Abstract: Intensive longitudinal method is a general term for a set of methods such as experience-sampling methodology, ecological momentary assessment, real-time data capture, and daily diary. This type of methods generally collect intensive longitudinal data with tens or hundreds of time points for each participant under real situation. Intensive longitudinal data was expected to contain detailed information regarding temporal, irregular dynamic changes, as well as time-varying effects of covariates on psychological outcome. While commonly used statistical methods of longitudinal data usually have a convention to assume a shape of change as a prespecified form (linear, quadratic or exponential), and the association between psychological outcome and covariate is constant over time. Although for some situations such two assumptions can be a convincing justified by a well-established theory, for most other situations, the actual course of changes might be quite complicated, interactions between psychological outcome and relevant covariates might also evolve over time. More advanced statistical models were needed for describing detailed development patterns and changing relationships between psychological outcome and relevant factors. In this paper, we present the Time-Varying Effect Model (TVEM) to analyze intensive longitudinal data to capture the temporal changes and time-varying effects of interest. We first introduced the mathematical formula and principle of TVEM, described process and technical details for fitting TVEM with penalty-spline method, then presented a simulation study to prove effectiveness and applicability of TVEM for intensive longitudinal data. In present simulation study, a model included one intercept function and two slope functions of two covariates was adopted. Two simulation conditions were considered, sample size condition was set to be 30,100,300,1000, knots of penalty-spline method condition was set to be 1,3,5,7,9. The maximum of absolute deviation(d), mean absolute deviation error(MADE) and 95% confidence interval were used as precision indexes of model fit，interaction effects between sample size and knots were investigated by MANOVA. Results showed that: (1) The accuracy of function estimation was affected by sample size, it would become more precise as sample size increased. (2) Selection of the number of knots is mainly depended on complexity of function, a simple function could be approximated with few spline functions, and a complicated function need more knots. When the number of knots is large enough, further increasing knots wouldn’t lead to significant change on d and MADE. (3) MANOVA results showed that, sample size had significant main effects on d and MADE of three functions, while number of knots only had a significant main effect on function , interaction effects between sample size and knots were not remarkable. In summary, the TVEM was proved to be an effective statistical method for analyzing intensive longitudinal data to study detail course of change and time-varying effects of covariates on psychological outcome. It could be used to explore inherently nonlinear longitudinal relationships, resulting in a model with temporal ups and downs. As the TVEM wasn’t been extensively studied, further researches should focus on problems about requirement of sample size, number of repeated assessments, impact of missing data, data distribution and error structure on the robustness of model estimation, and extension of TVEM to describe latent group heterogeneity on change process.

Key words: intensive longitudinal data, time-varying effect, time-varying effect model, penalty-spline method

摘要： 密集追踪数据通常蕴含了心理过程的详细变化信息，反映了某些心理的复杂变化过程。时变效应模型用函数替代恒定的系数,可描述密集追踪数据中随时间推移心理的动态变化过程和时变效应，是分析复杂心理过程的有效方法。在介绍时变效应模型的原理后，通过模拟研究考察模型的表现，结果显示：（1）样本量增加可降低函数估计的误差；（2）惩罚样条法的节点数选择与函数的复杂度有关，函数越复杂，所需节点越多；（3）样本量与节点数对函数估计误差的交互效应不显著。进一步应探讨测量次数、数据分布形态、数据缺失等如何影响模型的表现。

关键词: 密集追踪数据, 时变效应, 时变效应模型, 惩罚样条法