Identification of a latent change score model with categorical indicators
Timothy Wong
Hello lavaaners,
I am in the process of writing up syntax for a tri-variate latent change score model, measured at two time points. I would like each of the latent variables to have a measurement model, consisting of multiple categorical indicators. I’ve drawn a simplified diagram of the LCS model below (showing the measurement model for only one variable, and structural paths):
I am now wondering how to specify the lavaan syntax, to allow the model to be identified. In particular, I would like to know which item intercepts are constrained (to 0).
Rogier Kievit has written an excellent paper (and provided accompanying code) showing how to identify LCS models with continuous indicators. Going through his code (“2_MILCS.R”), it seems that at all time points, the intercept of the first indicator of the latent variable is constrained to zero while all other item intercepts are freely estimated.
On the other hand, Svetina et al. have also written an excellent paper, which shows how to identify a multi-group CFA with categorical indicators (and which builds upon Wu and Estabrook 2016). In Svetina's paper – and from my own experience of using measEq.syntax to establish measurement invariance in CFA models – one of the conditions of delta parameterization is that “ν = 0”. Unless I have interpreted the maths symbols incorrectly, I think this means that the intercepts of all indicators are constrained to zero.
Unfortunately, neither paper provides guidance on my situation specifically, as I want to combine the concepts from both papers (LCS structural model and categorical indicators measurement model). This makes me uncertain which parts of these papers are applicable for identifying my model.
If anyone has had experience running these kinds of model, or knows of a paper which addresses this issue, I would be very grateful for your help!
Rönkkö, Mikko
Hi,
The general form of LCS requires observations from four time points. If you apply some constraints, you can get identification with three time points. With two timepoints, you cannot really do an LCS. Consider the figure below (https://doi.org/10.1177/1094428120963788):
The key features of LCS model is that it allows you to fit time trends and dynamics in the same model. You are leaving out the time trend and are thus not doing an LCS. Your model is simply a two-period cross-lagged model that is specified in a complex way. You can simplify by eliminating the deltas and just regress man2 on man1, slp1, and dep1. The result will be the same except that the coefficient of man1 will be 1 larger.
That is, because
man2 = delta-man + man1
delta-man = beta0 + beta1 man1 + beta2 dep1+ beta3 slp1 + u
we can write
man2 = (beta0 + beta1 man1 + beta2 dep1+ beta3 slp1 + u) + man1
which simplifies to
man2 = beta0 + (beta1 +1) man1 + beta2 dep1+ beta3 slp1 + u
You can estimate this model without delta and just substract 1 from the coefficient of man1 to get the effect to delta-man
Mikko
From:
lav...@googlegroups.com <lav...@googlegroups.com> on behalf of Timothy Wong <thetimw...@gmail.com>
Date: Thursday, 4. May 2023 at 8.28
To: lavaan <lav...@googlegroups.com>
Subject: Identification of a latent change score model with categorical indicators
Hello lavaaners,
I am in the process of writing up syntax for a tri-variate latent change score model, measured at two time points. I would like each of the latent variables to have a measurement model, consisting of multiple categorical indicators. I’ve drawn a simplified diagram of the LCS model below (showing the measurement model for only one variable, and structural paths):
I am now wondering how to specify the lavaan syntax, to allow the model to be identified. In particular, I would like to know which item intercepts are constrained (to 0).
Rogier Kievit has written an excellent paper (and provided accompanying code) showing how to identify LCS models with continuous indicators. Going through his code (“2_MILCS.R”), it seems that at all time points, the intercept of the first indicator of the latent variable is constrained to zero while all other item intercepts are freely estimated.
On the other hand, Svetina et al. have also written an excellent paper, which shows how to identify a multi-group CFA with categorical indicators (and which builds upon Wu and Estabrook 2016). In Svetina's paper – and from my own experience of using measEq.syntax to establish measurement invariance in CFA models – one of the conditions of delta parameterization is that “ν = 0”. Unless I have interpreted the maths symbols incorrectly, I think this means that the intercepts of all indicators are constrained to zero.
Unfortunately, neither paper provides guidance on my situation specifically, as I want to combine the concepts from both papers (LCS structural model and categorical indicators measurement model). This makes me uncertain which parts of these papers are applicable for identifying my model.
