Embedded Files
Companion website to Approaches to Understanding Structural Equation Models: Relationships Between Individuals, Variables, and Occasions
Program Code and Additional Resources
Videos and Other Resources
Ωnyx Videos
Tutorial on Ωnyx and R
Doing Regressions in Ωnyx
Two Predictor Regression in Ωnyx
Mplus Videos
Introduction to Mplus Diagrammer
Diagrams for Useful SEM Components Using Deviation Score Matrix (Variance/Covariance Matrix)
Covariance and Single Predictor Regression
Calculating a Standard Deviation
Calculating a Correlation coefficient
Calculating a Partial Correlation Coefficient
Single Predictor Model with Error Variances or Error Standard Deviations & Standardized Error
Two Predictor Regression
Single Factor with Three Indicators
Path Models for the Adjoined Matrix
Calculation of variance (and mean) from an Adjoined Matrix
Two Predictor Regression with Intercept
Between Groups T-test
One Factor Model with Intercepts, Latent Variance fixed to 1
Mediation Model
Example of Dummy Coding for Three Groups
Mediation Model
Example of Dummy Coding for Three Groups
Orthogonal Coding Example for Three Groups
Path Diagram for Ancova (Dummy Coded)
Assumption Checking and Influence Diagnostics
Example Loess Curve
Mplus Influence Example
R program for examining Influence Diagnostics from Mplus
R program for examining Influence Diagnostics using various options in Influence.SEM in R
Reliability Models (Correction for Attenuation)
Correction for Attenuation for Single Predictor Regression Model
Correlation between Two Variables Corrected for Attenuation
Correlation Corrected for Attenuation for All Variables
Two Predictor Regression Corrected for Attenuation
Identification Strategies for Confirmatory Factor Models
SIngle Factor with Unit Variance, Unstandardized and Standardized Solutions
SEM Models of Growth and Change
(Expand to See Software Links/Descriptions)
(Expand to See Software Links/Descriptions)
Which Programs to Use?
Which Programs to Use?
The SEM program you use depends on:
The SEM program you use depends on:
Your needs. Will you be doing fairly standard "run of the mill" SEM, or does your work involve more complicated statistical models? This 2012 review by Narayanan is a bit dated now but can provide you with a description of some of the features the major software packages offer.
Your expertise and software platforms. Do you prefer/require a graphical user interface or, given your work, which may involve many variables, is a line-by-line or matrix algebra program better suited to your preferences?
Your existing software platform. If, for example, you do most of your work in R, then lavaan may be an appropriate choice for you or, if you require advanced modeling, MplusAutomation. On the other hand, if most of your work is in SAS, you may wish to familiarize yourself with SAS Institute's Proc Calis.
The Bottom Line for First Time Users
The Bottom Line for First Time Users
Currently, when I teach my classes on this topic, I introduce students to the Ωnyx platform, as it is free, students find the graphical interface easy to understand, and the program can autogenerate Mplus, OpenMx, and lavaan scripts to help students go on to those platforms.
From there, most students migrate to either lavaan or Mplus, given the wider availability of statistical model options and the fact that a graphical interface is cumbersome for models involving a large number of variables. Other students migrate to the Amos platform, because the quality of the graphics is publication-grade and the interface with SPSS is easy for them. Long-term, it is probably a good idea to beome somewhat "multilingual" in software, as some things are easier to do in some software packages and the reality is that sofgtware in this area is likely to keep changing even more rapidly than it has in the past and it is good not to become too platform dependent.
Statistical Software
Statistical Software
Ωnyx is a free public domain software package that employs a graphical user interface
lavaan is an SEM package available on the R platform
Mplus is a commercial software package available here.
