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I'm curious if there are graphical techniques particular, or more applicable, to structural equation modeling. I guess this could fall into categories for exploratory tools for covariance analysis or graphical diagnostics for SEM model evaluation. (I'm not really thinking of path/graph diagrams here.)

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The term "SEM" is vague. It could also mean "Search Engine Marketing", for instance, for someone looking for statistical analysis techniques for studying ad click data or evaluating advertising effectiveness. Consider making the title more verbose. – user87 Jul 26 '10 at 9:10
drpaulbrewer, thanks for the suggestion. – ars Jul 27 '10 at 0:49

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I met Laura Trinchera who contributed a nice R package for PLS-path modeling, plspm. It includes several graphical output for various kind of 2- and k-block data structures.

I just discovered the plotSEMM R package. It's more related to your second point, though, and is restricted to graphing bivariate relationships.

As for recent references on diagnostic plot for SEMs, here are two papers that may be interesting (for the second one, I just browsed the abstract recently but cannot find an ungated version):

  1. Sanchez BN, Houseman EA, and Ryan LM. Residual-Based Diagnostics for Structural Equation Models. Biometrics (2009) 65, 104–115
  2. Yuan KH and Hayashi K. Fitting data to model: Structural equation modeling diagnosis using two scatter plots, Psychological Methods (2010)
  3. Porzio GC and Vitale MP. Discovering interaction in Structural Equation Models through a diagnostic plot. ISI 58th World Congress (2011).
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@chl: thanks! I remember plspm being announced on the semnet list -- for some reason PLS isn't as big on this side of the Atlantic, not sure why. plotSEMM looks really interesting, can't wait to play with it. – ars Oct 20 '10 at 13:57
@chl: btw, I meant to add that it's shame PLS isn't more noted here, since there seems to be a lot of exciting stuff happening around it, especially with tools being developed (e.g. SmartPLS in addition to plspm). I read some of Wold's work a while back and some of his ideas are only just being realized (e.g. "having a conversation with your data"). I really need to set aside some time to explore it more. – ars Oct 20 '10 at 16:47
@ars Do you want a list of recommend readings? I also worked with Arthur Tenenhaus who submitted a nice paper with his father (yes, Michel Tenenhaus) to Psychometrika: They are unifying all two-block methods (PCA, CCA, PLS, inter-battery, etc.) thanks to a very neat rewrite of the argmax constraint. I've been playing myself with penalized PLS/CCA (L1/L2) in genomics, but I feel it will bring more interesting on my biomedical data. – chl♦ Oct 20 '10 at 17:14
@chl: wow, you keep good company; very cool! I would really appreciate some references, thanks! – ars Oct 21 '10 at 1:02
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@ars So, I'd like to suggest the following papers from Father & Son: j.mp/dvEDgb, j.mp/csD1Yf, j.mp/dkEHq5. – chl♦ Oct 24 '10 at 21:11
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up vote 4 down vote

This is a very interesting question. Suppose that we have a 2 dimensional covariance matrix (very unrealistic example for SEM but please bear with me). Then you can plot the iso-contours for the observed covariance matrix vis-a-vis the estimated covariance matrix to get a sense of model fit.

However, in reality you will a high-dimensional covariance matrix. In such a situation, you could probably do several 2 dimensional plots taking 2 variables at a time. Not the ideal solution but perhaps may help to some extent.

Edit

A slightly better method is to perform Principal Component Analysis (PCA) on the observed covariance matrix. Save the projection matrix from the PCA analysis on the observed covariance matrix. Use this projection matrix to transform the estimated covariance matrix.

We then plot iso-contours for the two highest variances of the rotated observed covariance matrix vis-a-vis the estimated covariance matrix. Depending on how many plots we want to do we can take the second and the third highest variances etc. We start from the highest variances as we want to explain as much variation in our data as possible.

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Srikant, thanks for the response! I'm not sure what you mean by contour plots of covariances (obs v est) -- could you elaborate? Thanks. – ars Jul 27 '10 at 8:52
See this: en.wikipedia.org/wiki/Level_set. Let Sigma be a a 2 dimensional covariance matrix and Y ~ N(0, Sigma). An iso-contour line would plot the set of points Y for which f(Y|sigma) = c where c is a constant. Note that Y is a 2-dimensional vector. You would choose various values of c and hence obtain different iso-contour lines which would give you a sens of the spread of the distribution. – user28 Jul 27 '10 at 10:54
@Srikant, thanks for the suggestion. I spent some time trying it out and it seems like a good start at getting a quick visual comparison, especially when the fit is bad. – ars Jul 28 '10 at 4:35
up vote 2 down vote

I suppose you could do a multidimensional scaling of the correlation or covariance matrix. It's not exactly structural equation modelling, but it might highlight patterns and structure in the correlation or covariance matrix. This could then be formalised with an appropriate model.

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Thanks Jeromy. Just read the Wikipedia entry for MDS -- seems like it could lead somewhere. – ars Aug 9 '10 at 16:24
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If there is an interaction effect(or even otherwise) you could use the sofware ITALASSI v1.2 (free software) to get 2D and 3D views

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