Accounting for Time-Varying Inter-Individual Differences in Trajectories when Assessing Cross-Lagged Models

ABSTRACT

This paper explores relationships amongst cross-lagged models allowing trajectories to be freely estimated, some accounting for time-varying differences amongst individuals (Autoregressive Latent Trajectory (ALT), General Cross-lagged Model (GCLM), and Latent Growth Curve Model with Structured Residuals and Unspecified Growth Trajectory (LGCM-SR-UGT)) and some not (Cross-lagged Panel Model (CLPM), Random Intercept Cross-lagged Panel Model (RI-CLPM), and Mean Stationary GCLM). An applied example using LSAY data demonstrates these models. Simulations examine (1) fit indices assessing “good” fit and Bayes Factor for model selection; (2) consequences of ignoring variability in trajectories on cross-lagged estimates. Findings were (1) RMSEA discerned “good” fit and Bayes Factor tended to select models closely related to true model over less related models; (2) various patterns of bias in path estimates and standard errors are found, in particular, causal dominance in conjunction with time-variant between-person variance and covariance were notably influential on bias in path estimates.

KEYWORDS:

• Cross-lagged panel model
• general cross-lagged model
• random intercept cross-lagged panel model
• autoregressive latent trajectory model
• latent growth curve model with structured residuals

Supplementary material

Supplemental data for this article can be accessed on the publisher’s website.

Notes

¹ It should be noted that there is evidence that under certain conditions (viz., time-invariant autoregressive paths which do not exhibit a random walk |AR| = 1 or explosive process |AR|>1), models fitting lagged relations on the residuals are algebraically equivalent to models fitting lagged relations to the observed variables (Hamaker, 2005). This is specifically demonstrated between ALT (Curran & Bollen, 2001) and the Latent Growth Curve with autocorrelated disturbances (Chi & Reinsel, 1989).

² It should be noted that technically, in theory, any two values can be used for identification allowing the remaining loadings to be freely estimated (McArdle & Hamagami, 1991).

³ As Usami et al. (2019) have aptly pointed out, these should rather be considered as accumulating factors since they work in concert with the AR & CL paths to establish the change over time. See Supplement A for details.

⁴ Setting trajectory variance ${\zeta }_i$=0 reduces this ALT to a reparametrized version of the CLPM as given in Equation 2a,b(2a) $y_{it}={\alpha }_t^{\left(y\right)}+{\rho }_{yy}{y}_{it-1}+{\rho }_{yx}{x}_{it-1}+u_{it}^{\left(y\right)}$(2a) with ${\alpha }_t^{\left(x\right)}=0\text{amp}{\alpha }_t^{\left(y\right)}=0$, and the mean structure as given in Equations 15a,b(15a) ${\mu }_t^{\left(x\right)}={\lambda }_t{\mu }_{{\beta }^{\left(x\right)}}+{\rho }_{xx}{\mu }_{t-1}^{\left(x\right)}+{\rho }_{xy}{\mu }_{t-1}^{\left(y\right)}$(15a) .

⁵ Note: for the predetermined ALT, the covariance at time point 1 are considered exogenous hence they are not constrained to be equal with the covariance at subsequent time points which are considered endogenous.

⁶ All simulation codes and complete results (including convergence information, model fit/selection, and cross-lagged path/standard error biases across simulation conditions for all combinations of generating and fitting models) are available upon request.