Latent Growth Models with Floors, Ceilings, and Random Knots

Abstract

In longitudinal/developmental studies, individual growth trajectories are sometimes bounded by a floor at the beginning of the observation period and/or a ceiling toward the end of the observation period (or vice versa), resulting in inherently nonlinear growth patterns. If the trajectories between the floor and ceiling are approximately linear, such longitudinal growth patterns can be described with a linear piecewise (spline) model in which segments join at knots. In these scenarios, it may be of specific interest for researchers to examine the timing when transition occurs, and in some occasions also to examine the levels of the floors and/or ceilings if they are not known and fixed. In the current study, we propose a reparameterized piecewise latent growth curve model so that a direct estimation of the random knots (and, if needed, a direct estimation of random floors and ceilings) is possible. We derive the model reparameterization using a 4-step structured latent curve modeling approach. We provide two illustrative examples to demonstrate how the proposed reparameterized models can be fitted to longitudinal growth data using the popular SEM software Mplus and we supply the full coding for applied researchers’ reference.

Keywords:

• Latent growth models
• floor and ceiling effects
• piecewise models
• random knots

Notes

1 As discussed earlier, our paper focused on the piecewise model with a linearly increasing middle segment ($\beta >0$). In some research scenarios, researchers may be interested in studying the decreasing trajectories for certain behaviors (e.g., aggression or drug use) and thus require a piecewise model with linearly decreasing growth for the middle component ($\beta <0$). For readers who are interested, the parameterization of such a model is provided in online supplemental Appendix G.

2 In this target function and in some of the functions presented later in the manuscript, we have included terms that take the square root of a squared quantity (e.g., $\sqrt{{\left({\gamma }_1-t\right)}^2}$). This may seem unnecessary and burdensome at first; however, they play a critical role in preserving the piecewise nature of the growth trajectories (detailed proofs are available upon request). Further, we chose to use the form of square root over a squared value instead of using the absolute value operator for computational convenience. It is much easier to take the derivatives in the current form; additionally, Mplus cannot handle absolute value operations in model constraints command.

3 In this study, we specify the model as an LGM given the advantages of latent variable modeling. But for readers who are interested, it can also be understood as a mixed effects model where time is the level-1 unit nested within person, the level-2 unit. The means and variance of the latent growth factors correspond to the fixed effects and random effects variances, respectively. The variance of time-specific residuals corresponds to the level-1 residual variance within the HLM framework. But in HLM, they are constrained to be homogenous across time points.

4 Theoretically, the errors can have a non-zero covariance structure. In most SLCM literature, however, it is more often assumed that the errors are independent over time (Blozis, 2007b; Preacher & Hancock, 2015). Further, within the SEM framework, it is not required that the residual variances are identical across time, although researchers can sometimes choose to constrain the residual variances to be equal, which makes an LGM analogous to a multilevel model.

5 When the lower censoring limit ${\tau }_l$ is more than 3 standard deviations below the mean of the parent normal distribution $\mu ,$ $\varphi \left(\frac{\mu -{\tau }_l}{\sigma }\right)$ approaches 0, $\Phi \left(\frac{{\tau }_l-\mu }{\sigma }\right)$ approaches 0, and $\Phi \left(\frac{\mu -{\tau }_l}{\sigma }\right)$ approaches 1; hence, the expectation of the censored normal distribution $\mathrm{E}\left(y\right)$ is approximately $\mu .$ Similarly, when the upper censoring limit ${\tau }_u$ is more than 3 standard deviations above the mean of the parent normal distribution $\mu ,$ the expectation of the censored normal distribution $\mathrm{E}\left(y\right)$ is approximately $\mu .$

6 The simulated data set used in this example is available for download at http://www.terpconnect.umd.edu/<yifeng94/

7 We acknowledge that it is somewhat arbitrary in terms of which variables are to be treated as censored, and which ones are censored from below or from above. Looking at descriptive statistics and spaghetti plots can be helpful, although the decision ultimately rests upon the judgment of the researcher.

8 As shown in Figure 5, in the current example, ${\mathrm{\gamma }}_{2g},{\mathrm{\gamma }}_{\mathit{fg}},{\mathrm{\gamma }}_{\mathit{cg}}$ were all constrained to be zero, assuming this is consistent with researchers’ prior knowledge and the substantive theory.

9 For simplicity in this illustration, we did not distinguish the endogenous latent factor ${\eta }_1$ from the other three exogenous latent factors (${\eta }_2,$ ${\eta }_f,$ ${\eta }_c$) by introducing different matrix notations. Here ${\mathbf{\eta }}_{\mathrm{i}}$ is a vector that contains both the endogenous and exogenous latent factors. Similarly, below we used $\mathbf{\Phi }$ to represent the variance-covariance matrix for all the latent factors (both endogenous and exogenous), instead of further introducing more symbols for this illustrative example.

10 For convergence reasons, the covariance between ${\eta }_1$ and ${\eta }_c\hspace{0.17em}$was constrained to be equal to the covariance between ${\eta }_2\hspace{0.17em}$ and ${\eta }_c.$

11 Based on the global model fit indices, however, the floor-linear-ceiling piecewise latent growth model (AIC = 17,798.813; BIC = 17,889.133) fitted better than both the linear model (AIC = 18,242.819; BIC = 18,301.723) and the Richards curve model (AIC = 17,958.923; BIC = 17,994.265).

12 In this example, under H₀, there is one parameter on the boundary (the variance of the random floor factor, k = 1) and three unconstrained parameters of interest (u = 3). According to Stoel et al. (2006), the resulting asymptotic distribution is a mixture of k + 1 = 2 chi-square distributions with u = 3 and u + 1 = 4 degrees of freedom.