1.1. The DT-VAR Model

The DT-VAR model is a single-subject time-series model that describes dynamic relationships between variables measured repeatedly over time. Lagged regression parameters encode the effect of a variable on itself (an auto-regressive effect) or another variable (a cross-lagged effect) at the next measurement occasion (i.e., at a lag of one). This model can be written as

(1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{Y}}_\tau = {\varvec{c}} + \varvec{\Phi } {\varvec{Y}}_{\tau -1} + \varvec{\epsilon }_\tau \end{aligned}$$\end{document}

where given p variables, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times 1$$\end{document} vector of random variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}$$\end{document} at occasion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is regressed on the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times 1$$\end{document} vector of those same variables at the previous occasion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}_{\tau -1}$$\end{document} . The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times p$$\end{document} matrix of lagged regression parameters is denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }$$\end{document} , while the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times 1$$\end{document} vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{c}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\epsilon }_\tau $$\end{document} denote the intercepts and random shocks, respectively, the latter being normally distributed with mean zero and variance-covariance matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Psi }$$\end{document} (Hamilton, Reference Hamilton1994). The multivariate mean of the DT-VAR model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\mu }$$\end{document} can be expressed as a function of the intercepts and lagged parameters ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\mu } = ({\varvec{I}} - \Phi )^{-1}{\varvec{c}}$$\end{document} ) and can be thought of as the equilibrium or attractor of the dynamical system. If we assume the data are centered, the intercept term can be omitted ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{c}} = {\varvec{0}}$$\end{document} ), a convention we will adopt throughout the remainder of the paper.

In qualitative terms, the model describes a system where the random shocks \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\epsilon }_\tau $$\end{document} push the system away from its equilibrium, and the lagged parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }$$\end{document} determine how the variables react to these shocks, eventually returning them to equilibrium over time (for more details, see, among others, Ryan et al., Reference Ryan, Kuiper, Hamaker, Montfort, Oud and Voelkle2018; Strogatz, Reference Singer2015; Haslbeck et al., in press). A distinction can be made between DT-VAR models which exhibit “positive” and “negative” auto-regression: in the former case, the system returns to equilibrium in an exponential fashion over time; In the latter, variables switch their sign (from positive to negative and vice versa) at each subsequent occasion. In the univariate case, this is determined by the sign of the auto-regressive parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} , but in the multivariate case by the sign of the eigenvalues of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }$$\end{document} . Positive auto-regression systems are found in many psychological time-series applications (e.g., Bringmann et al., Reference Bringmann, Vissers, Wichers, Geschwind, Kuppens, Peeters and Tuerlinckx2013; Koval & Kuppens, Reference Koval and Kuppens2012; Oravecz & Tuerlinckx, Reference Oravecz and Tuerlinckx2011). In this paper, we will assume that our system of interest exhibits positive auto-regression. Two crucial assumptions of the DT-VAR model in general are that the same amount of time (denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} ) elapses between two subsequent measurement occasions, and that the underlying process is stationary, which entails that the means, variance and covariances, and lagged regression parameters remain the same over time.Footnote 1

Figure. 1

Path model (left-hand side) and network (right-hand side) representations of two four-variable DT-VAR models. In the path models, the presence of an arrow linking two variables denotes some nonzero dependency between them, conditional on all variables at the previous wave. For the networks, the arrows represent auto-regressive and cross-lagged regression parameters in a first-order DT-VAR model. Solid red arrows denote parameters while dashed blue arrows represent negative parameters.

The DT-VAR model can be represented as either a path model, as shown in the left-hand panel of Fig. 1a, or as a dynamical network structure, as shown in the right-hand panel, where the nodes represent the random variables, and the edges represent the values of the lagged parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }$$\end{document} (Bringmann et al., Reference Bringmann, Vissers, Wichers, Geschwind, Kuppens, Peeters and Tuerlinckx2013; Epskamp et al., Reference Epskamp, van Borkulo, van der Veen, Servaas, Isvoranu, Riese and Cramer2018). The lagged parameters in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }$$\end{document} are typically interpreted as direct effects of these variables on each other over time. As an example, take it that the four variables in Fig. 1a represent (repeated measurements of) Stress ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_1$$\end{document} ), Anxiety ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_2$$\end{document} ), Self-Consciousness ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_3$$\end{document} ) and feelings of Physical Discomfort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_4$$\end{document} ). We will refer throughout to the dynamical system composed of these four time-varying processes as the Stress-Discomfort system. We can see from the parameter values in the dynamical network that all variables share reciprocal cross-lagged relationships with all other variables, resulting in a completely connected network. Typically, a cross-lagged parameter such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{41} = 0.05$$\end{document} would be interpreted as the direct effect of current Stress ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{1,\tau }$$\end{document} ) on Physical Discomfort at the next measurement occasion ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{4,{\tau + 1}}$$\end{document} ), conditional on (i.e., controlling for) current feelings of Anxiety, Self-Consciousness and Physical Discomfort \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Y_{2,\tau }, Y_{3,\tau }, Y_{4,{\tau }})$$\end{document} . This parameter is weakly positive, leading to the interpretation that a high level of current Stress has a small positive direct effect on feelings of Physical Discomfort at the next occasion.

1.2. Intervention Targets from DT-VAR Models

To identify which variables should be considered targets for an intervention based on a DT-VAR model, psychology researchers have mainly used two approaches: (a) path-specific effects, which are inspired by the SEM literature (Bollen, Reference Bollen1987); and (b) centrality measures, which come from the network analysis literature (Freeman, Reference Freeman1978; Opsahl et al., Reference Opsahl, Agneessens and Skvoretz2010).

Path-specific effects have been used to describe the total, direct and indirect effects of one variable on another, and can be calculated using the well-known path-tracing rules from the SEM literature (Bollen, Reference Bollen1987). For instance, the total effect of Stress levels now ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{1,\tau }$$\end{document} ) on Physical Discomfort two measurement occasions later ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{4,\tau +2}$$\end{document} ) is the sum of the direct effect pathways (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{1,\tau } \rightarrow Y_{4,\tau +1} \rightarrow Y_{4,\tau +2}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{1,\tau } \rightarrow Y_{1,\tau +1} \rightarrow Y_{4,\tau +2}$$\end{document} ), and the indirect effect pathways through the mediating variables Anxiety and Self-Consciousness (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{1,\tau } \rightarrow Y_{2,\tau +1} \rightarrow Y_{4,\tau +2}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{1,\tau } \rightarrow Y_{3,\tau +1} \rightarrow Y_{4,\tau +2}$$\end{document} , respectively; Cole & Maxwell, Reference Cole and Maxwell2003). If we interpret \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }$$\end{document} parameters as direct causal effects, we may suggest that interventions should target variables that have strong direct or total effects on others. Alternatively, we could search for those mediators through which the strongest indirect effects pass (Groen et al., Reference Groen, Ryan, Wigman, Riese, Penninx, Giltay, Wichers and Hartman2020; Bernat et al., Reference Bernat, August, Hektner and Bloomquist2007; Bramsen et al., Reference Bramsen, Lasgaard, Koss, Shevlin, Elklit and Banner2013). For instance, based on the parameters in Fig. 1a, we might suggest Anxiety as an intervention target due to the relatively strong lag-one direct effect on Physical Discomfort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{42} = .08$$\end{document} ), or because it is a mediator of the largest lag-two indirect effect, from Stress to Physical Discomfort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{1,\tau } \rightarrow Y_{2,\tau + 1} \rightarrow Y_{4,\tau +2} = -.005$$\end{document} ).

