Modeling Growth in Latent Variables Using a Piecewise Function

Abstract

Latent growth curve models with piecewise functions for continuous repeated measures data have become increasingly popular and versatile tools for investigating individual behavior that exhibits distinct phases of development in observed variables. As an extension of this framework, this research study considers a piecewise function for describing segmented change of a latent construct over time where the latent construct is itself measured by multiple indicators gathered at each measurement occasion. The time of transition from one phase to another is not known a priori and thus is a parameter to be estimated. Utility of the model is highlighted in 2 ways. First, a small Monte Carlo simulation is executed to show the ability of the model to recover true (known) growth parameters, including the location of the point of transition (or knot), under different manipulated conditions. Second, an empirical example using longitudinal reading data is fitted via maximum likelihood and results discussed. Mplus (Version 6.1) code is provided in Appendix C to aid in making this class of models accessible to practitioners.

Notes

¹Cohen's (1988) heuristic values were computed and reported for η². Justification for using the same benchmark for η² _p as for η² comes from Sapp (2006), who stated that the difference between the two effect-size indices becomes negligible as sample size increases.

²There was a small effect size (η² _p = 0.02) for the mean slope of the first segment.

^aThe parameter estimates are of the transformed model described in Equation 7.

^bThe variance of the random effect for β₂ and β₃ was determined to be zero. A subsequent model was fitted that allowed only variance components of random effects for β₁ to be estimated. The lone variance parameter was statistically significant at the .05 level.