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

Abstract

This study proposes transformation functions and matrices between coefficients in the original and reparameterized parameter spaces for an existing linear-linear piecewise model to derive the interpretable coefficients directly related to the underlying change pattern. Additionally, the study extends the existing model to allow individual measurement occasions and investigates predictors for individual differences in change patterns. We present the proposed methods with simulation studies and a real-world data analysis. Our simulation study demonstrates that the method can generally provide an unbiased and accurate point estimate and appropriate confidence interval coverage for each parameter. The empirical analysis shows that the model can estimate the growth factor coefficients and path coefficients directly related to the underlying developmental process, thereby providing meaningful interpretation.

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

“…The simplest linear spline function is a bilinear spline growth model (BLSGM, or a linear-linear piecewise model). This functional form helps identify a process that is theoretically two stages with different rates of change (Dumenci et al, 2019; Liu, 2019; Riddle et al, 2015); more importantly, it is also capable of approximating other nonlinear trajectories (Kohli, Hughes, et al, 2015; Kohli, Sullivan, et al, 2015; Liu, Perera, Kang, Kirkpatrick, et al, 2019; Sullivan et al, 2017). Harring et al (2006) developed a BLSGM to estimate a fixed knot with the assumption that the knot is at an identical point in time for all individuals.…”

“…By relaxing the assumption of the same knot location across all individuals, Preacher and Hancock (2015) extended the BLSGM to estimate a knot while considering variability so that the knot is a growth factor (i.e., a random coefficient in the multilevel model framework) in addition to the intercept and two slopes. For interpretation purposes, Grimm et al (2016a), Kohli (2011), Kohli et al (2013), Liu (2019), and Liu, Perera, Kang, Kirkpatrick, et al (2019) proposed to transform the mean vector and variance-covariance matrix of the reparameterized growth factors to the original setting, for the BLSGM with fixed and random knots, respectively.…”

Jenss-Bayley Latent Change Score Model with Individual Ratio of Growth Acceleration in the Framework of Individual Measurement Occasions

“…Mehta and West (2000); Mehta and Neale (2005) defined the 'definition variables' as manifested variables which adjust model parameters to individual-specific values. This approach has been widely employed in the latent growth curve modeling framework for nonlinear development, for example, Preacher and Hancock (2015); Sterba (2014); Liu et al (2019); Liu and Perera (2020). In the LCSM framework, Grimm and Jacobucci (2018) proposed to specify the latent true scores at individual measurement occasions to obtain the latent change scores during individually-varying time intervals.…”

“…In the proposed model, we consider the ratio of growth acceleration γ as the fourth growth factor in addition to η 0 , η 1 and η 2 . Earlier studies, for example, Hancock (2012, 2015); Liu et al (2019); Liu and Perera (2020); Grimm et al (2016b) have demonstrated how to obtain an additional growth factor by utilizing the first-order Taylor series expansion for latent growth curve models, and Liu et al (2019); Liu and Perera (2020) have shown that the approximation introduced by the Taylor series expansion only affects the model performance slightly by simulation studies. In this article, we extend the first-order Taylor series expansion to the latent change score modeling framework and estimate both fixed and random effects of the ratio of growth acceleration.…”

“…Note that Equation 4does not fit into the LCSM since it specifies a nonlinear relationship between the response dy i | t=t i(j− 1 2 ) and the growth factor γ i , which cannot be estimated in the SEM framework directly. We extend the first-order Taylor series expansion, an approach that has been utilized in the latent growth curve modeling framework to address such nonlinear relationships Hancock, 2012, 2015;Liu et al, 2019;Liu and Perera, 2020;Grimm et al, 2016b) to the LCSMs, and express the derivatives as a linear combination of the growth factors that related to the growth rate (see Appendix A.1 for details of Taylor series expansion). That is, the derivative specified in Equation 4 can be written as…”