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Am I understanding correctly that growth mixture model is just Gaussian mixtures applied to coefficients of polynomials fitted to the time-series data?

For example, we have 1000 cases, with 3 measurements each. We fit, say, a quadratic equation to each which gives us 3 extra values per case (quadratic, linear, constant). Then we fit Gaussian mixture model to those 3 coefficients, which gives us clustering of the trajectories.

Is that it, or are growth mixtures something more involved?

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It is often something more involved, though what you have said is not that far off for many applications.

In a mixture model, generally speaking, you can choose to freely estimate parameters across levels of an otherwise unobserved multinomial variable, the number of levels of which are specfied by the user.

The underlying model/parameters could just be a univariate mean and variance from an assumed normal distribution, which asks the question, "Is the observed distribution of my variable consistent with the idea that it is made up of a mixture of subpopulations, each with their specfic normal distribution described by a subgroup specfic mean and/or variance?"

Alternately, you could have something very complicated as the underlying model. In most things that get called growth mixture models (GMM), that underlying model is a latent growth curve model (or sometimes just latent curve model).

That said, the parameters that can be freely estimated across latent subgroups in a GMM will depend on how the LGCM is parameterized.

1) You will have $\ge 1$ number of latent growth factors, each with an estimated mean and variance, depending on the model implied funcational form (linear, quadratic, cubic, piece-wise, etc). Sometimes the means and variances are both freely estimated across groups; often the variances are constrained as zero, meaning that all of the variation in the observed measures is modelled as a function of group membership.

2) You will have time specfic error variances; these may be freely estimated across groups, or contrained as equal (or anything in between).

3) You might include autocorrelated errors, which could be estimated or assumed, and theoretically allowed to vary across latent subgroups.

4) You could estimate a freed-loading model, where the functional form of the curve is achieved by estaimting the factor loadings beween the "slope" factor(s) and observed variables. This is my favorite approach.

All of the above are fair game to be freely estimated across groups - or to be set at some values, or constrained as equal...and for all of this to happen in some groups but not others.

I have a draft chapter on GMM here, which cites some better overviews of the topic.

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    $\begingroup$ As a word of caution, and may be you cover that in your chapter: the interpretation of the model results as subpopulations may be way too strong. What a mixture model may be capturing instead is non-normality of the underlying distribution of random effects. E.g., you'd approximate a $t$-distribution with two normal distributions that have the same mean and different variance, and you'd approximate a skewed distribution by a sequence of normals with offset means and mixture shares that are monotone in the means. Dan Bauer had some papers on this equivalency (indeterminacy?) 5-10 years ago. $\endgroup$ – StasK Feb 12 '14 at 18:52
  • $\begingroup$ Absolutely...loads of issues to consider beyond the question posed above. $\endgroup$ – D L Dahly Feb 12 '14 at 19:53

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