Piecewise latent growth models: beyond modeling linear-linear processes

Abstract

Piecewise latent growth models (LGMs) for linear-linear processes have been well-documented and studied in recent years. However, in the latent growth modeling literature, advancements to other functional forms as well as to multiple changepoints or knots have been nearly non-existent. This manuscript deals with three extensions. The first is to a piecewise latent growth model incorporating higher-order polynomials. The second is to extend the basic framework to three phases. The last extension is to inherently nonlinear functions. In these extensions, the changepoint(s) is a parameter to be estimated and may be fixed or allowed to vary across subjects as an application warrants. The approaches are developed and two illustrative empirical examples from psychology are used to highlight the methodological nuances. Annotated statistical software is provided to make these elaborations accessible to practitioners and methodologists.Change that occurs in distinct phases or regimes can be modeled using piecewise (spline) functions (see, e.g., Gallant & Fuller, 1973; Seber & Wild, 1989, Chapter 9) where the overall piecewise regression model is a complex amalgamation of submodels, each of which is associated with a distinct interval of time (see, e.g., Cudeck & Klebe, 2002). Because the functional form in each segment can be tailored to fit the localized data (Cudeck & Harring, 2010), the piecewise growth model is quite flexible and can accommodate a variety of modeling scenarios that are not adequately captured by mathematical functions for singlestage change processes (Grimm et al., 2011;Sterba, 2014) and/or often yield parameters with more meaningful and substantive interpretations.For example, consider the two-phase linear-linear piecewise model depicted in Fig. 1. The changepoint distinguishes between two phases comprising the overall change process. On the left-hand side of the changepoint are the

The Long-Term Changes in Dynamic Risk and Protective Factors Over Time in a Nationwide Sample of Dutch Forensic Psychiatric Patients

“…Consequently, an alternative so-called the linear piecewise model was tested. Linear piecewise models are used for modeling changes that deviate from a simple linear trajectory; when the rate of change during the specific time window differs from the rate of change during another time window ( 54 ). The simplest variant of the linear piecewise model is the two-phase model with two linear slopes and a single change point ( 55 ).…”

“…In this model, the first linear slope represents the changes that occur during the first phase of the study, and the second linear slope describes the trajectories during the second phase. The change point represents the fixed time point where these two linear slopes are to be joined ( 54 ). Based on the plot (see Figure 1 ; clinical scale), the change point was assumed to be at T3, and therefore the unconditional two-phase linear piecewise model was tested.…”

Extending growth mixture model to assess heterogeneity in joint development with piecewise linear trajectories in the framework of individual measurement occasions.

“…Second, the linear-linear piecewise functional form can be utilized to approximate nonlinear change patterns in a variety of situations in practice, for example, earlier and later development stages in psychology, treatment and follow-up phases in psychotherapy, and adolescent and adulthood periods in behavioral science. It could be extended further to analyze a phenomenon with three stages and estimate fixed knots (Harring et al, 2021). Thus, the proposed model can also be extended for joint development in cases where three phases are scientifically justified.…”

Extending Latent Basis Growth Model to Explore Joint Development in the Framework of Individual Measurement Occasions

“…Multiple parametric functional forms, such as polynomial, exponential, logistic, and Jenss-Bayley growth curves, have been proposed to capture characteristics of nonlinear change patterns. Earlier studies, for example, Harring et al (2006); Flora (2008); Dumenci et al (2019); Kohli (2011); ; ; Kohli et al (2015a,b); Liu and Perera (2021); Harring et al (2021) have also proposed and employed several piecewise models (i.e., the growth curve model with semi-parametric functions), such as linear-linear piecewise, linear-quadratic piecewise, and piecewise functions with three segments, to describe nonlinear curves which have different change rates to different stages in longitudinal processes. These studies have also demonstrated that the piecewise functions are versatile and valuable in modeling repeated outcomes in multiple domains, including developmental, cognitive, and biomedical.…”