An alternative approach to finding a target for intervention comes from the network approach and is based on centrality measures (e.g., Bringmann et al., Reference Bringmann, Vissers, Wichers, Geschwind, Kuppens, Peeters and Tuerlinckx2013; Fisher & Boswell, Reference Fisher and Boswell2016; Kroeze et al., Reference Kroeze, van der Veen, Servaas, Bastiaansen, Oude Voshaar, Borsboom and Riese2017; Epskamp et al., Reference Epskamp, van Borkulo, van der Veen, Servaas, Isvoranu, Riese and Cramer2018; Rubel et al., Reference Rubel, Fisher, Husen and Lutz2018; Bak et al., Reference Bak, Drukker, Hasmi and van Os2016; Bringmann et al., Reference Bringmann, Lemmens, Huibers, Borsboom and Tuerlinckx2015; Bastiaansen et al., Reference Bastiaansen, Kunkels, Blaauw, Boker, Ceulemans and Chen2019). Centrality measures are used to summarize the relations a particular variable has with the network as a whole, typically summing over the individual relations that variable has with all other variables in the network. While the precise connection between path-specific effects and centrality measures for DT-VAR models has not yet been described in the literature a close inspection of the computation and interpretation of many popular centrality measures reveals that they are very similar to path-tracing effects: specifically, many centrality measures are interpreted as capturing either total, direct or indirect effects, and in turn, these measures are often closely related to summaries of the corresponding path-specific effects. Here, we will mention three such measures; for the exact connection between these measures and path-tracing effects, the reader is referred to Appendix A.

First, the Two-Step Expected Influence measure ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{EI}}^{(2)}_{i}$$\end{document} ; Robinaugh et al., Reference Robinaugh, Millner and McNally2016; Kaiser & Laireiter, Reference Kaiser and Laireiter2018) is typically interpreted as a summary of total effects emanating from the variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} . In path-tracing terms, it is the sum of lag-one direct effects and lag-two total effects. As such, variables with a high Two-Step Expected Influence could be expected to exert a high overall influence on the system, making it an attractive intervention target. Second, the One-Step Expected Influence ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{EI}}^{(1)}_{i}$$\end{document} ; Robinaugh et al., Reference Robinaugh, Millner and McNally2016; Kaiser & Laireiter, Reference Kaiser and Laireiter2018) and Out-Strength centrality (Opsahl et al., Reference Opsahl, Agneessens and Skvoretz2010) measures are interpreted as summarizing direct effects. They are both sums of lag-one direct effects, with the latter taking the absolute value (and so, we will calculate only the expected influence measure in the remainder). Third, Betweenness Centrality (BC) is interpreted as indicating the degree to which a variable funnels information flow, similar to how mediating variables funnel indirect effects (e.g., Bringmann et al., Reference Bringmann, Vissers, Wichers, Geschwind, Kuppens, Peeters and Tuerlinckx2013; Opsahl et al., Reference Opsahl, Agneessens and Skvoretz2010; Freeman, Reference Freeman1977). This measure is conceptually similar to determining which variables are strong mediators, although paths are calculated by summing, rather than multiplying parameters, as in path-tracing rules. The first column of Table 1 contains the value of these three centrality metrics for each node in the Stress-Discomfort network shown in Fig. 1a.

Table 1

Two-Step Expected Influence ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{EI}}^{(2)}$$\end{document} ), One-Step Expected Influence ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{EI}}^{(1)}$$\end{document} ) and Betweenness Centrality ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{BC}}$$\end{document} ) for each of the four variables in the 1-h ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }(\Delta t = 1)$$\end{document} , Fig. 1a) and half-hour ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }(\Delta t = 0.5)$$\end{document} , Fig. 1b) networks.

In each column, the largest centrality values are highlighted in bold.

1.3. The Time-Interval Problem and its Consequences

From this review, it is clear that network analysis based on the DT-VAR model relies critically on the interpretation of a cross-lagged regression parameter as the direct effect of one process on another over time (for similar interpretations of DT-VAR models in the time series and panel data literature, see Cole & Maxwell, Reference Cole and Maxwell2003; Hamaker et al., Reference Hamaker, Kuiper and Grasman2015; Bulteel et al., Reference Bulteel, Tuerlinckx, Brose and Ceulemans2016). Of course, the interpretation of any model parameter estimated from observational data as a causal effect or as informative about hypothetical interventions should be approached with due caution. Developments in the causal inference literature have shown that such interpretations are highly dependent on the validity of assumptions regarding, among others, the ignorability of unobserved confounding variables, our ability to intervene on the system of interest in a modular way (i.e., without altering the rest of the causal system), and of course, the correct specification of the statistical model itself (Pearl, Reference Pearl2009; Robins, Reference Robins, Green, Hjort and Richardson2003; Eichler & Didelez, Reference Eichler and Didelez2010 VanderWeele, Reference Suls, Green and Hillis2015). As such we can understand the use of path-tracing and centrality measures as a first approximation to a possible underlying causal effect, an approximation which is seemingly valid under highly idealized conditions.

However, a well-known critique of DT-VAR models casts doubt on the veracity of such an interpretation even in such ideal conditions. This critique focuses on the property that lagged regression parameters exhibit time-interval dependency, hereby referred to as the time-interval problem (Gollob & Reichardt, Reference Gollob and Reichardt1987; Oud & Jansen, Reference Oud and Jansen2000; Reichardt, Reference Reichardt2011; Voelkle et al., Reference Voelkle and Oud2012; Deboeck & Preacher, Reference Deboeck and Preacher2016; Kuiper & Ryan, Reference Kuiper and Ryan2018).Footnote 2 Gollob and Reichardt (Reference Gollob and Reichardt1987) offer a classic example of this problem regarding the effect of taking aspirin on headache levels. This effect is negligible 2 min after ingestion, moderate after 30 min, strong after two hours and zero 24 hour later.

