# Structural Equation Models II - Advanced Methods

**Prerequisites (knowledge of topic)**

St. Gallen Summer School in Empirical Research Structural Equation Models I (Introduction) Course or equivalent, and either i) the course Regression II: Linear Models or ii) a course on Logistic Regression are required. Participants should have a working knowledge of structural equation model basic topics, including model estimation, identification, testing nested models, assessing model fit, model assumptions, parameter constraints, multiple group models, construct and measurement equation equivalence across multiple groups. They should also have a working familiarity with regression models for categorical dependent variables (logistic regression), including the estimation and interpretation of parameters, Wald and LR testing, “fitted values” (expected probabilities) and the implications of parameterization in log-odds/logit form.

**Hardware**

Course exercises can be completed with desktop computers provided in a computer lab at the University of St. Gallen. Participants who can bring their own laptop computers loaded with the software identified below are encouraged to do so.

**Software**

We conduct the class using the lavaan package in R, the sem (and to a lesser extent the GSERM) procedure in STATA and with MPlus for a a small number of models which cannot be estimated using R/lavaan or STATA. Due to the high expense associated with MPlus, participants considering software purchases may wish to forego purchasing the software and simply use the computer lab at the University of St. Gallen which runs MPlus or at least wait until after the course is over to determine whether their own research program might require it.

**Course content**

1. Very brief overview of SEM I topics: missing data (including multiple imputation, which is not covered in SEM), alternative estimators, scaled test statistics and bootstrapping for non-normal data.

2. SEM models with ordinal latent variable indicators, including polychoric correlation approaches.

3. SEM models with binary indicator latent variables.

4. SEM models with binary, ordinal and multinomial outcomes: mixture models.

5. Detailed overview, with an emphasis on interpreting multiple-group models with non-parallel slopes: Means and intercepts in structural equation models

6. Dealing with scalar invariance in multiple-group models.

7. Interactions in structural equation models: product indicator, LMS and other approaches

8. Growth curve models for single-indicator variables

9. Growth curve models for multiple-indicator latent variables

10. Growth curve models for binary outcomes

11. Categorical latent variables: latent class models

12. The integration of latent class and continuous-variable SEM models

13. Multi-level SEM (introduction)

Structure*Day 1 Morning*

Missing data and non-normal data in SEM models. Class exercises (R and STATA for missing and non-normal data). SEM models with ordered latent variable indicators. *Day 1 Afternoon*

SEM models with ordinal latent variable indicators. Threshold parameters in ordinal variable models. SEM models with binary indicators. Limitations. Lab: Introduction to MPlus software. MPlus for non-normal data. MPlus for categorized (ordinal) variables.*Day 2 Morning*

Mixture models: SEM models with binary, ordinal and multinomial outcomes. Interpreting coefficients with logit or probit parameterization. Limitations. Comparisons with polychoric correlation approaches. Exercise: interpreting program results (MPlus, STATA).*Day 2 Afternoon*

Multiple group mean and intercept models for 2 groups. Mean and intercept models for k groups. Parallel and non-parallel slope models. Scalar invariance. Mean and intercept models in R/lavaan and STATA. (Optional lab: mean and intercept models in MPlus).*Day 3 Morning*

Interactions in SEM models: multiple group, product-indicator and LMS approaches.

Panel data: models for latent variable mean change. Panel data causal models, including cross-lagged models.*Day 3 Afternoon*

Growth curve models for single-indicator variables. Class exercise: growth curve models using R and STATA. Brief comparisons with MPlus.*Day 4 Morning*

Growth curve model extensions: piecewise models, dynamic models.

Growth curve models for multiple-indicator latent variables. Class exercise: latent variable growth curve models using R and STATA. Brief comparisons with MPlus.*Day 4 Afternoon*

Growth curve models for binary outcomes (brief overview).

Multilevel SEM (examples using STATA and MPlus).*Day 5 Morning*

Day 4 topics, continued.

Latent class models with categorical indicators.

Latent class models with categorical and continuous indicators.*Day 5 Afternoon*

If time permits: latent class growth curve models. The integration of latent class and continuous variable SEM models. Lab: latent class models using MPlus* (*note: this lab may be replaced with an exercise using alternative software, depending on class interest).

Literature

**Mandatory**

Four PDF files to be made available to participants.

**Mandatory readings before course start**

For participants who have not taken the SEM I course at GSERM, it is strongly recommended that the 8 PDF files constituting the reading materials for this first course be reviewed. These materials will be made available to SEM II. registrants. For participants who are not very familiar with generalized linear models (models for binary logit/probit, ordered logit/probit and multinomial logit/probit models), please read the PDF entitled, “A Primer on Categorical Dependent Variable Models for SEM Researchers,” which will be made available to participants in advance of the course. In addition, for those who have not taken a prior course on categorical data, if possible, browse through a categorical dependent variable model textbook (e.g., the STATA-based text by Scott Long and Jeremy Freese,__ Regression Models for Categorical Dependent Variables Using STATA__, though this text is more detailed than we need for SEM II; Scott Menard’s __Logistic Regression__ would also work).

**Examination part**

Two computer exercises: 20% x 2 = 40%

Two highly structured assignments using computer files to be provided to participants. One assignment is due on Thursday, while the other is due on the Monday following the course.

One short written exercise, based on questions covering course material: 10%

One major project: 50%

The major project involves an original SEM analysis to be conducted using a dataset of the participant’s choice, following consultation with the instructor. (Some datasets can be provided for those participants who do not have access to their own data).The project should employ at least one and preferably more than one of the advanced techniques covered in the course (i.e., beyond that materials covered in an introductory SEM course) – for example, a panel data model, a model with ordinal indicators for latent variables, a latent class model or a complex multiple-group model with factor means and intercepts.

**Supplementary aids**

Completion of project will require access to SEM software.

**Examination content**

Students will be expected to display a detailed knowledge of at least one of the topics covered in the course and preferably more than one for the project:

1. Multiple group models for means and intercepts.

2. Panel data models for latent variable change scores and cross-lagged effects.

3. Growth curve models with covariates

4. Models for non-normally distributed data

5. Models for coarsely-categorized ordinal variables.

6. SEM models with binary, ordered and multinomial outcomes.

7. Latent class models

**Literature**

Powerpoint and PDF materials, distributed to class participants (PDF files on SEM models and software guide files (MPlus, STATA, R) also distributed to the class.