Relaxing the Proportionality Assumption in Latent Basis Models for Nonlinear Growth

ABSTRACT

Change over time is frequently nonlinear, which can present unique statistical challenges. Generally, different approaches for nonlinear growth engage in a tradeoff between interpretable parameters, expedient estimation, or how specific the model must be about the nature of the nonlinearity. Latent basis models are one method that can circumvent tradeoffs that other methods necessitate: it is quick to estimate, simple to interpret, and does not require specification of a particular trajectory. However, latent basis models require a restrictive proportionality assumption that is not required with other methods, which can limit its applicability with empirical data. This paper discusses this proportionality assumption and shows how it can be relaxed by reparameterizing the latent basis model as a multilevel structural equation model. We provide an example to show how relaxing proportionality can improve parameter estimates and person-specific growth curves. We also walkthrough Mplus code to facilitate fitting the model to empirical data.

KEYWORDS:

• Growth modeling
• latent growth
• longitudinal data analysis
• nonlinear growth

Notes

¹ Mixed effects models can be coerced to accommodate estimated values for time (e.g., Grimm et al., 2016, p. 243), but the programmatic manipulation requires that data be balanced and time-structured, which effectively limits it to the models that can be transformed into structural equation framework and that lack the flexibility typically associated with mixed effect models.

² For those not familiar with the Bayesian Markov Chain Monte Carlo (MCMC) method used to obtain these results, MCMC provides a distribution for each parameter, rather than a single point estimate as in maximum likelihood. The estimates listed in Table 2 are the medians of the posterior distribution for each parameter, rather than single-point estimates.

³ Fitting both models with MCMC estimation and comparing the DIC would similarly not be an option because the models differ in the number of latent variables. Asparouhov, Hamaker, and Muthén (2018) note that DIC values from Mplus are not comparable when the number of latent variables differ between models (p. 367).