Modeling Nonlinear Change via Latent Change and Latent Acceleration Frameworks: Examining Velocity and Acceleration of Growth Trajectories

Abstract

We propose the use of the latent change and latent acceleration frameworks for modeling nonlinear growth in structural equation models. Moving to these frameworks allows for the direct identification of rates of change and acceleration in latent growth curves—information available indirectly through traditional growth curve models when change patterns are nonlinear with respect to time. To illustrate this approach, exponential growth models in the three frameworks are fit to longitudinal response time data from the Math Skills Development Project (Mazzocco & Meyers, 2002, 2003). We highlight the additional information gained from fitting growth curves in these frameworks as well as limitations and extensions of these approaches.

Notes

¹This material can be found on the first author's website as well as information on how commonly fit growth models (e.g., linear, quadratic, cubic, latent basis) can be fit in the latent change and latent acceleration frameworks.

²The rotation of x _1n from the latent change model to the traditional model in the exponential model is

where μ₁(lcm) is the mean of x _1n in the traditional exponential model, μ₁(lcs) is the mean of x _1n in the latent change exponential model, μ₂ · exp(−μ₂) is the factor loadings from x _1n to the latent change scores at the second occasion in the latent change score model, and 1 – exp(−μ₂) is the factor loading from x _1n to the true score at the second occasion in the traditional specification. Additionally, we note that the correlations involving x _1n are invariant across frameworks.

³Estimates of the rate of change at the first occasion and estimates of acceleration at the first and second occasion were generated by including latent variables before the first measurement occasion in kindergarten. Including latent variables before the first measurement occasion enabled the inclusion of a latent change score for the kindergarten occasions and latent acceleration scores for the kindergarten and first-grade occasions. The inclusion of these latent variables does not change model fit or the model implied trajectories but provides information regarding rate of change and acceleration at the first two occasions.