Proc Calis is a commercial software package available on the SAS platform
Books Introducing Ideas on one software platform
Books Introducing Ideas on one software platform
There are a number of good books which discuss doing structural equation modeling using a preferred software platform. As mentioned in later chapters, I employ a variety of software platforms, as each platform as advantages and disadvantages for various types of models. New resources and books emerge annually, but a number of texts I have found helpful is here:
There are a number of good books which discuss doing structural equation modeling using a preferred software platform. As mentioned in later chapters, I employ a variety of software platforms, as each platform as advantages and disadvantages for various types of models. New resources and books emerge annually, but a number of texts I have found helpful is here:
Latent Variable Models: An Introduction to Factor, Path, and Structural Equation Analysis, Fifth Ed. by John C. Loehlin and A. Alexander Beaujean which features a website with program code in Mplus and R here
Latent Variable Modeling Using R: A Step-by-Step Guide by Alexander Beaujean which also features a website containing program syntax here
Web Resources
Web Resources
IDRE Stats Statistical Consulting Center at UCLA (features a variety of software examples, syntax for textbooks, annotated output examples and video presentations from workshops)
The Mplus Youtube Channel which features a variety of video introductions to basic and advanced topics
The Mplus web site also contains several examples.
The lavaan project resources page and the lavaan tutorial page
Chapter 1 Why Another Book on Structural Equation Modeling?
Chapter 1 Why Another Book on Structural Equation Modeling?
Why Another Book on Structural Equation Modeling?
Why Another Book on Structural Equation Modeling?
As mentioned in the text, this text has two goals:
As mentioned in the text, this text has two goals:
To discuss the major topics of SEM across the ways they are represented: scatterplots, equations, matrix algebra, path diagrams, and vector diagrams.
To present the use of structural models from a falsificationist perspective.
To some people the idea that "there is no true model" may be somewhat novel. There is a nice discussion of how to understand models within the context of philosophy of science at https://plato.stanford.edu/entries/models-science/. In addition, Karl Popper has written quite a bit about how any scientific model, to be useful, must not completely correspond to the phenomenon being explained. There is a nice article on his views at https://iep.utm.edu/pop-sci/. One of his more popular quotes is frequentlly cited:
"Science may be described as the art of systematic over-simplification — the art of discerning what we may with advantage omit." The Open Universe : An Argument for Indeterminism (1992), p. 44
Chapters 3 & 4: Path Notation Basics
Chapters 3 & 4: Path Notation Basics
Basic Path Diagrams
Basic Path Diagrams
- One Predictor Regression Deviation Score Form Galton Data
- Calculating Standard Deviation from Deviation Score Matrix
- Calculating Correlation Matrix using a Deviation Score Matrix
- Calculating Error Standard Deviation vs. Error Variance
- Calculation of Partial Correlation Fig. 4.6:
- Calculating Mean and Variance from Adjoined Matrix (SSCP)
- Section 4.2, 4.3 & 4.4: Standardized Two Predictor Model (Uses Summary Data)
- Between Groups t-test (Ex. 3 in Chapter 3)
9. Dummy Coding Three Group Between Design
9. Dummy Coding Three Group Between Design
10. Orthogonal Coding Three Group Between Design
10. Orthogonal Coding Three Group Between Design
11. Ancova Three Group Between Design
11. Ancova Three Group Between Design
12. Single Factor with Three Indicators
12. Single Factor with Three Indicators
Based on Deviation Scores Latent Variance fixed to 1 (Includes standardized and unstandardized values) Ωnyx Program Files Ωnyx Diagram Data
Based on Deviation Scores Latent Variance fixed to 1 (Includes standardized and unstandardized values) Ωnyx Program Files Ωnyx Diagram Data
