Abstract: When conducting the multivariate longitudinal studies, reciprocal relationship and latent trajectory are two of the focusing issues. The reciprocal relationship is often examined by a cross-lagged model that could build autoregressive influence and the multivariate influence between target variables, while the latent trajectory is usually defined by a latent growth model that explores the growth pattern simultaneously with individual difference. These two kinds of models are easily built under the SEM framework, at the same time could be flexibly combined by other research questions, such as the measurement error, the random factor, as well as the combination of the above issues. Such a combination yields a more complex model definition exploring the longitudinal relations, such as factor cross-lagged model, random-intercept cross-lagged model, trait-state-error model, autoregressive trajectory model, latent change score model, etc.
In the study, we built a unified framework to analyze the above series of models according to the variance decomposition. First, the between-person difference was built by the latent trajectory often modeled as the latent growth. Second, the within-person difference was further decomposed as the within-person carry-over and the reciprocal relations between variables, which is the key question in the cross-lagged model series. Finally, the measurement error could be added to increase the measuring accuracy, where the trait-state-error model usually answers such a question. Since the research question of interest could be easily drawn from any above components, in summary, a “factor latent curve model with structured reciprocals” model was built as an extension and unified framework including all the components discussed above.
We also used an empirical dataset to compare the above models. The data was driven from the Early Childhood Longitudinal Survey-Kindergarten (ECLS-K) project. There were 21,049 participants selected from 6 waves of measures from kindergarten to Grade 8. Reading and mathematics abilities IRT scores were used calibrated on the same scale. We first decided on the shape of the growth trajectory, where a series of alternative models indicated that the piecewise growth model best fit the data. Followed, longitudinal models suggested in our unified framework were adopted, i.e., (random intercept) cross-lagged model, trait-state-error model, latent growth model, (latent variable) autoregressive latent trajectory model, as well as (factor) latent curve model with structured residuals/reciprocals.
Results indicated that the trait-state-error model best described the data. It showed that after controlling for the between-person difference (the trait factor—reading and mathematics ability), individually carry-over effects were significantly influential typically for students in the early elementary years. The significant reciprocal effect between reading and mathematics was also obtained showing these two domains of subjects influenced one another. Finally, we summarized how the results could be interpreted and offered suggestions on model selection for the researchers.

Key words: longitudinal study, reciprocal effect, growth trajectory, factor latent curve model with structured reciprocals