# Introduction to structural equation modeling

I am asked by colleagues some help in this subject, that I don’t really know. They made hypotheses on the role of some latent variables in one study, and a referee asked them to formalize this in SEM. As what they need doesn’t seem too difficult, I think I’ll give it a shot... for now, I am just looking for a good introduction to the subject!

PS: I read Structural Equation Modeling With the sem Package in R by John Fox, and this text by the same author. I think this can be sufficient for my purpose, anyway any other references are welcome.

• Do you want some key textbooks on SEM in an applied perspective, or more general and formal textbooks? – chl Jan 19 '12 at 23:05
• @chl Thanks for your attention. Now that I got the basics, I’d like to see explicit writing of the likelihood, and conditions for identifiability. Something about binary and ordinal indicators would be much welcome also: Fox uses polychoric correlations, which seems simple and efficient, but obviously more complex solutions are possible... I found some interesting papers but I lack time to go into an extensive bibliographic search; a textbook or a "paper of reference" would be welcome. – Elvis Jan 20 '12 at 22:19

I would go for some papers by Múthen and Múthen, who authored the Mplus software, especially

1. Múthen, B.O. (1984). A general structural equation model with dichotomous, ordered categorical and continuous latent indicators. Psychometrika, 49, 115–132.
2. Muthén, B., du Toit, S.H.C. & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes. Unpublished technical report.

(Available as PDFs from here: Weighted Least Squares for Categorical Variables.)

There is a lot more to see on Mplus wiki, e.g. WLS vs. WLSMV results with ordinal data; the two authors are very responsive and always provide detailed answers with accompanying references when possible. Some comparisons of robust weighted least squares vs. ML-based methods of analyzing polychoric or polyserial correlation matrices can be found in:

Lei, P.W. (2009). Evaluating estimation methods for ordinal data in structural equation modeling. Quality & Quantity, 43, 495–507.

For other mathematical development, you can have a look at:

Jöreskog, K.G. (1994) On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika, 59(3), 381-389. (See also S-Y Lee's papers.)

Sophia Rabe-Hesketh and her colleagues also have good papers on SEM. Some relevant references include:

1. Rabe-Hesketh, S. Skrondal, A., and Pickles, A. (2004b). Generalized multilevel structural equation modeling. Psychometrika, 69, 167–190.
2. Skrondal, A. and Rabe-Hesketh, S. (2004). Generalized Latent Variable Modeling: Multilevel, Longitudinal, and Structural Equation Models. Chapman & Hall/CRC, Boca Raton, FL. (This is the reference textbook for understanding/working with Stata gllamm.)

Other good resources are probably listed on John Uebersax's excellent website, in particular Introduction to the Tetrachoric and Polychoric Correlation Coefficients. Given that you are also interested in applied work, I would suggest taking a look at OpenMx (yet another software package for modeling covariance structure) and lavaan (which aims at delivering output similar to those of EQS or Mplus), both available under R.

• Many thanks for all these references, including R packages. – Elvis Jan 22 '12 at 19:59

While only tangent to your goals at this point, if you continue on projects using latent variables I would highly suggest you read Denny Boorsboom's Measuring the Mind. Don't be fooled by the title, it is mainly a detailed essay on the logic of latent variables, and a large critique of classical test theory. I would say it is necessary reading if you are utilizing latent variables in a longitudinal framework. It is only about the logic of latent variables though, it has nothing about actually estimating models.

Do post back with your experiences, I have some of the references given here already, although I would like to expand my library as well. FWIW, Ken Bollen's Structural equations with latent variables was the next on my reading list (although that is only based on my opinion of his scholarly work).

Besides that I would say I enjoy the work of Bengt Muthén as well. The MPlus software is incredibly popular, and you can see all of the types of analysis that can be accomplished on the Mplus website (link to the user's guide). He also has a series of mp3 postings of his course on statistical analysis with latent variables at UCLA. I haven't listened to them all, but I suspect all are thorough introductions to whatever particular topic is covered for that weeks lecture.

• (+1) I'm really a big fan of Denny Boorsboom's papers. – chl Jan 23 '12 at 14:35
• Does the Borsboom book cover item response theory? I'm trying to do investigative work using Rasch analysis on social science surveys, and I'm interested in adding books to my library that critique CTT and recommend IRT for social science work. – Michelle Jan 23 '12 at 17:49
• @Michelle, the Borsboom book is not specific to ways in which we represent latent variables (either through IRT or other Factor analysis type models). It is simply a detailed essay on what latent variables are, and also in large part on how CTT is silly as a scientific endeavor. – Andy W Jan 23 '12 at 18:05
• @AndyW thanks for the extra information, it sounds like the book will still be a good addition to my library. – Michelle Jan 23 '12 at 18:17
• @Michelle CTT is often used as a preliminary analytical stage (see e.g. Bechger et al., Using Classical Test Theory in Combination with Item Response Theory, APM 2003 27:319) during scale construction, in order to discard items that are badly behaving. The main criticism is about the fact that CTT statistics are sample-dependent (and holds some axiomatic definition of the true score), but not all IRT models are truly 'measurement model', for some authors. – chl Jan 23 '12 at 18:21

This was the recommended text on the course I took: P.B.Kline, Principles and Practice of Structural Equation Modeling, The Guilford Press. It is an introductory text, and not heavily mathematical.

For a more mathematical, Bayesian, treatment, you could try: S-Y. Lee, Structural Equation Modeling: A Bayesian Approach, Wiley.

I'm studying SEM at the moment, using LISREL. We're using these two books:

Dr Schumaker is the instructor on my course. The first book is really good at introducing SEM, as it takes you through the process of model specification, identification, and so forth. While it is based on the LISREL software, I would expect that the general methods and interpretation of results will be independent of software.

• I would recommend Loehlin's Latent Variable Models: An Introduction to Factor, Path, and Structural Equation Analysis (2003, 4th ed., Lawrence Erlbaum Associates). That's a very good book with lot of illustrations and references. – chl Jan 22 '12 at 20:25
• The first book is excellent at stepping you through the decisions around how to trim variables from your SEM results, so that you end up with a correctly specified model. In the course I am doing, I spend a lot of time trying to work out the correct model specification, and we're using illustrative datasets. The specification -> identification -> estimation -> testing -> modification process is well covered in the first book. – Michelle Jan 22 '12 at 20:31

Kline's book is excellent. For a quick intro as a paper see

Gefen, D. 2000. Structural equation modeling and regression: Guidelines for research practice. CAIS. Volume 4. http://aisel.aisnet.org/cais/vol4/iss1/7/

Hox, J.J. and Bechger, T.M. An introduction to structural equation modeling. Family Science Review. 11:354-373. http://joophox.net/publist/semfamre.pdf

Lei, P.W. and Wu, Q. 2007. Introduction to Structural Equation Modeling: Issues and Practical Considerations. Educational Measurement: Issues and Practice. http://dx.doi.org/10.1111/j.1745-3992.2007.00099.x

Grace, J. 2010. Structural Equation Modeling for Observational Studies. The Journal of Wildlife Management. 72:14-22 http://dx.doi.org/10.2193/2007-307