Estimating knots and their association in parallel bilinear spline growth curve models in the framework of individual measurement occasions.

Abstract

Multiple existing studies have developed multivariate growth models with nonlinear functional forms to explore joint development where two longitudinal records are associated over time. However, multiple repeated outcomes are not necessarily synchronous. Accordingly, it is of interest to investigate an association between two repeated variables on different occasions, for example, how a short-term change of one variable affects a long-term change of the other(s). One statistical tool for such analyses is longitudinal mediation models. In this study, we extend latent growth mediation models with linear trajectories (Cheong et al., 2003) and develop two models to evaluate mediational processes where the bilinear spline (i.e., the linear-linear piecewise) growth model is utilized to capture the change patterns. We define the mediational process as either the baseline covariate or the change of covariate influencing the change of the mediator, which, in turn, affects the change of the outcome. We present the proposed models by simulation studies. Our simulation studies demonstrate that the proposed mediational models can provide unbiased and accurate point estimates with target coverage probabilities with a 95% confidence interval. To illustrate modeling procedures, we analyze empirical longitudinal records of multiple disciplinary subjects, including reading, mathematics, and science test scores, from Grade K to Grade 5. The empirical analyses demonstrate that the proposed model can estimate covariates' direct and indirect effects on the change of the outcome. Through the real-world data analyses, we also provide a set of feasible recommendations for empirical researchers. We also provide the corresponding code for the proposed models.

Obtaining Interpretable Parameters From Reparameterized Longitudinal Models: Transformation Matrices Between Growth Factors in Two Parameter Spaces

“…Second, the model in Equation 1 specifies a nonlinear relationship between the y i j and the growth factor (or random coefficient in the mixed-effects modeling framework) γ i and then cannot be estimated in the SEM framework directly (Grimm et al, 2016, Chapter 12). As shown in Tishler and Zang (1981), Seber and Wild (2003), Grimm et al (2016, Chapter 11), and Liu (2019), the expression of repeated outcome in Equation 1 before and after the knot can be unified by reparameterization (see Online Appendix A.1 for the detailed derivation of reparameterization). Following Browne and du Toit (1991), Grimm et al (2016, Chapter 12), and Liu (2019), we then write the repeated outcome as a linear combination of all four growth factors by the Taylor series expansion (see Online Appendix A.2 for the detailed derivation of Taylor series expansion).…”

“…As shown in Tishler and Zang (1981), Seber and Wild (2003), Grimm et al (2016, Chapter 11), and Liu (2019), the expression of repeated outcome in Equation 1 before and after the knot can be unified by reparameterization (see Online Appendix A.1 for the detailed derivation of reparameterization). Following Browne and du Toit (1991), Grimm et al (2016, Chapter 12), and Liu (2019), we then write the repeated outcome as a linear combination of all four growth factors by the Taylor series expansion (see Online Appendix A.2 for the detailed derivation of Taylor series expansion). Accordingly, the model specified in Equation 1 can be written as a standard LGC model with reparameterized growth factors…”

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.…”

“…In longitudinal studies, researchers often collect measurements of two or more repeated outcomes with interest in how each outcome changes over the study duration. Longitudinal processes in multiple domains, such as development (Shin et al, 2013;Liu and Perera, 2021;Peralta et al, 2020), behavioral Duncan, 1994, 1996), and biomedical (Dumenci et al, 2019), rarely occur in isolation, although most studies of change have focused on analyses of a univariate repeated outcome. Some recent applications considered the joint observation of multiple longitudinal outcomes.…”

“…In one situation, the same cohort of individuals provides longitudinal records of the outcomes of interest. For example, achievement scores of multiple disciplinary subjects are obtained in developmental studies (Shin et al, 2013;Liu and Perera, 2021;Peralta et al, 2020), so that researchers are able to determine whether increased achievement in one subject is related to increased (or decreased) performance in another. Similarly, multiple endpoints are collected in clinical trials (Dumenci et al, 2019), which allows investigators to evaluate treatment effects with consideration of multiple endpoints comprehensively.…”