Studying Between-Subject Differences in Trends and Dynamics: Introducing the Random Coefficients Continuous-Time Latent Curve Model with Structured Residuals

Abstract

The recently proposed continuous-time latent curve model with structured residuals (CT-LCM-SR) addresses several challenges associated with longitudinal data analysis in the behavioral sciences. First, it provides information about process trends and dynamics. Second, using the continuous-time framework, the CT-LCM-SR can handle unequally spaced measurement occasions and describes processes independently of the length of the time intervals used in a given study. Third, it is a hierarchical model. Thus, multiple subjects can be analyzed simultaneously. However, subjects might also differ in dynamics and trends. Therefore, in the present paper, we extend the CT-LCM-SR to capture these differences as well. This newly proposed random coefficients continuous-time latent curve model with structured residuals (RC-CT-LCM-SR) is introduced theoretically and technically. Additionally, we provide an illustrative example with data from the Health and Retirement Study (HRS), and we show how the RC-CT-LCM-SR can be used to study multiple sources of between-subject differences over time.

Keywords:

• Continuous-time modeling
• longitudinal data analysis
• random effects

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Additionally, it should be noted that if there is no systematic growth/decline over time (i.e., the trend parameter is zero), the intercept reduces to the stable, trait-like mean level (e.g., average stress level) of a given individual (Delsing & Oud, 2008). Stable trait-like differences have been discussed intensively in the context of longitudinal analysis in recent years (e.g., Hamaker et al., 2015).

2 In contrast to the well-known (discrete-time) autoregressive effect, the special feature of the CT auto-effect is that this carry-over effect is modeled as a function of the length of time interval. In the present article, we limited our presentation to so-called “stable” processes, corresponding to discrete-time autoregressive effects in the range of (0, 1) and CT auto-effects in the range of (−$\infty ,$ 0), respectively (cf. e.g., Ryan et al., 2018).

3 However, the combination nopriors = TRUE and optimize = FALSE is not possible in ctsem (version 3.7.4) and thus priors are always used with MCMC estimation by default (but the priors can of course be adapted by the user—see, Driver & Voelkle, 2021).

4 When using optimization-based estimation, one strategy for assuring that results are trustworthy is re-estimating the models several times with different starting values (cf. Gelman et al., 2013). This can be done by just rerunning the ctStanFit function repeatedly because starting values are always randomly initialized or by setting a new seed before running ctStanFit. This helps to identify whether a stable solution has been found. Via the stanoptimis function, it is also possible to switch the optimization algorithm or use importance sampling for further robustness checks. With smaller sample sizes than in the data example presented in the following section but models with equivalent complexity, MCMC estimation might provide more robust solutions (cf. Driver & Voelkle, 2021). However, run time can be (very) high with complex models and huge datasets using MCMC in ctsem (several hours to many days; Hecht & Zitzmann, 2020).

5 This subject-specific asymptotic diffusion can be interpreted according to Jongerling et al. (2015) as the subject-specific “sensitivity, reactivity, and exposure to [unmeasured, temporal] external events that influence the process under investigation.”

6 A significant positive effect of Education on the diffusion parameter would have implied that fluctuation/variability of cognitive functioning states are higher for people with higher education status. A significant negative effect on the CT auto parameter would have implied that people with a higher status of education tended to exhibit less inert cognitive functioning in later life.