Abstract:

In recent years, dynamic research has become an important topic in longitudinal studies, focusing on the temporal changes of psychological constructs across repeated measurements. Dynamic effects typically include autoregressive and cross-lagged effects, emphasizing the dependence of individual’s current state on their past states. With advances in data collecting techniques, dynamic effects could be investigated through intensive longitudinal data (ILD), which obtains data through rapid and frequent assessments.
The ILD framework typically employs two time scales for research design and data analyses: discrete time (DT) and continuous time (CT). These two approaches differ substantially in their theoretical assumptions and modeling approaches. DT models are based on the assumption of equal time intervals, analyzing equally spaced dynamic processes. This can limit their use when observations are not equally spaced. In contrast, CT models are based on an underlying continuous process, allowing for unequal time intervals and enabling the modeling of dynamic effects across different time intervals within a cohesive framework. Therefore, CT models offer greater flexibility regarding time intervals, making it necessary to compare and organize the two approaches within a unified framework.
Based on this background, the present study uses ILD and takes a first-order autoregressive model as an example to systematically compare DT and CT modeling frameworks. We begin by reviewing classical models within both the DT and CT frameworks separately, including the autoregressive model, multilevel autoregressive model, and (residual) dynamic equation models. Next, we compare the construction of models between DT and CT, establishing a transformation relationship between the two approaches. Additionally, we address a crucial issue: the time interval. This includes defining how the time interval is determined and how to manage unequal intervals within each framework.
In addition, we use an emotional dataset from an experience sampling method study as an empirical example to demonstrate the modeling procedures under both approaches. Our findings show that, before data analysis, DT models require imputation of missing time points within equally spaced intervals, whereas CT models use the original observations. As a result, the number of usable time points differs the two approaches. In terms of parameter estimation, the two models produce marginally identical results for trend parameters and dynamic parameters (i.e., autoregressive and cross-lagged effects). For the present dataset, although the assumption of equal time intervals is not met, the imputation procedure in the DT model performs well, and the two models yield comparable results.
In conclusion, we provide several practical recommendations. First, it is essential to determine whether higher-order dynamic effects are defined. The parameters in CT models can more accurately represent such sophisticated dynamic processes. Second, model selection should be guided by the research design and the characteristics of the data. When data is collected at random time intervals or has significant missing values, CT models are more suitable. In contrast, DT models can serve as a more parsimonious alternative when the violation of the equal-interval assumption is not severe. Third, the choice of time intervals should align with the sampling frequency. In this case, CT models offer greater flexibility in rescaling time. Finally, for better clarity, results from both CT and DT models should be presented on a DT scale, allowing for a clearer understanding of the dynamic parameters’ practical implications.

Key words: dynamics, intensive longitudinal data, discrete time, continuous time