Psychology 454: Psychological Measurement
An introduction to latent variable modeling
William Revelle
Swift 315
email: revelle@northwestern.edu
November 21, 2016

1Objectives

To understand the fundamental concepts in latent variable modeling in order to make you a better consumer
and producer of latent variable models in your research.
To understand how to evaluate the quality of models when applied to data by understanding various sourcesof variabilty of goodness of fit tests.
To learn how to apply these concepts to real data sets using a variety of standard statistical packages (e.g.,R, Mx, EQS, Lisrel, Prelis, Amos, MPlus)

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Text, readings, and requirements

2.1

Text

Loehlin, J. C. Latent Variable Models (4th ed). Lawrence Erlbaum Associates, Mahwah, N.J. 2004
Revelle, W. (in preparation) An introduction to psychometric theory with applications in R. Springer. Draft
chapters available at http://personality-project.org/r/book.
Revelle, W. (in preparation) Adventures in Latent Variable Modeling using R. Draft chapters available at
http://personality-project.org/r/book.

2.2

Readings

Multiple web based readings including, but not limited to the ones listed in the references. This list will beadded to throughout the quarter.
Syllabus and handouts available at http://personality-project.org/revelle/syllabi/454/454.syllabus.pdf

2.3

Requirements

Some basic knowledge of psychometric theory (to be reviewed in week 1). This course is a natural sequel toPsychology 405: Psychometric Theory. Some of the web readings will be taken from the 405 syllabus.
Familiarity with linear (matrix) algebra (to be reviewed in week 1)

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Willingness to use computer packages that allow for structural equation modeling. These can either bedownloaded to your computer (e.g, the open source packages R, and Mx), run on the Social Science Computer
Cluster (e.g., MPlus), or ones that you have a license for (e.g., EQS?)
Willingness to ask questions and add to the class discussion.
As might be expected, most of my examples and lectures will make use of the powerful statistical system,
R (R Core Team, 2016). This is because the open source nature of R allows us to see (if we want) how thecalculations are actually done, and to add new features to existing packages. We will use two packages agreat deal. The R packages psych (Revelle, 2016) and lavaan (Rosseel, 2012).

2.4

Evaluation

Homework assignments will be given weekly. These are your benefit and will be graded on a completed, notcompleted basis.
Students will be expected to write a short paper demonstrating the use of structural equation techniquesapplied to their particular research interests. They will also be asked to present their use of latent variable
models in short (15-30 minute) presentations in the last few weeks of the course.
This is a hands on course. You will be expected to try the various programs on simulated and real datasets.

3Outline (to be added to frequently – keep checking)Week
1

TopicReview
Correlation/Regression
Reliability

Reading
Loehlin Chaper 1
Correlation/Regression
Reliability
Linear Algebra

2

Basic Model fitting

Loehlin Chaper 2

3Simple models

Loehlin Chapter 3

4

Exploratory FA

5

EFA (continued)

Loehlin Chapter 5
Hierarchical factor models
Loehlin Chapter 6
Goodness of fit

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CFA

Multiple Groups

Loehlin Chapter 4
one good and one bad example
of modeling complex structures
R and LISREL models
R and lavaan models

Lecture Notes
Correlation and Regression
Introduction to R
Introduction to R – appendix
Linear Algebra
inverse of a matrix.
Using the SSCC
Path models
Week 2 slides
Model fitting – regression vs. latent variables
regression/mediation/latent
Using R
OLS/WLS/MLE
Even more on regression
Simulating structural data.
A general factor of personality?
EFA/CFA – psych, sem and lavaan
EFA/CFA continued
items vs continuous measures
Item Response Models
more on skew
changeand more on change
SEM = regression + reliability
How to define a model
analysis and critique
R and LISREL models

Homework
Problem set 1
Exercises from Loehlin
1-12 (page 32-34)

Problem set 2

Problem set 3

Problem set 4
Problem set 5

Problem set 6

types of variables, types of models
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Goodness of Fit
Evaluating Alternatives

Loehlin Chapter 7
More on change
latent change Ghisletta and McArdle, 2014Loehlin Chapter

Problem set 7
Advanced modeling with Lavaan
Problem set 8
Review of SEM

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Detailed Notes

4.1

Week 1Introduction to latent variables (405 in a week).
Review of Correlation and Regression and classical reliability theory. See also Chapter 4 on Correlation andregression as well as Chapter 5 on multiple correlation and regression.
Review of linear/matrix algebra (Appendix E)

4.2

Week 2

Application of matrix algebra to pattern and structure. Chapter 6: Exploratory factor analysis as a basic latent variable model. Finding the inverse of a matrix. For a review of factor analysis, see http://personality-project.org/revelle/syllabi/405/405-efa.pdf.

