Adding measurement error covariances over time? #64
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Hi, First of all, big thanks for the developer of the ctsem package and the excellent accompanying tutorials, published recently. They’ve made it possible for more basic guys like me to start learning and working with continuous-time models—really impressive and much appreciated! I’m genuinely fascinated by this "world," even if I’m naturally struggling a lot and feel really humble in the face of the complexities that come with understanding dynamic systems... Lately, I’ve been working on estimating a basic bivariate ctsem model with cross-effects, auto-effects, intercepts, diffusion variances and covariances, and T0 values. For each latent variable, I have three or four indicators in the measurement model. Attached you can find the equations for the model. My question concerns the possibility of specifying associations between indicator-level measurement errors across time (e.g., indicator 1 T-->indicator 1 TΔ). For example, in longitudinal CFA, it’s common practice to allow the measurement errors of the same indicator to correlate over time (this seems theoretically a plausible assumption in many cases). I’m wondering whether there’s a way to implement something similar in ctsem. My intuition (and some supporting data patterns) is that if there are substantial correlations in item-level measurement error over time—which seems like a reasonable assumption in my case based on conducted longitudinal CFAs—but these are not modeled, particularly the auto-effects of the latent variables might be inflated, thus, biasing the estimates of whole Drift matrix. I came across Oud, Voelkle, and Driver (2018), where you mention that this kind of modeling might be possible using the MANIFESTTRAIT option ctsemOMX, but I’m struggling to figure out how to implement it in newer version. So, is there a specific way to specify only the over-time correlations in measurement error, without introducing other cross-variable trait structures? Thanks again for your work on this fantastic tool! Best regards, |
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Replies: 3 comments 7 replies
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Hi Jaako, thanks for the nice words, they are always welcome -- and feeling humble is a really good start when it comes to making sense of dynamic systems in psychology, I think! the original ctsem (now packaged as ctsemOMX) included specific matrices to manage individual differences in the latent intercept (cint) and manifest intercept (manifestmeans). Modern ctsem allows individual differences in every parameter, and the old formulation stopped making sense, so those two specific matrices (TRAITVAR and MANIFESTTRAITVAR) were dropped in favour of the current more general setup. By default, ctsem now freely estimates the variance / correlations of individual differences in T0MEANS, CINT, and MANIFESTMEANS, whenever the parameters are estimated. You can see this in your model.pdf, you have individual differences in the latent t0 and latent intercept, and because the MANIFESTMEANS are fixed to zero, no other differences. This brings us to the first important point -- generally it's a very strong assumption to assume that the intercepts of all your measures for a single factor will be the same, in typical factor analysis these would be freely estimated. Because you are also estimating the latent intercept freely, you can't currently estimate all of them freely -- one would need to be restricted, generally to zero, and the others estimated. However, based on your (very reasonable) worries about measurement errors correlating over time, a better approach might be to fix the cint to zero (the default), and estimate all the manifestmeans (the default). This will essentially estimate the stable component of each of your measurements (and the correlation to other measurements stable factors), and ensure the remaining part of the data is fluctuating around zero (i.e., you don't need a latent intercept). This approach assumes an autocorrelation of 1 across measurement occasions -- more sophisticated approaches where you model each measurement as a separate process and estimate the autocorrelation, as well as the link to a higher level factor process, are possible, but probably beyond what you need at this point. Good luck :) |
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That's pretty close I think! You would need to free the diffusion variance, but leave the correlations zero, to force the covariance into the latent (latent latent?) factors. I also think you probably want to estimate the initial mean for the latent latents. |
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Thank you again for your precise help! Excellent — I think I follow the logic behind freeing the diffusion variances of the indicators. My (evidently bad) initial intuition was to fix the T0means of latent latent to zero as well (like when modeling separate trends alongside auto-cross dynamics). At least, when modeling T0 means of latent-latent I guess it is important to not include their random effects (or fix the correlations of these random effects with other T0 and intercept random effects to zero). Naturally, the next step is to expect some cross-dynamics between the indicators :).... I'm now excited to see what happens in the estimation phase! |
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Yeah you're basically dependent on some covariance for identification. Be careful with the interpretation, the individual processes will absorb a lot of the variation I expect... |
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I will! I think that in general, if i am lucky, I see quite similar results as with the "sum variables" based the indicators. My intuition says that summing the indicators and then estimating 2-dimensional model with drift, diffusion (co-)variances, intercepts and measurement errors etc. is also one (yet less sophisticated) approach to avoid the challenges of "time-evolving indicator dynamics". Naturally, then I make much assumptions about the dimensionality and the phenomena... |
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sensitivity checks are pretty critical for this kind of modelling, so no bad thing :) |
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Oh, I just noticed -- do you really want to assume GAD6/7 has the same factor loading as GAD3/4? If not, you need to give them a different label! |
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Thank you Charles for your excellent and educative answer, I get it right away I think! And I apologize, I run the model in a secure platform from which you cannot get anything out without the secure permission procedure, just wrote the model code quickly, ofc, I do not expect the same factor loadings :)! BTW, I think you could become a rich (or at least highly popular) man by arranging a summer school etc. on continuous-time modeling in behavioral sciences. best regards, |
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I have seen many many wild looking models by now, I take little for granted ;) Definitely thinking of a summer school like setup here in Zurich or surrounds at some point, but for now I'm contributing to this offering (edit: which I find out is already overbooked unfortunately): Maybe next time around it can make me rich haha. |
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Hi Jaako, thanks for the nice words, they are always welcome -- and feeling humble is a really good start when it comes to making sense of dynamic systems in psychology, I think!
the original ctsem (now packaged as ctsemOMX) included specific matrices to manage individual differences in the latent intercept (cint) and manifest intercept (manifestmeans). Modern ctsem allows individual differences in every parameter, and the old formulation stopped making sense, so those two specific matrices (TRAITVAR and MANIFESTTRAITVAR) were dropped in favour of the current more general setup.
By default, ctsem now freely estimates the variance / correlations of individual differences in T0MEANS, CINT, …