# Introduction to Structural Equation Models

**Prerequisites (knowledge of topic)**

A course in regression (e.g., GSERM Regression I) is essential. A second course in regression (e.g., GSERM Regression II) is recommended. Regression topics that are particularly important: i) assessing and dealing with non-linearity ii) dummy variables (including block F-tests) iii) standardization.

**Hardware**

Participants should bring laptops loaded with the software identified below.

**Software**

We will make primary use of the **lavaan **package in **R** but will also demonstrate the **sem **procedure in STATA. The following **R **packages should be installed on participant laptops: lavaan, haven, semTools. STATA will be available in a computer lab at the University of St. Gallen for participants who do not have it installed on their own laptops.

**Learning objectives**

The course will provide a conceptual introduction to structural equation models, provide a thorough outline of model “fitting” and assessment, teach how to effectively program structural equation models using available software, demonstrate how to extend basic models into multiple group situations, and provide an introduction to models where common model assumptions regarding missing and non-normal data are not met.

**Course content**

1. Introduction to latent variable models, measurement error, path diagrams.

2. Estimation, identification, interpretation of model parameters.

3. Scaling and interpretation issues

4. Scalar programming for structural equation models in R-lavaan and STATA.

5. Mediation models in the structural equation framework.

6. Model fit and model improvement

7. General linear parameter constraints

8. Multiple-group models

9. Introduction to models for means and intercepts

10. The FIML approach to analysis with missing data

11. Alternative estimators for non-normal data.

**Structure**

Schedule may vary slightly according to class progress.

*Day 1 Morning*

Path models, mediation. Introduction to latent variable conceptualization. Diagrams, equations and model parameters. Moving from equations to diagrams and vice versa; listing model parameters.

*Day 1 Afternoon*

Introduction to computer SEM software. Computer exercises: A simple single-indicator model. A latent variable measurement model.

*Day 2 Morning*

Identification. Variances, scaling. Covariance algebra for structural equation models. Applications. Class exercises: a) identification b) covariance algebra. Equality constraints and dummy variables in SEM models.

*Day 2 Afternoon*

Computer exercises (R, STATA): A latent variable measurement model with covariates. Model diagnostics, fit improvement approaches. Mediation with manifest and latent variables.

*Day 3 Morning*

Nested models, Wald and LM tests, mixing single- and multiple-indicator measurement models. Fit functions. Estimation. Dealing with estimation problems, including negative variance estimates and non-convergence.

*Day 3 Afternoon*

Computer exercise (R/lavaan): SEM model with multiple latent variables, single-indicator and multiple-indicator covariates. Improving model fit, assessing diagnostics. Non-standard models. Multiple group models: conceptual introduction.

*Day 4 Morning*

Multiple Group Models. Measurement equation equivalence across groups (tests, assessment). Construct equation equivalence. Software applications, formal versus substantive comparisons. Reporting SEM model results. Computer exercise (R): a multiple-group model.

*Day 4 Afternoon*

Computer exercise: multiple-group models in STATA. Alternative estimators and scaled variance estimators: dealing with missing data and non-normal data. Item parcels (pro and con).

*Day 5 Morning*

Computer exercise (R/lavaan) for datasets with missing and/or non-normal data. An introduction to models for means and intercepts.

*Day 5 Afternoon*

Computer exercises (R/lavaan and STATA): a model for means and intercepts

**Literature**

**Mandatory**

Nine PDF files will be made available to participants as reading materials for this course, titled Notes(Section1) through Notes(Section9).

**Supplementary / voluntary**

Randall Schumacker and Richard Lomax, A Beginner’s Guide to Structural Equation Modeling. 4th edition (Routledge, 2016). This reading is helpful but not essential. Earlier versions of this text can be used.

**Mandatory readings before course start**

There are no mandatory pre-course readings. Participants are encouraged to red through section 1 of the course notes in advance of the class, but may choose to read this while the class is in progress.

**Examination part***Two computer exercises, 20% each: 40%.*

First exercise is due Thursday during the course. Second exercise is due Monday immediately following the course.

One major exercise: 60%.

This exercise will consist of a series of 5-7 questions requiring essay-style responses (approx. 8-14 pp. total). Some questions will involve the interpretation of computer output listings, while other questions will deal with conceptual issues discussed in the course. The exercise is due within 2 weeks of the end of the course.

**Supplementary aids**

For computer exercises, the following materials will be helpful: a) lab exercise materials and descriptions and b) an abbreviated software user manual/guide (one available for each of STATA and lavaan), c) PDF course text files. For the major project, the PDF course files will be very helpful.

**Examination content**

For the final exercise, students will need to understand the following subject matter:

1. Converting equations to path diagrams and vice versa.

2. Principles of mediation assessment: total, direct and indirect effects in structural equation path models

3. Determining whether a model is identified or not

4. Dealing with estimation difficulties

5. Interpreting model parameters in the metric of the manifest variables

6. Interpreting standardized model parameters

7. Determining whether the fit of a model is acceptable

8. Hypothesis testing: simultaneous tests for b=0; tests for equality

9. Interpreting models with parameter constraints

10. Testing measurement model equivalence in multiple-group models

11. Testing construct equation equivalence in multiple group models; assessing individual parameters and groups of parameters for cross-group differences

12. Dummy exogenous variables in structural equation models

13. Approaches to missing data in SEM models.

14. Dealing with non-normal data: ADF, DWLS estimators, Bentler-Satorra and other variance adjustment approaches.

**Examination relevant literature**

For the major assignment exercise, students should have access to the course powerpoint slide materials and the course text PDF files.