The phenomenon of time-interval dependency is a straightforward implication of assuming an underlying DT-VAR model, as can be shown with a simple example. Take it that the parameters that are introduced in Fig. 1a represent the lagged relationships of the Stress-Discomfort system based on measurements taken at one-hour intervals; we denote these parameters as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }(\Delta t = 1)$$\end{document} . In theory, we could also have observed all variables in the Stress-Discomfort system at twice that rate, that is, at half-hour intervals. The path model for this is depicted in the left panel of Fig. 1b, where the half-hour measurements that could have been observed are depicted as latent variables (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}(t = 1\frac{1}{2})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}(t = 2\frac{1}{2})$$\end{document} ). The effects matrix relating the half-hour realizations of the process is denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }(\Delta t = \frac{1}{2})$$\end{document} . From the time-series literature, it is known that the parameters of these two models are related by the expression

(2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{}[\varvec{\Phi }\left( {1/2}\right) ]^2 = \varvec{\Phi }(1) \end{aligned}$$\end{document}

that is, by squaring the matrix of parameters at the shorter interval, we obtain the parameters at twice that interval (Hamilton, Reference Hamilton1994).Footnote 3 It is important to note here that squaring a matrix is not equivalent to squaring the parameters of that matrix: instead, any given parameter in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }(1)$$\end{document} is a function of multiple parameters in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }(\frac{1}{2})$$\end{document} . For instance, the cross-lagged parameter which regresses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{4, \tau + 1}$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{1,\tau }$$\end{document} can be re-written in terms of the shorter-interval parameters as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{42}(1) = \phi _{22}(1/2)\phi _{42}(1/2) + \phi _{42}(1/2)\phi _{44}(1/2) + \phi _{12}(1/2)\phi _{41}(1/2) + \phi _{32}(1/2)\phi _{43}(1/2)$$\end{document} .

When we compare the dynamical network based on the one-hour and half-hour parameters (i.e., Fig. 1a, b, respectively), three consequences of the time-interval problem for network researchers become apparent. A first consequence is that networks based on different time-intervals can lead to seemingly contradictory conclusions regarding the sign, size and relative ordering of effects. For example, in the one-hour network, Stress and Anxiety both have positive lagged effects on Physical Discomfort, with the effect of Anxiety being slightly larger; yet, in the half-hour network, the corresponding lagged relations are both strongly negative, with the effect of Stress being larger (cf. Kuiper & Ryan, Reference Kuiper and Ryan2018). Since centrality measures are mere summaries of lagged parameters, this implies that different time-intervals between the observations are likely to lead to different centrality measures and, as a result, to different suggestions for intervention targets.

A second consequence of the time-interval problem is that, if data were obtained with unequal intervals and this is not accounted for, then the estimated parameters are a blend of the lagged relationships at different intervals present in the data. Although inserting missing observations can somewhat correct for unequal intervals (such as implemented in the DSEM module in Mplus; Asparouhov et al., Reference Asparouhov, Hamaker and Muthén2018), the results of these techniques can at best only approximate the lagged parameters for a single target time-interval (De Haan-Rietdijk et al., Reference De Haan-Rietdijk, Voelkle, Keijsers and Hamaker2017).

A third consequence of the time-interval problem is that the interpretation of any lagged parameter as a direct effect becomes questionable. Specifically, based on the relationship in Eq. (2), the lagged parameters of the one-hour network \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }(1)$$\end{document} should be interpreted as total rather than direct effects (Deboeck & Preacher, Reference Deboeck and Preacher2016; Aalen et al., Reference Aalen, Røysland, Gran, Kouyos and Lange2016).Footnote 4 Take for example in the one-hour path model the cross-lagged relation from current Anxiety ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{2,\tau }$$\end{document} ) to Physical Discomfort an hour later ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{4,\tau +1}$$\end{document} ), controlling for current values of all other variables. This parameter ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{42}(1) = 0.077$$\end{document} ) has a seemingly intuitive interpretation as a direct effect when we consider only observed values of the Stress-Discomfort system. However, when we examine how these variables are related to one another at half-hour intervals, we see that this relationship is in fact made up of a number of different pathways through latent values of our processes in between measurement occasions. These include direct paths ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{2,\tau } \rightarrow Y_2(1\frac{1}{2}) \rightarrow Y_{4,\tau + 1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{2,\tau } \rightarrow Y_4(1\frac{1}{2}) \rightarrow Y_{4,\tau + 1}$$\end{document} ) as well as indirect paths through latent values of Stress ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{2,\tau } \rightarrow Y_1(1\frac{1}{2}) \rightarrow Y_{4,\tau + 1}$$\end{document} ) and Self-Consciousness ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{2,\tau } \rightarrow Y_3(1\frac{1}{2}) \rightarrow Y_{4,\tau + 1}$$\end{document} ).

Taken together, this shows current practice in dynamical network analysis—using summaries of DT-VAR parameters to find intervention targets—is flawed due to the time-interval problem. However, our presentation here also highlighted one potential solution to the time-interval problem: decomposing lagged relationships between observations into truly direct and indirect effects operating over a shorter time-interval. This decomposition opens up a new perspective on how lagged relationships should be interpreted, a perspective which we can use to explore time-interval dependency, and avoid coming to misleading or contradictory choices regarding intervention targets.

2.1. Continuous-Time Processes and Differential Equations

In the previous section, we have shown how a single latent measurement wave between consecutive observations changes the way we should interpret DT-VAR parameters. Taking this approach one step further, it can be argued that for many psychological processes there can be infinitely many latent waves in-between two measurement occasions, and that such processes should be characterized as evolving continuously over time rather than in discrete “jumps” (cf. Boker, Reference Boker2002; Coleman, 1968; Deboeck & Preacher, Reference Deboeck and Preacher2016; Driver et al., Reference Driver, Oud and Voelkle2017; van Montfort et al., Reference van Elteren and Quax2018; Ou et al., Reference Ou, Hunter and Chow2019; Oud & Jansen, Reference Oud and Jansen2000; Oravecz et al., Reference Oravecz, Tuerlinckx and Vandekerckhove2011; Ryan et al., Reference Ryan, Kuiper, Hamaker, Montfort, Oud and Voelkle2018; Voelkle et al., Reference Voelkle and Oud2012). For example, it is reasonable to think that processes like stress and anxiety continue to vary in-between measurement occasions, and that, if those processes influence one another, they also do so in a continuous manner over time (for an extended discussion see Boker, Reference Boker2002). Popular methods like experience sampling, which are based on measuring individuals at random points in time, seem to adhere to this notion that we are dealing with CT processes (at least while the participant is awake). Hence, it seems reasonable to suggest that many of the target processes being studied by dynamical network researchers in psychology, can be conceptualized as CT processes (e.g., Bringmann et al., Reference Bringmann, Vissers, Wichers, Geschwind, Kuppens, Peeters and Tuerlinckx2013; Groen et al., Reference Groen, Ryan, Wigman, Riese, Penninx, Giltay, Wichers and Hartman2020; Pe et al., Reference Pe, Kircanski, Thompson, Bringmann, Tuerlinckx and Mestdagh2015; Fisher & Boswell, Reference Fisher and Boswell2016; Rubel et al., Reference Rubel, Fisher, Husen and Lutz2018; Bak et al., Reference Bak, Drukker, Hasmi and van Os2016).

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 Fig. 2 (see also Singer, Reference Voelkle and Oud2012; Deboeck & Preacher, Reference Deboeck and Preacher2016). Modeling CT processes is based on breaking down the relations between observed measurement waves into their fundamental building blocks, to obtain the truly direct lagged relationships operating over an infinitesimally small time-interval, which we will refer to as moment-to-moment effects. These continuous moment-to-moment dynamics are captured by differential equation models.

