Drawing Conclusions from Cross-Lagged Relationships: Re-Considering the Role of the Time-Interval

Abstract

The cross-lagged panel model (CLPM), a discrete-time (DT) SEM model, is frequently used to gather evidence for (reciprocal) Granger-causal relationships when lacking an experimental design. However, it is well known that CLPMs can lead to different parameter estimates depending on the time-interval of observation. Consequently, this can lead to researchers drawing conflicting conclusions regarding the sign and/or dominance of relationships. Multiple authors have suggested the use of continuous-time models to address this issue.In this article, we demonstrate the exact circumstances under which such conflicting conclusions occur. Specifically, we show that such conflicts are only avoided in general in the case of bivariate, stable, nonoscillating, first-order systems, when comparing models with uniform time-intervals between observations. In addition, we provide a range of tools, proofs, and guidelines regarding the comparison of discrete-and continuous-time parameter estimates.

The Differential Time-Varying Effect Model (DTVEM): A tool for diagnosing and modeling time lags in intensive longitudinal data

“…Most often, the measurement interval in a study is assumed by design to correspond to Δt = 1 in the underlying difference or differential equation model. The measurement intervals utilized for data collection have direct implications on the strengths of the regression coefficients linking the lagged responses to the current responses at time t-a point that has been brought up by several researchers in advocating direct use of continuous-time models over discrete-time models (Kuiper & Ryan, 2018;Voelkle & Oud, 2013). We will return to this point in the Discussion section.…”

Recovering Bistable Systems from Psychological Time Series

“…In theory, to recover the data generating parameters, we would need to fit the integral solution form of the differential equation (Strogatz, 2015), as this describes the relationships between observed variables spaced ∆t apart, as implied by the differential equation. It is well known that this integral solution may contain a seemingly different set of dependency relationships than the differential equation: variables which are independent in the DE form may be dependent in the integral form, and the signs and relative orderings of these dependencies may change depending on the value of ∆t (Aalen, Røysland, Gran, & Ledergerber, 2012;Kuiper & Ryan, 2018;Ryan, 2018). Because methods based on approximating integral solutions are expected to suffer from similar problems as the two-step DE estimation procedure, and because these methods are difficult to apply in practice, we limit ourselves to the two-step approach in this paper (see Discussion section 5.3 for further details).…”

Meta-analysis of lagged regression models: A continuous-time approach

“…A major obstacle for any meta-analytic study of lagged effects is the well-known timeinterval dependency problem. This refers to the phenomenon that studying the same underlying dynamic process can result in autoregressive and cross-lagged parameter estimates of different signs, size, and relative ordering due only to the use of different time intervals between measurements (Chatfield, 2004;Dormann & Griffin, 2015;Gollob & Reichardt, 1987;Hamilton, 1994;Kuiper & Ryan, 2018;Oud, 2002). Stated otherwise, these lagged parameters are a function of the time interval and are therefore denoted by Φ(∆t) in the remainder.…”

“…Equation 3is unique (Hamerle et al, 1991;Kuiper & Ryan, 2018). This means that, using the lagged parameters which are obtained at a particular time interval ∆t can be said to directly imply a set of lagged parameters at any different time interval ∆t * :…”

“…Notably, if the eigenvalues of Φ(∆t) are complex or when at least one of them is negative, then this transformation is only unique and possible, respectively, when ∆t * is a whole-numbered multiple of ∆t. A more detailed discussion on the conditions under which this is (uniquely) possible is given by Kuiper and Ryan (2018).…”