Path Diagram for a Latent Class Regression

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rabil's picture
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I cannot find a clear example of a path diagram for a latent class regression where the manifest variables are longitudinal (at varying time points for each subject). The slope and intercept for time for each response are different for each latent class. The closest thing seems to be a longitudinal latent growth curve models but they seem to require the same time points for all subjects. I would greatly appreciate information on this if only where to look.

rabil's picture
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Path Diagram for a Latent Class Regression

I think possibly growth mixture modeling might be the same thing as a latent class regression. Growth Modeling With Latent Variables Using Mplus by Muthen & Muthen shows some diagrams.

neale's picture
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Mixture Distributions

To my knowledge, there is no formal diagram syntax for structural equation models with mixture distributions - of which latent class analysis, latent profile analysis, factor mixture models, growth curve mixture models and regime switching models are all examples. I have represented these models by drawing different path diagrams for each mixture component, with a multiplier for the or class membership probability. In some software, such as classic Mx, two diagrams would imply two groups with different data attached to each. If the diagrams are drawn correctly (with double-headed loops to represent variances - even if fixed to 1 as in many latent exogenous variables) they are mathematically complete. This suggests that if the same data are represented in two different diagrams by, say, the same labels for the observed variables, then a mixture distribution is implied. It remains unclear how to represent the class membership probabilities in this format. I quite like the idea of scaling their axes according the class membership probabilities, for which zooming in and out would suffice to inspect and edit on a computer screen, but in print this could prove illegible. Another approach might be to display the class membership probability or formula for it in an object other than a circle, triangle or square. Pentagons anyone? Or a value inside an outline of a large multiplication sign?

rabil's picture
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This is the diagram I

This is the diagram I constructed based on "Multilevel Growth Mixture Models for Classifying Groups" by Palardy and Vermunt, 2010. It doesn't include the class probabilities but I think it conceptually describes the model. But I'm not sure whether a latent class regression is the same thing as a growth mixture model. They would seem to do the same thing but I'm not sure whether LCR uses latent variables for the intercepts and slopes.

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rabil's picture
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I should mention that in the

I should mention that in the diagram, there are two longitudinal responses Y1 and Y2 that are functions of time t. The alpha's denote intercepts and the beta's slopes (all latent). My understanding is that a crude approach would be to compute intercepts and slopes for each subject using simple regression and then use a clustering technique on them. I'm using flexmix to estimate the model (which has 4 longitudinal responses) although I would like to know how to use OpenMx for this also.

rabil's picture
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LCR versus growth mixture model

I think the diagram I constructed is reasonable for a latent class regression. The alpha's and beta's have no disturbance terms so there is no variation in the latent profile among subjects. I think if such terms were added we would have a growth mixture model.

Ryne's picture
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Your comment about variances

Your comment about variances is consistent with my understanding of the distinction between latent class and mixture models, though I'd be careful to say that there is no variation within a particular latent profile.

Diagrams for multiple group, latent class and mixture models are tough because they are unstandardized. Mplus has pushed for diagrams including categorical latent variables, with the caveat that most categorical latent variables are nominal, which aren't otherwise modeled using structural diagrams. Others use a separate diagram for each group/class, separate them with lines or boxes, and label each box by group or class number. There have been similar issues with thresholds (diamonds or triangles on paths) and definition variables (paths pointing directly to paths, diamonds again). Once one gets past circles squares and triangles for intercepts/means, it's safe to assume that you'll have to explain all other symbols.

I'll close with a relevant xkcd on the proliferation of standards:
http://xkcd.com/927/