In the current paper, we will limit ourselves to considering a very simple type of differential equation model, known as a first-order stochastic differential equation, which can be thought of as the CT counterpart of a DT-VAR model which exhibits positive auto-regression.Footnote 5 It can be written as

(3) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\mathrm{d}{\varvec{Y}}(t)}{\mathrm{\mathrm{d}}t} = {\varvec{A}}{\varvec{Y}}(t) + {\varvec{W}}(t) \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\mathrm{d}{\varvec{Y}}(t)}{\mathrm{d}t}$$\end{document} on the left is the first derivative or the rate of change of the variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}$$\end{document} at time t (denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}(t)$$\end{document} ). We can think of this derivative as being equivalent to a (scaled) change score \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}(t+s) - {\varvec{Y}}(t)$$\end{document} over the shortest possible time-interval ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim s \rightarrow 0$$\end{document} ). This derivative is dependent on the current value of the variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}(t)$$\end{document} , and the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times p$$\end{document} matrix of regression parameters which relates these two is called the drift matrix, denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{A}}$$\end{document} . The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{W}}(t)$$\end{document} term represents a Wiener process, a special kind of mean-zero white noise residual term (described in greater detail by, among others, Oud & Jansen, Reference Oud and Jansen2000; Voelkle et al., Reference Voelkle and Oud2012; Voelkle & Oud, Reference van Montfort, Oud and Voelkle2013).

Like the DT-VAR model, the stochastic differential equation has an equilibrium value, here defined by its mean of zero. While the Wiener process pushes the system away from this equilibrium, the drift parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{A}}$$\end{document} determine how the variables react to these shocks, eventually returning the system back to equilibrium over time (Strogatz, Reference Singer2015; Ryan et al., Reference Ryan, Kuiper, Hamaker, Montfort, Oud and Voelkle2018). The key difference between the two models is in how this behavior is encoded. The DT-VAR describes this behavior in terms of discrete jumps, current process values determining future process values some fixed time-interval later. The stochastic differential equation describes the same behavior in terms of moment-to-moment changes, current process values determining the instantaneous rate of change of each process, over the smallest imaginable interval.

2.2. CT Networks and Interpretation

As the critical effects matrix in the differential equation model, we can interpret the drift parameters as representing truly direct moment-to-moment dependencies between our processes of interest. This interpretation of the drift matrix parameters makes them a natural choice to use as edges in a network-representation of the CT system. This type of network is known as a local dependence graph (Schweder, Reference Schweder1970; Aalen et al., Reference Aalen, Borgan, Keiding and Thormann1980; Didelez, Reference Didelez2008) with local here referring to the idea that these relationships are locally spaced in time. As such, if there is a direct moment-to-moment relationship from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{d}Y_j/\mathrm{d}t$$\end{document} ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{ji} \ne 0$$\end{document} ) we say that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_j$$\end{document} is locally dependent on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} and draw an arrow \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i \rightarrow Y_j$$\end{document} ; if there is no such relationship ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{ji} = 0$$\end{document} ) we say that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_j$$\end{document} is locally independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} and omit that arrow. By assigning a weight to these local dependencies based on the value of the drift parameters, we create a weighted local dependence graph, hereafter referred to as a CT network. The drift matrix parameters of the Stress-Discomfort system are plotted as a CT network on the right-hand side of Fig. 2, using hexagonal nodes to distinguish these from DT-VAR networks.

Figure. 2

CT-VAR path model (left-hand side) and CT network in the form of a weighted local independence graph representing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{A}}$$\end{document} (right-hand side). In the path model, the latent variables and ellipses represent an infinite number of latent unobserved process values in-between measurement occasions, spaced an infinitesimally small time-interval apart; the presence of an arrow linking two variables in the path model denotes some nonzero dependency between them, conditional on all variables at the previous “wave,” that is, a local dependency. For the networks, the arrows represent auto- and cross-effect parameters of the drift matrix in a CT-VAR model. Solid red arrows denote positive parameters and dashed blue arrows represent negative parameters (Color figure online).

The interpretation of individual drift matrix parameters is similar to that of a change-score model from the time-series literature. For researchers familiar with DT-VAR models, however, some care should be taken when interpreting the diagonal parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{ii}$$\end{document} , known as auto-effects, which encode the relationship a variable has with its own rate of change. These auto-effects are typically negative, which ensures the system tends to move back toward its equilibrium as positive deviations are followed by negative changes, and negative deviations are followed by positive changes. Moreover, the auto-effects are not bounded, and can thus run from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\infty $$\end{document} (i.e., auto-regression close to 0), to 0 (i.e., auto-regression close to 1). In our example, we see a larger negative auto-effect for Stress ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{11} = -6$$\end{document} ) than for Anxiety ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{22} = -2.5$$\end{document} ), meaning Anxiety moves back toward equilibrium less quickly than Stress after a perturbation (for more details, see for example Oravecz et al., Reference Oravecz, Tuerlinckx and Vandekerckhove2011; Ryan et al., Reference Ryan, Kuiper, Hamaker, Montfort, Oud and Voelkle2018).

From a network perspective, what is likely to be of primary interest for applied researchers is the interpretation of the off-diagonal elements of the drift matrix ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{ji}, j \ne i$$\end{document} ). These are the parameters we think of as direct moment-to-moment relationships between different processes. These off-diagonal parameters are also referred to as cross-effects, and have a very similar interpretation to cross-lagged parameters from a DT-VAR model: the negative cross-effect of Anxiety on the rate of change of Physical Discomfort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{42} = -7.3$$\end{document} ) means that an increase in the value of Anxiety will produce a decrease in the rate of change, and thus, the value of Physical Discomfort a moment later. The equivalence of these two statements can be seen by re-arranging the differential equation as an auto-regressive model over an infinitely small time-interval, as shown in Appendix B. As is the case for cross-lagged parameters, the higher the absolute value of the parameter, the greater the magnitude of the effect (for more details on the interpretation of these parameters, see also Oravecz et al., Reference Oravecz, Tuerlinckx and Vandekerckhove2011; Ryan et al., Reference Ryan, Kuiper, Hamaker, Montfort, Oud and Voelkle2018; Voelkle et al., Reference Voelkle and Oud2012). When comparing this CT network to the DT networks presented in Fig. 1, we can see that there are many fewer direct dependencies. For instance, Stress has no direct moment-to-moment effect on Physical Discomfort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{41} = 0$$\end{document} ). Furthermore, the nonzero connections are all positive, apart from the strong negative effect of Anxiety on Physical Discomfort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{42} = -7.3$$\end{document} ). Hence, this raises the question where such differences between the two networks stem from.

