A Continuous-Time Approach to Intensive Longitudinal Data: What, Why, and How?

Time to Intervene: A Continuous-Time Approach to Network Analysis and Centrality

“…Current anxiety levels in part determine feelings of stress a second, a minute, ten minutes and an hour from now to different degrees, meaning that there are likely different lagged relationships between those variables over a whole range of different time-intervals. This perspective is consistent with viewing psychological phenomena as continuoustime processes, a perspective described in detail by Boker (2002) and promoted by proponents of CT statistical models in psychology (e.g., Coleman, 1968;Deboeck & Preacher, 2016;Driver et al, 2017;van Montfort et al, 2018;Ou et al, 2019;Oud & Jansen, 2000;Oravecz et al, 2011;Ryan et al, 2018;Voelkle et al, 2012). In SEM terms, we can represent a CT process as a path model in which there are infinitely many latent variable values in-between any two measurement occasions, spaced an infinitesimally small time-interval apart, as depicted in the left-hand panel of Figure 2 (see also Deboeck & Preacher, 2016).…”

“…Typically, the variables themselves are centered around this mean value (e.g., Asparouhov, Hamaker, & Muthén, 2018), and so the intercept term is often omitted for notational simplicity (c = 0), a convention we will adopt throughout the remainder of the paper. The random shocks τ push the system away from equilibrium, and the lagged parameters Φ determine how the variables react to these shocks, eventually returning them to equilibrium over time (for more details see for instance Ryan et al, 2018;Strogatz, 2015;.…”

“…In the previous section we have detailed the potentially dire implications that the time-interval problem has for researchers using a DT-VAR based approach to network analysis. In this section we will present a potential solution to this problem in the form of a Continuous-Time (CT) approach to dynamical network analysis, based on CT lagged regression models (e.g., Oud & Jansen, 2000;Oravecz, Tuerlinckx, & Vandekerckhove, 2011;Ryan et al, 2018;van Montfort et al, 2018;Ou, Hunter, & Chow, 2019;Driver, Oud, & Voelkle, 2017;Voelkle et al, 2012). We will begin by introducing the basic notion behind continuous-time models, describing how a simple CT process can be modeled using a stochastic differential equation, and how the parameters of that model can be interpreted as encoding moment-to-moment direct effects.…”

“…The precise connection between the first-order SDE and the DT-VAR model will be made clearer below, but for now it suffices to note that these two models describe precisely the same type of dynamic behavior: Each variable Y (t) has an equilibrium value, here defined by its mean; the Wiener process pushes the system away from this equilibrium; and the drift parameters A determine how the variables react to these shocks, eventually returning the system back to equilibrium over time (Strogatz, 2015;Ryan et al, 2018). 2 The key difference between the two models is in how this behavior is encoded.…”

“…Numerous authors have argued that this problem could be resolved by mod-eling psychological processes as unfolding continuously over time, that is, using continuous-time (CT) models (e.g., Boker, 2002;Oud & Delsing, 2010;van Montfort, Oud, & Voelkle, 2018;Ryan, Kuiper, & Hamaker, 2018;Voelkle, Oud, Davidov, & Schmidt, 2012). CT models deal with this issue by explicitly modeling the timeinterval dependency of parameters.…”