# Structural Equation Models I

**Prerequisites (knowledge of topic)**

The content of the St. Gallen Summer School in Empirical Research Methods Regression I (Introduction to Regression) Course or the Pre-Session course on Regression or equivalent is very important. The Regression II Course (Linear Models) or its equivalent is strongly recommended. Participants should be thoroughly familiar with the following linear regression topics: dummy variable models; contrasts; regression model assumptions and diagnostics; dealing with outliers; collinearity; specification error; non-linearity; non-normality; relationship between regression and ANOVA; Y-variable and X-variable transformations; models for quadratics; interactions. A basic knowledge of matrix algebra for regression models and regression models for limited dependent variables (logistic, poisson, etc.) would be helpful.

**Hardware**

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

**Software**

We will make use of STATA and the **lavaan **package in **R**.

**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 STATA and R-lavaan.

5. Scalar programming for structural equation models: an overview of other software

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 analysis with missing data

11. Alternative estimators for non-normal data.*Schedule may vary slightly according to class progress.*

**Structure***Day 1 Morning*: Path models, mediation. Introduction to latent variable conceptualization. Diagrams, equations and model parameters. Class exercise: Moving from equations to diagrams and vice versa; listing model parameters.*Day 1 Afternoon:* Computer exercises: Introduction to SEM software (STATA and R-lavaan). 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 (STATA and R): A latent variable measurement model. Model diagnostics, fit improvement approaches. *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. Class exercise: interpreting software output.*Day 3 Afternoon: * Computer exercises (STATA and R). A full structural equation model. Improving model fit, assessing diagnostics. Non-standard models. If time permits: effects decomposition; assessing linearity with single-indicator exogenous variables.*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. Class exercise: interpreting software output for multiple-group models.*Day 4 Afternoon:* Computer exercises (STATA and R). A multiple group model. Alternative estimators and scaled variance estimators: dealing with missing data and non-normal data. Item parcels (pro and con).*Day 5 Morning:* Computer exercises for datasets with missing and/or non-normal data. An introduction to models for means and intercepts. *Day 5 Afternoon:* Class exercise: a model for means and intercepts (software output interpretation). Programming mean/intercept models in STATA. Comparisons in complex models.

**Literature****Mandatory**

Eight PDF files will be made available to participants as reading materials for this course, titled Notes(Section1) through Notes(Section8).

**Supplementary / voluntary**

Additional supplementary readings: 1. Randall Schumacker and Richard Lomax, *A Beginnier’s Guide to Structural Equation Modeling*, 4th edition (Routledge, 2016) 2. Alan Acock, *Discovering Structural Equation Modeling Using STATA* (STATA Press, 2013). These additional readings are helpful but not essential. Earlier editions of these texts are fine.

**Mandatory readings before course start**

There are no mandatory pre-course readings.

Participants are encouraged to read through section 1 of the course notes in advance of the class, but may choose to read sections after the material is covered in class.

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

**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. 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. Estimating models with single-indicator latent variables

8. Determining whether the fit of a model is acceptable

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

10. Interpreting models with parameter constraints

11. Testing measurement model equivalence in multiple-group models

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

13. Dummy exogenous variables in structural equation models

14. The FIML estimator.

15. Bentler-Satorra and other variance adjustment approaches.

**Literature**

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