatory Factor Analysis for Applied Research. 2006

Item Response TheoryI Embretson and Reise, Item Response Theory for Psychologists. 2000

Path AnalysisI Kerlinger and Pedhazur, Multiple Regression in Behavioral Research.

1973

Structural Equation ModelingI Jöreskog and Sörbom, LISREL 8: User's Reference Guide. 1996

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Introduction What you have to look forward to

Mplus Work�ow

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Introduction But life can be better

Mplus Work�ow

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Introduction Weaving with Stata and a good Text Editor is Fun and Easy

Mplus Work�ow - Weaving = Reproducible Research

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Introduction Path Diagrams and Equations - Variables

Multivariate Regression

y = ν + ΓX + ε

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Introduction Path Diagrams and Equations - Variables

Multivariate Regression

Mplus Syntax

TITLE: Multivariate regression

DATA: FILE = data.dat ;

VARIABLE: NAMES = y1 y2 x1 x2 ;

MODEL: y1 y2 on x1 x2 ;

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Introduction Path Diagrams and Equations - Variables

Con�rmatory Factor Analysis

y = ν + Λη + ε

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Introduction Path Diagrams and Equations - Variables

Con�rmatory Factor Analysis

Mplus Syntax

TITLE: Confirmatory Factor Analysis

DATA: FILE = data.dat ;

VARIABLE: NAMES = y1 y2 y3 ;

MODEL: eta by y1 y2 y3 ;

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Introduction Path Diagrams and Equations - Variables

Structural Equation Modeling

y = ν + Λη +KX + ε

η = α +Bη + ΓX + ζ

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Introduction Path Diagrams and Equations - Variables

Structural Equation Modeling

Mplus Syntax

TITLE: Structural Equation Model

DATA: FILE = data.dat ;

VARIABLE: NAMES = y1-y6 x1 ;

MODEL: eta1 by y1-y3 ; ! measurement model for eta1

eta2 by y4-y6 ; ! measurement model for eta2

eta2 on eta1 ; ! a structural regression

eta1 on x1 ; ! an "indirect effect"

y1 on x1 ; ! a "direct effect"

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Introduction Path Diagrams and Equations - Variables

Structural Equation Modeling

Mplus Syntax

. runmplus y1-y6 x1 , model(eta1 by y1-y3 ; eta2 by y4-y6 ; ///eta2 on eta1 ; eta1 on x1 ; y1 on x1 ;)

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Latent Variables What is a Latent Variable?

What Latent Variables Are

Latent variables are mathematical abstractions

that account for covariation among observed

variables

Latent variables may be continuous or

categorical

But what do they mean?

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Latent Variables What is a Latent Variable?

What is the Meaning Behind a Latent Variable?

The answer depends on theI scienti�c questionI philosophical position 2

Two broad classes of latent variable (LV) applicationsI Instrumentalist

F the LV is a mathematical abstraction

I RealistF the LV existsF the LV re�ects some unmeasurable quantity or quality that really exists

in natureF the LV exists independently of our measurement of it

Realist or Instrumentalist interpretations are a matter of statistical

inference

2Borsboom, Mellenbergh et al., 2003 Psychol Rev 110:203-18

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Conclusion

Parting Words

Why use SEM?I More easily specify analysis to answer research questionI Gain statistical power

Not sure if SEM is right for you?I Stata Corp recently added SEM to Stata (version 12)I ... and pitching SEM to EconomistsI http://www.stata.com/news/statanews.26.3.pdf

Why use Mplus?I Regression-based framework for exogenous variablesI Categorical dependent variablesI Categorical latent variablesI Complex sampling weightsI Survival analysisI Bayesian data analysis

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Conclusion Penetrance of SEM Concepts and Software

Google Scholar Hits 1998-2011 (5 Oct 2011)

Keyword phrase NEJM JAMA+ AJE

analysis 10,000 16,000 4,500

logistic regression 1,000 2,000 2,000linear regression 620 1,700 1,700cox proportional 610 760 650random e�ects 100 540 340generalized estimating 110 280 320

factor analysis 90 190 170structural equation 21 25 49pro�le mixture OR latent class 13 12 21item response theory 2 12 3path analysis 5 10 19latent growth curve 0 1 4

SAS Institute 440 2,500 1,400SPSS 210 1,100 240Stata 190 800 590R Foundation OR R Development 36 100 87

Mplus 0 12 15LISREL 1 3 13

Note: Hits include the text matches in the reference list.

Values greater than 100 rounded to two signi�cant digits.

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