2.3. The Integral form: CT-VAR

The key to establishing a link between the CT and DT approaches is that the first-order stochastic differential equation defined above can also be expressed in its integral form. The latter is known as the CT-VAR or Ornstein–Uhlenbeck process (Oud & Jansen, Reference Oud and Jansen2000; Oravecz et al., Reference Oravecz, Tuerlinckx and Vandekerckhove2009; Voelkle et al., Reference Voelkle and Oud2012), and can be written as

(4) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{Y}}(t_\tau )={\varvec{e}}^{{\varvec{A}} \Delta t_\tau } {\varvec{Y}}(t_{\tau -1}) + \varvec{\epsilon }(\Delta t_\tau ) \end{aligned}$$\end{document}

where variables at the current measurement occasion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}(t_\tau )$$\end{document} are regressed on variables at the previous measurement occasion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{Y}}(t_{\tau -1})$$\end{document} . Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} refers to the measurement occasion, whereas t refers to the actual time when this measurement took place. Hence, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t_\tau = t_{\tau } - t_{\tau -1}$$\end{document} indicates the time-interval between two consecutive measurement occasions, which may differ across pairs of observations.

The above expression of the CT-VAR model is very similar to the DT-VAR model that was presented in Eq. (1). The variables are centered, and so an intercept term is again omitted, while the vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\epsilon }(\Delta t_\tau )$$\end{document} contains the residuals, which are normally distributed with mean zero and variance-covariance matrix that is also a function of the time-interval (for more details, see Oud & Jansen, Reference Oud and Jansen2000; Voelkle et al., Reference Voelkle and Oud2012; Voelkle & Oud, Reference van Montfort, Oud and Voelkle2013). In place of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }$$\end{document} matrix in Eq. (1), these lagged variables are related by the matrix exponential of the drift matrix multiplied by the time-interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{e}}^{{\varvec{A}} \Delta t_\tau }$$\end{document} . It follows from this that the key effects matrices from the CT- and DT-VAR model are related to each other by the expression

(5) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{e}}^{{\varvec{A}} \Delta t} = ({\varvec{e}}^{{\varvec{A}}})^{\Delta t} = \varvec{\Phi }(\Delta t) \end{aligned}$$\end{document}

which shows that the lagged parameters for any particular time-interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }(\Delta t)$$\end{document} can be found by taking the matrix exponential of the moment-to-moment drift matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{A}}$$\end{document} to the power of the length of that time-interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} (cf. Oud & Jansen, Reference Oud and Jansen2000; Voelkle et al., Reference Voelkle and Oud2012). Recall that to relate the half-hour and one-hour parameter matrices in Eq. (2), we also took the matrix associated with the half-hour interval and raised it to the power two to get the matrix for a one-hour interval, a very similar operation to that defined here.

2.4. Consequences of the CT-VAR for Dynamical Network Analysis

The connection that the CT-VAR model establishes between the DT-VAR and differential equation parameters has a number of important implications. First, if we know the drift matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{A}}$$\end{document} we can use the above relationship to derive how the DT lagged parameters are expected to evolve as a function of the time-interval. Figure 3 depicts the lagged relationships of the Stress-Discomfort system over a range of zero to two hours. It shows that the lagged parameters change continuously and smoothly. It also shows that the parameters reach their peak value at different time-intervals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} , and that some parameters even change sign (positive/negative) over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} . For instance, from Fig. 3b we can see that the effect of Anxiety on Physical Discomfort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{42}(\Delta t)$$\end{document} ) is strongly negative at very short intervals (around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t = 0.2$$\end{document} ) before becoming weakly positive at longer intervals (from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t = 0.6$$\end{document} to 1.2). In comparison, the peak effect of Anxiety on Stress ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{12}(\Delta t)$$\end{document} ) occurs both at a longer time-interval and with a smaller magnitude. This information yields novel insights into the underlying dynamics which cannot be obtained by inspecting a single set of DT-VAR parameters, nor can it be easily read off from the local dependencies in the CT network.

Second, the time-interval dependency of the lagged parameters shown above also implies that the network will change as we consider different time-intervals (see the Online Supplementary Materials 1 for an animation of this). As a result, the centrality measures—which are based on the network—also change as a function of the time-interval. This is shown in Fig. 4, where we see that rather different conclusions regarding intervention targets are liable to be made depending on the time-interval under consideration.

Figure. 3

Lagged regression parameters as a function of the time-interval for the Stress-Discomfort system. Black dotted lines indicate the parameter values of the half-hour and one-hour networks in Fig. 1a, b.

Figure. 4

Centrality measures as a function of the time-interval for the Stress-Discomfort system. Black dotted lines indicate the centrality values of the half-hour and one-hour networks in Fig. 1a, b.

Third, it becomes clear that the lagged regression parameters at any time-interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} should be interpreted as total rather than direct effects (Deboeck & Preacher, Reference Deboeck and Preacher2016). When the drift matrix contains the truly direct relationships, then the matrix exponent, as in both Eqs. (2) and (5), should be understood as a path-tracing operation (see Appendix B for an elaboration on how the matrix exponential relationship can be derived by re-arranging the first-order stochastic differential equation, and applying a path-tracing operation through \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim n \rightarrow \infty $$\end{document} latent measurement waves). This interpretation also explains the greater density of the DT-VAR networks in comparison to the CT networks, as the former is a sum of the direct and indirect effects defined by the latter over a particular time-interval. Yet, it is impossible to tell to what extent these are direct or indirect effects without knowledge or an estimate of the underlying drift matrix parameters.

3.1. Pulse and Press Interventions

One of the fundamental conceptual building blocks of modern causal inference is the notion of an intervention. In the framework of Pearl (Reference Pearl2009), an intervention is defined as an operation by which we manipulate the value of a variable in our causal system. This intervention is denoted using the do-operator, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$do(X = 1)$$\end{document} denoting that we intervene to set the variable X to a constant value of one. In this paper, we will consider two different basic types of do-operation, reflecting two of the most basic types of intervention often discussed in relation to dynamical systems (Bender et al., Reference Bender, Case and Gilpin1984).

A pulse intervention is an operation by which we change the value of a variable at one particular point in time. Taking the Stress-Discomfort system as an example, we can imagine that it is possible to induce a momentarily high experience of Anxiety in our participant (for instance by making the participant view a unpleasant photograph, a manipulation which has been shown to increase state anxiety in laboratory studies; Richards & Whittaker, Reference Richards and Whittaker1990; Richards et al., Reference Richards, French, Johnson, Naparstek and Williams1992). Using the do operator, we would denote such an intervention \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$do(Y_2(t) = 1)$$\end{document} meaning we intervene to set Anxiety to a value of 1 at time t. The effect of this pulse intervention on the other processes in the system depends on the time since that impulse was applied; hence, we can visualize the effect of this intervention by plotting the expected trajectories of the different variables in our system.Footnote 7 Figure 5a shows the effect that an initial intervention on Anxiety has on our Stress-Discomfort system: the other three variables leave their equilibrium, and eventually, the effect of the intervention fades and all variables return to their resting state.