Based on Adjoined Matrix Latent Variance fixed to 1 (Includes standardized and unstandardized values) Onyx Program Files Ωnyx Diagram Data
Based on Adjoined Matrix Latent Variance fixed to 1 (Includes standardized and unstandardized values) Onyx Program Files Ωnyx Diagram Data
13. Mediational Models
13. Mediational Models
Chapter 5 Assumption Checking
Chapter 5 Assumption Checking
Mplus Exploration of Influential Observations (Basic Plots) Mplus Program Mplus Program Output Mplus Data
Mplus Exploration of Influential Observations (Basic Plots) Mplus Program Mplus Program Output Mplus Data
Examination of Influential Observations in R using Influence.SEM package in R. (Various options) R Program Example Data
Examination of Influential Observations in R using Influence.SEM package in R. (Various options) R Program Example Data
Chapter 6 Identification of Confirmatory Factor models
Chapter 6 Identification of Confirmatory Factor models
Amos
Amos
Mplus
Mplus
Unit Factor Variance Scaling Mplus Program Data File Output File Mplus Graphics Diagram Saved Diagram (Note that this also calculates and saves influential diagnostics)
Unit Factor Variance Scaling Mplus Program Data File Output File Mplus Graphics Diagram Saved Diagram (Note that this also calculates and saves influential diagnostics)
Referent Indicator Scaling Mplus Program Data File Output File Mplus Graphics Diagram (Note that this also calculates and saves influential diagnostics)
Referent Indicator Scaling Mplus Program Data File Output File Mplus Graphics Diagram (Note that this also calculates and saves influential diagnostics)
Latent Reference Scaling (Little, Slegers, and Card (2006) Mplus Program Data File Output File Mplus Graphics Diagram (Note that this also calculates and saves influential diagnostics)
Latent Reference Scaling (Little, Slegers, and Card (2006) Mplus Program Data File Output File Mplus Graphics Diagram (Note that this also calculates and saves influential diagnostics)
Chapter 7 Reliability Models (Correction for Attenuation)
Chapter 7 Reliability Models (Correction for Attenuation)
Chapter 8 Confirmatory Factor Model
Chapter 8 Confirmatory Factor Model
- One Factor Model (Uses Raw Data)
Identification using latent variable scaling (i.e., factor variance-1)
Identification using latent variable scaling (i.e., factor variance-1)
Chapter 14 Multiple Groups
Chapter 14 Multiple Groups
Multi-Group Analysis Using Amos (Diagram in Figure 14.3 "Two Group Relative Metric Invariance Model" is the Measurement Intercepts Model.
Multi-Group Analysis Using Amos (Diagram in Figure 14.3 "Two Group Relative Metric Invariance Model" is the Measurement Intercepts Model.
Amos Program File Amos Data Set
Multi-Group Analysis Using R
Multi-Group Analysis Using R
Multi-Group Analysis Using Ωnyx
Loadings & Intercepts Invariant, Factor Means, Variances, and Errors Vary
Chapter 15 Exploratory Factor analysis
Chapter 15 Exploratory Factor analysis
- Reasons for Drinking EFA
Geomin Rotation for Reasons for Drinking (Table 15.1)
Geomin Rotation for Reasons for Drinking (Table 15.1)
Multi-Group ESEM (Exploratory Factor Analysis (Follows Table 24 in Chapter 16)
Multi-Group ESEM (Exploratory Factor Analysis (Follows Table 24 in Chapter 16)
Model A (All Parameters Totally Variant) Program Listing Diagram
Model B (Loadings Invariant) Program Listing Diagram
Model C (Loadings & Intercepts Invariant) Program Listing Diagram
Model D (loadings, intercepts factor variances and covariances invariant) Program Listing Diagram
Model E (Factor Loadings, Intercepts, Factor Variances & Covariances, and Errors Invariant, Factor Means Variant) Program Listing Diagram
Model F (Factor Loadings, Intercepts, Factor Variances & Covariances, and Errors Factor Means Invariant) Program Listing Diagram
Model G (Factor Loadings, Intercepts, Factor Means, Errors, Factor Variances & Covariances Invariant) Program Listing Diagram
Chapter 17 Categorical Dependent Variables
Chapter 17 Categorical Dependent Variables
- One Predictor Regression Models Using Mplus (Section 17.4)
Resources and Program Code by Chapter
Resources and Program Code by Chapter
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