ISSN 1671-3710
CN 11-4766/R

心理科学进展 ›› 2021, Vol. 29 ›› Issue (10): 1755-1772.doi: 10.3724/SP.J.1042.2021.01755

• 研究方法 • 上一篇    下一篇



  1. 西南大学心理学部, 认知与人格教育部重点实验室, 重庆 400715
  • 收稿日期:2020-10-18 出版日期:2021-10-15 发布日期:2021-08-23
  • 通讯作者: 刘源
  • 基金资助:

A unification and extension on the multivariate longitudinal models: Examining reciprocal effect and growth trajectory

LIU Yuan()   

  1. School of Psychology, Southwest University; Key Laboratory of Cognition and Personality (Southwest University), Ministry of Education, Chongqing 400715, China
  • Received:2020-10-18 Online:2021-10-15 Published:2021-08-23
  • Contact: LIU Yuan


追踪研究当中, 交叉滞后模型可以探究多变量之间往复式影响, 潜增长模型可以探究个体增长趋势。对两类模型进行整合, 例如同时关注往复式影响与个体增长趋势, 同时可以定义测量误差、随机截距等变异成分, 衍生出随机截距交叉滞后模型、特质-状态-误差模型、自回归潜增长模型、结构化残差潜增长模型等。以交叉滞后模型和潜增长模型分别作为基础模型, 从个体间/个体内变异分解的角度对上述各类模型梳理, 整合出此类模型的分析框架, 并拓展建立“因子结构化潜增长模型(factor latent curve model with structured reciprocals)”作为统合框架。通过实证研究(早期儿童的追踪研究-幼儿园版, ECLS-K), 建立21049名儿童的阅读和数学能力的往复式影响与增长趋势。研究发现, 分离了稳定特质的模型拟合最优。研究也对模型建模思路和模型选择提供了建议。

关键词: 追踪研究, 往复式影响, 增长趋势, 因子结构化潜增长模型


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