A second type of intervention is the press intervention, which consists of changing the value a variable over an interval of time. For example, we may produce a longer-lasting state of high-anxiety by having participants prepare to give a public speech (e.g., Moscovitch et al., Reference Moscovitch, Suvak and Hofmann2010; Azevedo et al., Reference Azevedo, Bennett, Bilicki, Hooper, Markopoulou and Tsakiris2017), or induce longer-lasting changes in stress levels by prompting participants to engage in mindfulness meditation (e.g., Hoge et al., Reference Hoge, Bui, Marques, Metcalf, Morris, Robinaugh, Worthington, Pollack and Simon2013). Using the do-operator, we denote such an intervention for the time-interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$do(\overline{Y_2(t+\Delta t)}=1)$$\end{document} (i.e., intervening to set Anxiety to the value 1 starting from time t and lasting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} ). The effect of this press intervention is shown in Fig. 5b: we see that the other three variables leave their equilibrium, and that they eventually settle to a new equilibrium. They will only return to their old equilibrium value if the intervention is removed. Of course, the effect of this press intervention depends on the value the manipulated variable is set to: if the intervened-on variable takes on its equilibrium value, no new equilibrium will appear.

3.2. Path-Specific Effects and Interventions

We can show that CT path-specific effects, originally described by Deboeck and Preacher (Reference Deboeck and Preacher2016) and Aalen et al. (Reference Aalen, Røysland, Gran, Kouyos and Lange2016) and generalized in Appendix C, describe the effect on a target variable of either a pulse intervention, or a combination of pulse and press interventions. The effects themselves are expressed as a difference in expected value, with this difference contrasting two sets of intervention values.Footnote 8

First, a CT total effect simply describes the effect that applying a pulse intervention has on some other variable in the system. In Fig. 5a, we already saw the effect of setting Anxiety to a value of one, in that it pushed the other variables in the system away from their equilibrium. If we instead set Anxiety to its equilibrium value, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$do(Y_2(t) = 0)$$\end{document} , none of the variables would move away from equilibrium. Let’s say we are primarily interested in the effect of pulses to Anxiety on the expected value of Physical Discomfort some time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} later. We define this as the total effect

(6)

To be able to actually compute the above expression, we need to plug in a model for the expected values. Here, we use the CT-VAR model, which results in

(7) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textit{TE}}_{24}(\Delta t) = {{\varvec{e}}^{{\varvec{A}} \Delta t}} _{\left[ 42\right] } \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{e}}^{{\varvec{A}} \Delta t}} _{[42]}$$\end{document} denotes the parameter in the fourth row, second column of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{e}}^{{\varvec{A}} \Delta t}$$\end{document} . This expression for the total effect is actually identical to the path-tracing definition of a total effect given by Deboeck and Preacher (Reference Deboeck and Preacher2016) and Aalen et al. (Reference Aalen, Røysland, Gran, Kouyos and Lange2016) (for details, see Appendices C and D).

Figure. 5

Illustration of pulse (denoted by an empty diamond) and press (denoted by a filled diamond) interventions and their relationship with total effects (TE), direct effects (DE) and indirect effects (IE). a Shows the effect of a pulse intervention on Anxiety ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_2(t)$$\end{document} ) on the values of Stress ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_1(t)$$\end{document} ), Self-Consciousness ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_3(t)$$\end{document} ) and Physical Discomfort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_4(t)$$\end{document} ). b Shows a corresponding press intervention on Anxiety. c shows a combination of pulse intervention to Anxiety and press interventions to keep Stress and Self-Consciousness fixed at their equilibrium of zero. The consequence of this combination of interventions d shows the total, direct and indirect effect of Anxiety on Physical Discomfort. The total effect ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TE}}(\Delta t)$$\end{document} ) is simply the trajectory of Physical Discomfort in (a). The direct effect ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{DE}}(\Delta t)$$\end{document} ) is the trajectory of Physical Discomfort in (c). The indirect effect ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{IE}}(\Delta t)$$\end{document} ) is the difference between those two trajectories ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TE}}(\Delta t) - {\textit{DE}}(\Delta t)$$\end{document} ).

Second, CT direct effects simply described the consequence of a pair of interventions: a pulse intervention to change the cause variable, just as we did for the total effect, and another press intervention to keep fixed one or more mediating variables. Take for example the effect of applying a pulse to increase Anxiety, while simultaneously applying a press intervention to keep Stress and Self-Consciousness fixed at equilibrium. This is visualized by the trajectories in Fig. 5c. As before, Anxiety starts at a high level—as it was set to 1—and dissipates back to equilibrium. Because Stress and Self-Consciousness are kept fixed at all moments in time, Physical Discomfort is pushed even further from equilibrium than before; by press-intervening on the mediators, we no longer activate the compensating positive feedback loops that are present in the total effect (see also Fig. 2). We can express this direct effect as the difference between two conditional expectations

(8)

By plugging in the CT-VAR parameters for each conditional expectation, we can express the effect of this intervention as

(9)

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{A}}^{(D[-1,-3])}$$\end{document} denotes the drift matrix in which the indirect pathway parameters that link Anxiety to Physical Discomfort are set to zero (that is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{12}=a_{32}=a_{43}=0$$\end{document} ). Hence, in this drift matrix only the direct links between Anxiety and Physical Discomfort are retained (see Appendix D for a proof). Again, we can see that this expression is exactly equivalent to the path-tracing definition of a direct effect described in Appendix C.

Finally, the indirect effect \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{IE}}(\Delta t)$$\end{document} describes how the effect of applying a pulse to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i(t)$$\end{document} changes when we also press-intervene to keep the mediator(s) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_k$$\end{document} fixed. Suppose we are interested in the mediating roles that both Stress and Self-Consciousness play in the effect of Anxiety on Physical Discomfort. To quantify this, we would define the indirect effect of Anxiety on Physical Discomfort (relative to Stress and Self-Consciousness) as

(10)

The indirect effect is thus the difference in value of Physical Discomfort at a particular time-interval t in Fig. 5a (i.e., the total effect \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TE}}(\Delta t)$$\end{document} ), and the value at the same time-interval in Fig. 5c (i.e., the direct effect \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{DE}}(\Delta t)$$\end{document} ). Each of the total, direct and indirect effects are shown together in Fig. 5d. If the indirect effect is positive, it means that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TE}} > {\textit{DE}}$$\end{document} , that is, applying a press intervention on the mediators decreases the effect of the pulse intervention. If the indirect effect is negative ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TE}} < {\textit{DE}}$$\end{document} ) the press intervention increases the effect of the pulse intervention. Of course, when giving any further substantive interpretation, careful attention should be paid to the signs of the component direct and total effects. In this case, keeping Stress and Self-Consciousness both fixed makes Physical Discomfort take on a stronger negative value at shorter intervals, and the indirect effect quantifies that difference. As such, this indirect effect describes the mediating role of the variables Stress and Self-Consciousness combined. We can express this indirect effect using the expressions we found before for the total and the direct effects in terms of the CT-VAR parameters. This gives us

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It follows that the effect of this combination of interventions is equivalent to the path-tracing definition of the indirect effect (see Appendix C).