4.3

Week 3Structural models and goodness of fit tests. Barrett (2007), Examples with simulated data.
How to simulate structural data. This has been revised with a correction for two factor simulations andwith a more extensive analysis of the effects of sample size on estimating parameters in the two factor
model.
Using basic sem programs to find structure and apply goodness of fit tests. Using the sem (Fox et al., 2013)and lavaan (Rosseel, 2012) packages.

4.4

Week 4

Analysis of hierarchical factor models using hierarchical and bifactor solutions. The lecture notes for week
4 are here and prior notes are prior year notes

4.5

Week 5Exploratory and confirmatory factor analysis, continued. The lecture notes for week 5 are here.
Considering issues of using items rather than continuous measures. items vs continuous measures. Unfortunately, items have serious problems with skew.
One of the most powerful applications of sem is the analysis of change.

4.6

Week 6Comparing three examples from the literature: a short example (Erdle et al., 2009) of how not to reportfactor analysis, a sem paper which which actually fails to identify the model correctly (Erdle et al., 2010) andanother (Marsh et al., 2010) which systematically compares models. This last one includes a good discussionof how to do measurement invariance.

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Weeks 7-9

lavaan uses many examples from the MPlus manual (http://www.statmodel.com/ugexcerpts.shtml. Seein particular the example data sets at http://www.statmodel.com/usersguide/chapter5.shtml. The
notes describing lavaan output for these examples are available here.
Comparing sem in R and LISREL (Jöreskog and Sörbom, 1999). Consideration of goodness of fit tests(Barratt et al., 2007) (Click on Issue 5 in the left had column). R and LISREL lecture notes
Commercial software for structural equation modeling: EQS Bentler (1995), LISREL (Jöreskog and Sörbom,1999) MPlus (Muthén and Muthén, 2007).

4.8

R advice

The R tutorial gives a short introduction to the use of R.
• (Macs and PCs) For this, or any other package to work, you must activate it by either using thePackage Manager or the “library” command:– type library(psych)– If loading the psych package works, function such as ”describe” and ”pairs.panels” should work
(or at least give an error message that is NOT ”could not find function”).
– entering ?psych will give a list of the functions available in the psych package.

4.9

Using the Social Science Computer Cluster

1. Get an account and then to log in as a remote user.
2. Log on to the system using SSH (see the “how to” for doing this)
3. upload the appropriate batch command file using a sftp connection.
4. Issue a batch command (e.g., lisrel8 infile outfile)
5. sftp the outfile back to your desktop
6. repeat (3-5) until satisfied

References
Barrett, P. (2007). Structural equation modelling: Adjudging model fit. Personality and Individual Differences, 42(5):815–824.
Bentler, P. M. (1995). EQS structural equations program manual. Multivariate Software, Inc., Encino, CA.
Erdle, S., Gosling, S. D., and Potter, J. (2009). Does self-esteem account for the higher-order factors of thebig five? Journal of Research in Personality, 43(5):921 – 922.
Erdle, S., Irwing, P., Rushton, J. P., and Park, J. (2010). The general factor of personality and its relation
to self-esteem in 628,640 internet respondents. Personality and Individual Differences, 48(3):343–346.
Fox, J., Nie, Z., and Byrnes, J. (2013). sem: Structural Equation Models. R package version 3.1-3.
Jöreskog, K. G. and Sörbom, D. (1999). LISREL 8: Structural equation modeling with the SIMPLIS commandlanguage. Scientific Software International, Lincolnwood.

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Loehlin, J. C. (2004). Latent variable models: an introduction to factor, path, and structural equation
analysis. L. Erlbaum Associates, Mahwah, N.J., 4th edition.
Marsh, H. W., Scalas, L. F., and Nagengast, B. (2010). Longitudinal tests of competing factor structures
for the rosenberg self-esteem scale: Traits, ephemeral artifacts, and stable response styles. PsychologicalAssessment, 22(2):366 – 381.
Muthén, L. and Muthén, B. (2007). Mplus User’s Guide. Muthén & Muthén, Los Angeles, CA, fifth edition
edition.
R Core Team (2016). R: A Language and Environment for Statistical Computing. R Foundation for StatisticalComputing, Vienna, Austria.
Raykov, T. and Marcoulides, G. A. (2006). A first course in structural equation modeling. Lawrence Erlbaum
Associates, Mawwah,N.J, 2nd edition.
Revelle, W. (2016). psych: Procedures for Personality and Psychological Research. Northwestern University,
Evanston, http://cran.r-project.org/web/packages/psych/. R package version 1.6.9.
Revelle, W. (in prep). An introduction to psychometric theory with applications in R. Springer.
Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software,48(2):1–36.

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