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| This model is used for panel analysis. Notably there must be a significant number of time periods; 3 at minimum. | This model is used for panel analysis. Notably there must be a significant number of time periods; 3 at minimum. Additionally, this type of model cannot handle 'gaps' in measurements over time, or non-uniform time periods. |
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| Classical models are inappropriate for these circumstances because the repeated measurements are strongly correlated, and because they are stationary. [[Statistics/FixedEffectsModel|fixed effects models]] can correct for the former but not the latter. | Classical models are inappropriate for these circumstances because the repeated measurements are expected to be correlated, and because they are stationary. [[Statistics/FixedEffectsModel|Fixed effects models]] can correct for the former but not the latter. |
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| The second level model estimates the individuals' latent initial status and latent growth rate in terms of individual-level predictors. |
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| It is possible to extend the first level model with quadratic or cubic growth rates (i.e., ''T^2^'', ''T^3^'') if there are high numbers of time periods. | These loadings are expressed as '''''Λ''' = [1 1 1 1; 0 1 2 3]''. Then the covariance of error terms is introduced like: {{attachment:sem2.svg}} It is possible to extend the first level model with quadratic or cubic growth rates (i.e., ''T^2^'', ''T^3^'') if there are high numbers of time periods. The quadratic term's loadings would be expressed as ''[0 1 4 9]''. The second level model estimates the individuals' latent initial status and latent growth rate in terms of individual-level predictors. |
Latent Growth Model
A latent growth model is a specialized latent variable model for application to panel data. Alternative names for these methods are growth curve modeling and latent growth curve analysis.
Contents
Description
This model is used for panel analysis. Notably there must be a significant number of time periods; 3 at minimum. Additionally, this type of model cannot handle 'gaps' in measurements over time, or non-uniform time periods.
This model is useful when the measured outcomes feature time-dependent accumulation or inertia. A classical model is designed to be stationary; if all predictors are 0, then the estimated outcome reverts to the intercept as a reversionary level. In contract, some outcomes are expected to carry some growth or at least directionality from prior time periods. This growth pattern will be modeled as a latent variable.
Classical models are inappropriate for these circumstances because the repeated measurements are expected to be correlated, and because they are stationary. Fixed effects models can correct for the former but not the latter.
The first level model estimates individuals' growth over time as yit = biT + ai. Every individual i has a different intercept a and slope b. In the first time period (t=0), biT cancels out and the estimate ŷ is simply the intercept. For every subsequent time period, the slope is accumulated.
These slopes and intercepts vary across individuals, and can be described by overall averages and variances.
This is fit into the more general framework of SEM like:
These loadings are expressed as Λ = [1 1 1 1; 0 1 2 3].
Then the covariance of error terms is introduced like:
It is possible to extend the first level model with quadratic or cubic growth rates (i.e., T2, T3) if there are high numbers of time periods. The quadratic term's loadings would be expressed as [0 1 4 9].
The second level model estimates the individuals' latent initial status and latent growth rate in terms of individual-level predictors.
Latent growth models are essentially the same as multilevel models with random intercepts and random slopes.
Reading Notes
Democratic Trajectories in the Third Wave: Aligning Theory and Methods, Aníbal Pérez-Liñán and Scott Mainwaring, 2025