3.3. Centrality Measures to Identify Intervention Targets

Having established the link between CT path-specific effects and highly idealized interventions in the dynamic system, we here propose two new centrality measures for CT networks. Each centrality measure is explicitly defined as a summary of a path-specific effect, and as such, as a kind of network-wide summary of the consequences of a particular intervention. This means that these centrality measures are functions of the time-interval and have a clear link to a particular type of variable intervention: the first and second measure can be used to identify the optimal target for a pulse and press, respectively.

3.3.1. CT Total Effect Centrality

We define our first new centrality measure as the Total Effect Centrality ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TEC}}$$\end{document} ) of a variable, which can be calculated by summing the total effect of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i(t)$$\end{document} on all other variables, at a particular time-interval

(12) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textit{TEC}}_{i}(\Delta t) = \sum _{j\ne i}^{p} {\textit{TE}}_{ij}(\Delta t). \end{aligned}$$\end{document}

Hence, we sum over all the total effects of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} on other variables in the network (excluding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} itself). The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TEC}}$$\end{document} thus summarizes the effect of an impulse intervention to change \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i(t)$$\end{document} , on the system as a whole, that is, the cumulative effect on the network, some time-interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} later. Since we explicitly make this centrality measure a function of the time-interval, we can examine how the cumulative effect of this intervention evolves following the pulse.

Figure 6a shows the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TEC}}$$\end{document} of each variable in the Stress-Discomfort system over a range of intervals, from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t = 0$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t = 1.5$$\end{document} . From this, we can see that at short intervals, pulse intervention to increase Physical Discomfort has the biggest cumulative effect on the network: overall, this intervention on Physical Discomfort results in the other variables increasing in value over the next half an hour or so, before eventually the effect of this intervention fades away. Notably, an intervention to increase Anxiety has a weak net negative effect on the system at shorter intervals, and a weak net positive effect at longer intervals: we would expect this based on our visualization of that intervention in Fig. 5a, where a pulse to Anxiety resulted in Stress and Physical Discomfort taking on negative values at short intervals.

The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TEC}}$$\end{document} measure allows us to see that, for this system, Physical Discomfort is the optimal target for a pulse intervention, assuming that we can set Physical Discomfort to a low or negative value (e.g., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$do(Y_4(t)) = -1$$\end{document} ). Such an intervention would be expected to result in a short-lived decrease in value of the other processes in the model. Note that, as a consequence of DT-VAR parameters reflecting total effects from a CT perspective the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{TEC}}$$\end{document} measure is actually equivalent to calculating the first-order expected influence measure using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }(\Delta t)$$\end{document} over a range of values for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} (as depicted in Fig. 6b). Recall however that the latter is erroneously interpreted as a measure of direct rather than total influence.

Figure. 6

Illustration of the new total and indirect centrality measures for CT networks, applied to the Stress-Discomfort system.

3.3.2. CT Indirect Effect Centrality

The second centrality measure we propose quantifies the role a particular variable plays as a mediator of other relationships between variables in the network. To define this measure, we use the indirect effect measure described in Eq. (10). Recall that the CT indirect effect captures the change in the effect of pulses to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_j(t)$$\end{document} on the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_k(t+\Delta t)$$\end{document} , if we press-intervene to keep the mediator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} fixed at every moment in time ( ). Hence, we define the Indirect Effect Centrality ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{IEC}}$$\end{document} ) of a mediator variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} as

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that is, it represents the sum of all possible indirect effects between different pairs of variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_j(t)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_k(t+\Delta t)$$\end{document} , in which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} serves as the only mediator. Note here that the notation is chosen to reflect that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{IEC}}$$\end{document} is a property of a mediator, instead of a property of one particular cause-effect relationship. The summation denotes that we omit auto-regressive relationships ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \ne k$$\end{document} ) and pairs of variables where the mediator is either the cause or effect variable ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \ne i$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ne i$$\end{document} ).

The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{IEC}}$$\end{document} quantifies how a press intervention on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_i$$\end{document} changes the effects that other variables have on each other. This may be a very useful concept in clinical practice. For instance, suppose that in our Stress-Discomfort system we want to avoid a high value on all four variables as much as possible. The current measure can be used to determine which of these variables is most important in terms of mediating the effects of one variable on another in the system, such that by intervening on this variable, these indirect paths become blocked and the flow of activation from one variable to another is (partly) interrupted.

Figure 6c shows the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{IEC}}$$\end{document} of each variable over a range of intervals. It shows that Physical Discomfort has the strongest indirect effect centrality in absolute terms. A strong negative value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{IEC}}$$\end{document} means that keeping Physical Discomfort fixed at an equilibrium value actually increases the effects of pulses to other variables on each other, since the component direct effects are greater than the corresponding total effects. This happens because Physical Discomfort plays a key role in the only negative feedback loop in the network: since an increase in Anxiety actually decreases Physical Discomfort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{42} = - 7.3$$\end{document} ), the total effect of Anxiety on Stress is less strong than its direct effect. If, however, we intervene to keep Physical Discomfort fixed, then this negative compensating effect is not activated, meaning an increase to Anxiety in fact has a greater effect on the network as a whole. Stress has the largest positive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textit{IEC}}$$\end{document} , meaning that keeping Stress fixed decreases the effects of other variables on one another.

From this, we would conclude that we should choose Stress as a target for a press intervention, as it decreases the short-term impact of other variables in the network on each other. Moreover, we should avoid applying a press intervention on Physical Discomfort: such an intervention would in fact increase the strength of positive relationships between the other variables.

3.4. Other Ways of Identifying Intervention Targets

We have presented a selection of measures for CT models that have a clear conceptual link with path-tracing and centrality measures as well as hypothetical variable interventions. The centrality measures we introduce summarize both total and indirect effects in a CT network, but we did not develop a centrality metric based on the direct effects: such a measure would require us to add different direct effects such that each requires a different set of variables to be held constant, and so, are not directly informative about any particular variable intervention.Footnote 9

The total and indirect measures introduced here allow us, respectively, to choose a target for a pulse intervention which will result in the largest “shock” to the system, and choose a target for a press intervention which will result in the largest change in how other variables activate one another. As is typical for psychological network analyses, we have chosen variables such that positive values have a negative connotation: we want to avoid high values of Stress, Anxiety, Self-Consciousness and Physical Discomfort, so we want to apply negative-valued pulses, and in general, press interventions that lower the degree to which variables activate one another. We believe that our approach of visualizing the effects of these interventions, and how they change depending on the time-interval, is the most informative approach for researchers, but these measures can also be summarized (for example, by averaging or taking the “area under the curve” over a particular time-interval).

Of course, in addition to the centrality measures described above, researchers may wish to apply interventions in order to affect different types of change in the system. For example, we may wish to find a press intervention such that the equilibrium values of the other variables in the system change in a particular way, as we saw in Fig. 5b. We provide R code in Online Supplementary Materials 2 (and accompanying R package ctnet) which can be used to simulate the effect of different press interventions on the equilibrium positions and the stability of a system (see also Appendix D.3). Of course, many different types of interventions in a system are also possible: Driver and Voelkle (Reference Driver, Voelkle, Montfort, Oud and Voelkle2018b) focus on different interventions than the ones considered here, and show how these can be simulated from CT models. However, from a causal modeling perspective, we believe that, while any operation on a model’s parameters can be defined in principle, typically if we want to learn about the effects of those interventions, or understand how we might bring about those changes in the system, we will likely need to define those manipulations in terms of interventions on variables in our causal model, as we have done in the current paper.