If anyone has had experience running these kinds of model, or knows of a paper which addresses this issue, I would be very grateful for your help!
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Timothy Wong
Rönkkö, Mikko
Hi,
If you take out the time component, the model will be identified. But calling it an LCS model would be misleading because it is just an ARCL model where the cross-lagged effects are expressed as difference from 1. See my explanation in the original email and Usami’s work (10.1080/10705511.2015.1066680) and the reference therein.
Mikko
From: lav...@googlegroups.com <lav...@googlegroups.com> on behalf of Timothy Wong <thetimw...@gmail.com>
Date: Monday, 8. May 2023 at 8.41
To: lavaan <lav...@googlegroups.com>
Subject: Re: Identification of a latent change score model with categorical indicators
Hi Mikko,
Thanks a lot for the suggestion. I've been advised to look at ARCL models before, and have read Usami et al. (2015) comparing ARCL and LCS models.
I agree that the ARCL model seems easier to identify, as there is no change score and hence no need to estimate the variance or intercept of the change score.
However, my understanding is that the change score variance and intercept do carry meaningful information - the former, about whether change in a latent variable is uniform or heterogenous between time points, and the latter, the "average" change in the level of a latent variable between time points. So I would prefer to use a LCS model if possible.
If I exclude the time trend component of the LCS, will my model be identifiable?
Tim
On Thursday, May 4, 2023 at 4:12:24 PM UTC+10 mikko....@jyu.fi wrote:
Hi,
The general form of LCS requires observations from four time points. If you apply some constraints, you can get identification with three time points. With two timepoints, you cannot really do an LCS. Consider the figure below (https://doi.org/10.1177/1094428120963788):
The key features of LCS model is that it allows you to fit time trends and dynamics in the same model. You are leaving out the time trend and are thus not doing an LCS. Your model is simply a two-period cross-lagged model that is specified in a complex way. You can simplify by eliminating the deltas and just regress man2 on man1, slp1, and dep1. The result will be the same except that the coefficient of man1 will be 1 larger.
That is, because
man2 = delta-man + man1
delta-man = beta0 + beta1 man1 + beta2 dep1+ beta3 slp1 + u
we can write
man2 = (beta0 + beta1 man1 + beta2 dep1+ beta3 slp1 + u) + man1
which simplifies to
man2 = beta0 + (beta1 +1) man1 + beta2 dep1+ beta3 slp1 + u
You can estimate this model without delta and just substract 1 from the coefficient of man1 to get the effect to delta-man
Mikko
From: lav...@googlegroups.com <lav...@googlegroups.com> on behalf of Timothy Wong <thetimw...@gmail.com>
Date: Thursday, 4. May 2023 at 8.28
To: lavaan <lav...@googlegroups.com>
Subject: Identification of a latent change score model with categorical indicatorsHello lavaaners,
I am in the process of writing up syntax for a tri-variate latent change score model, measured at two time points. I would like each of the latent variables to have a measurement model, consisting of multiple categorical indicators. I’ve drawn a simplified diagram of the LCS model below (showing the measurement model for only one variable, and structural paths):
I am now wondering how to specify the lavaan syntax, to allow the model to be identified. In particular, I would like to know which item intercepts are constrained (to 0).
Rogier Kievit has written an excellent paper (and provided accompanying code) showing how to identify LCS models with continuous indicators. Going through his code (“2_MILCS.R”), it seems that at all time points, the intercept of the first indicator of the latent variable is constrained to zero while all other item intercepts are freely estimated.
On the other hand, Svetina et al. have also written an excellent paper, which shows how to identify a multi-group CFA with categorical indicators (and which builds upon Wu and Estabrook 2016). In Svetina's paper – and from my own experience of using measEq.syntax to establish measurement invariance in CFA models – one of the conditions of delta parameterization is that “ν = 0”. Unless I have interpreted the maths symbols incorrectly, I think this means that the intercepts of all indicators are constrained to zero.
Unfortunately, neither paper provides guidance on my situation specifically, as I want to combine the concepts from both papers (LCS structural model and categorical indicators measurement model). This makes me uncertain which parts of these papers are applicable for identifying my model.
If anyone has had experience running these kinds of model, or knows of a paper which addresses this issue, I would be very grateful for your help!
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