No Need to be Discrete: A Method for Continuous Time Mediation Analysis

Abstract

Mediation is one concept that has shaped numerous theories. The list of problems associated with mediation models, however, has been growing. Mediation models based on cross-sectional data can produce unexpected estimates, so much so that making longitudinal or causal inferences is inadvisable. Even longitudinal mediation models have faults, as parameter estimates produced by these models are specific to the lag between observations, leading to much debate over appropriate lag selection. Using continuous time models (CTMs) rather than commonly employed discrete time models, one can estimate lag-independent parameters. We demonstrate methodology that allows for continuous time mediation analyses, with attention to concepts such as indirect and direct effects, partial mediation, the effect of lag, and the lags at which relations become maximal. A simulation compares common longitudinal mediation methods with CTMs. Reanalysis of a published covariance matrix demonstrates that CTMs can be fit to data used in longitudinal mediation studies.

Keywords:

• continuous time models
• cross-lagged panel model
• exact discrete model
• longitudinal mediation
• mediation

Notes

¹ In this context, stationary refers to the definition commonly used in the time series literature. A stationary process has a joint probability distribution that is constant. Consequently, the expected values of the mean and variance are constant over time and finite. Furthermore, the covariance between lagged observations is also constant and finite, which means the covariance between lagged observations can depend on the lag between observations but not on time. The first-order differential equation model presented inherits these assumptions, unless nonstationary components are explicitly modeled (e.g., continuous time autoregressive latent trajectory model; Delsing & Oud, 2008).

² Equation 6 is provided as a method for understanding the values of A; in practice taking the logarithm of a matrix can be problematic as the solutions are not always unique (Higham, 2008).

³ Online appendices are available online through both authors’ Web sites.

⁴ As the derivation of this equation depends on the calculation of the natural log of a matrix (see Footnote 2), we encourage users of this formula to compare results with the previous graphical interpretation to ensure a valid result.

⁵ Online Appendix A provides estimates of the maximal direct and indirect effects and the lags at which they occur.

⁶ Often in implementation of the CLPM, parameters are allowed to change suddenly at exactly the times at which observations have been made; this creates a step-like function of the parameters between pairs of measurement occasions. This would be an unusual hypothesis for a continuous time model. The EDM can be extended, however, to allow the relationships between variables and the error variance to change as a function of time; for example, one could allow the parameters in A to change linearly with time (Oud & Jansen, 2000).

⁷ It is the authors’ perspective that the EDM and CLPM_A do not represent incompatible or confounding causal explanations for the same observed data, which is a common consideration in discussions of equivalent models in SEM. Other individuals familiar with equivalent models do not share the authors’ viewpoint.