4.1. Data

To illustrate the CT network approach, we use a single-subject open-source ESM dataset.Footnote 10 A subset of this dataset was originally the subject of a DT-VAR network analysis by Wichers et al. (Reference Voelkle, Oud, Davidov and Schmidt2016) and is described in full in Kossakowski et al. (Reference Kossakowski, Groot, Haslbeck, Borsboom and Wichers2017). For illustrative purposes, we chose to fit a CT-VAR model using four ESM items measured on a 7-point Likert scale from low to high agreement: Self-Doubt (S; “I doubt myself”), Fatigue (F; “I am tired”), Irritated (I; “I feel irritated”) and Restless (R; “I feel restless”). Prior to analysis the variables were standardized. A subset of the full time series is used, consisting of 1476 measurements taken over 239 consecutive days (reflecting a period preceding a blinded medication reduction). The randomized sampling scheme results in time-intervals between consecutive measurements ranging from 13.5 min to 42.1 h with a median of 2.04 h. Figure 7 displays the distribution of time-intervals up to the 97.5 percentile. Inferences made from the CT model beyond the time-intervals used in data collection represent a form of model extrapolation and should be approached with caution, so we will focus our analysis in the following on the observed time-interval range ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t = 0$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t = 5$$\end{document} ).

Figure. 7

Histogram showing the distribution of time-intervals between subsequent measurement occasions in the empirical dataset.

4.2. Estimated Networks

As a first step, we can inspect and interpret the estimated drift matrix parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\hat{A}}$$\end{document} . These are displayed as a CT network in Fig. 8a, with accompanying confidence intervals for these parameters given in Appendix E. From this, we can see that, for instance, Irritated and Restless have the highest auto-effects, implying that, given a shock to the system, these processes would be expected to return to baseline quicker than feelings of Self-Doubt or Fatigue. Furthermore, the strongest cross-effects in the network are the positive reinforcing relationships between Irritated and Restless: Feeling Irritated is likely to increase your feelings of Restlessness a moment later, and vice versa.

Figure. 8

Estimated networks based on the CT-VAR and DT-VAR analysis of empirical data, respectively. S (Self-Doubt); F (Fatigue); I (Irritated) and R (Restless). a Shows a weighted local dependence graph based on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\hat{A}}$$\end{document} while b shows a DT-VAR network based on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\hat{\Phi }}$$\end{document} . In the latter case, time-interval information was ignored in estimation.

The estimated DT-VAR model displayed in Fig. 8b shows a somewhat similar pattern of relationships between processes, but there are some key differences. Most notably, the signs of several cross-lagged relationships are different from the corresponding drift matrix estimates: there are positive cross-lagged effects from Self-Doubt to Irritated, from Restless to Fatigue and from Fatigue to Self-Doubt. Furthermore, the relative ordering of cross-lagged parameters is different than that of the drift matrix parameters.

4.3. Exploring Time-Interval Dependency

To see how the system evolves over time, we can use the drift matrix to derive how we would expect the lagged regression parameters to change as a function of the time-interval (that is, we use \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\hat{A}}$$\end{document} to derive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\hat{\Phi }}(\Delta t)$$\end{document} ). This is shown in Fig. 9.

Figure. 9

Expected lagged regression parameter values as a function of the time-interval, derived from drift matrix estimated from empirical data. Here, the subscripts S (Self-Doubt), F (Fatigue), I (Irritated) and R (Restless) refer to the first, second, third and fourth dimension of the drift matrix, respectively.

First, we can see that the auto-regressive relationships in Fig. 9a are quite similar over a range of intervals. Second, from Fig. 9b, c we can see that most cross-lagged relationships are expected to reach their peak between a half hour ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t =0.5$$\end{document} ) and one and a half hours ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t = 1.5$$\end{document} ). Third, we can see that the ordering of some effects changes depending on the interval: for instance, panel (c) shows that at short intervals the effect of Irritated on Restless ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{RI}(\Delta t)$$\end{document} ) is larger than that of Self-Doubt on Restless ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{RS}(\Delta t)$$\end{document} ), but at longer intervals ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t > 1.5$$\end{document} ) this is reversed. Fourth, we can see from Fig. 9c that the cross-lagged effect from Self-Doubt to Irritated ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\phi }_{IS}(\Delta t)$$\end{document} ) changes sign over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t$$\end{document} : it is negative at very short time-intervals, and becomes positive at longer intervals (around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t = 1$$\end{document} ). This pattern results from the direct negative moment-to-moment dependency ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{a}_{IS} = -0.47$$\end{document} ) which is dominant at shorter intervals, while the effect at longer intervals is mostly driven by positive indirect relations through Restless.

4.4. Centrality Analysis

The values of the estimated CT centrality metrics are shown in Fig. 10. For comparison, we also include the corresponding centrality measures one would compute based on the DT-VAR results. Figure 10a shows that Restless has the highest total effect centrality value over the entire range of intervals from zero to five hours. This implies that (making the same idealizing assumptions outlined in previous sections) it is the optimal target for a pulse intervention: we would expect such an intervention to have a large impact on the network as a whole, peaking around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t = 0.75$$\end{document} after the impulse. Since this centrality measure is positive in this interval range, we would recommend applying a negative impulse to Restless, which will lead to a decrease (rather than increase) of the other variables in the network.

From Fig. 10b, we see that Restless alone has a high positive indirect effect centrality value, indicating that applying a press intervention to Restless is expected to decrease the degree to which positive impulses in one part of the network results in positive activation in other parts of the network. Furthermore, we can see that a press intervention on Irritation is expected to actually increase activation levels at short intervals (i.e., make the other variables more-connected), and decrease them at longer intervals (i.e., making them less-connected). From this, we would conclude that Restless appears to be the most attractive target for a press intervention, and that press interventions on Tired and Self-Doubt should be avoided.

Figure. 10

Total, Direct and Indirect centrality metrics for S(t) (Self-Doubt), F(t) (Fatigue), I(t) (Irritated) and R(t) (Restless) based on the CT Network (left-hand side) and DT-VAR Network (right-hand side), respectively.

In the right-hand side of Fig. 10a, b, we show the related DT centrality measures based on the estimated DT-VAR parameters. In Fig. 10a, we see that Restless has the highest two-step expected influence scores, closely followed by Irritated and Self-Doubt. In Fig. 10b, we see that Restless and Irritated have the joint highest Betweenness scores. In this case, applying current practice, we may very well have concluded that Restless and Irritated are candidate intervention targets. However, only with the CT approach developed here could we make specific recommendations regarding the type of intervention to apply, understand the effect such an intervention is expected to have and describe the subtle differences in intervention effects across short and longer time-intervals we would expect to see.