The collection of longitudinal data in the social and behavioral sciences has been revolutionized by the widespread availability of information technologies such as smartphones, wearable technology, social media, digital learning, online games, and the Internet more generally. We use the term intensive longitudinal data (ILD) inclusively, to refer to the types of data available from such sources. ILD can arise from a broad range of data collection methods and research designs, and typically result in multivariate observations collected from multiple respondents over a relatively large number of time points. Examples of ILD from this collection of papers include experience sampling of alcohol and substance use, daily diaries of emotional states, students’ interactions with a web-based math tutoring app, and university attendance records.
ILD are typically collected on an ongoing basis over an extended duration of time, and in some applications data collection may continue indefinitely. This introduces the prospect of analyzing and acting upon ILD as they arrive, rather than waiting for data collection to be completed. This “online” approach to data analysis is common in domains such as engineering and machine learning, where it is usual for data to arrive on an ongoing basis. However, its potential application in the social and behavioral sciences remains largely unexplored. An initial step in this direction is to address the problem of forecasting with ILD, which is the focus of this special collection.
The statistical analysis of ILD has motivated the development of novel modeling approaches. Examples from previous literature that are considered in this collection include multilevel/random-effects extensions of vector auto-regression (VAR), dynamical structural equation modeling (SEM), and (non-) linear dynamical systems models. Hunter et al. (Reference Hunter, Fatimah and Bornovalova2022) provide an overview (and accompanying software) for filtering and forecasting techniques used in dynamical systems. Chow et al. (Reference Chow, Lee, Hofman, van der Maas, Pearl and Molenaar2022) consider how dynamical systems can be combined with control theory to “steer” a system towards a desired state, and consider the implications of this approach for personalized education. Lafit et al. (Reference Lafit, Meers and Ceulemans2021) provide data-driven insights into several factors that affect the predictive accuracy of multilevel VAR. This collection of papers also introduces some new modeling approaches. Li et al. (Reference Li, Oravecz, Zhou, Bodovski, Barnett, Chi, Zhou, Friedman, Vrieze and Chow2022) introduce a multilevel zero-inflated Poisson (ZIP) model in which both the Poisson counts and the ZIP regimes have person-specific auto-regressive time dependency. Fisher et al. (Reference Fisher, Kim, Fredrickson and Pipiras2022) introduce an alternative to the random-effects approach, instead using regularization (adaptive LASSO) to extend VAR to multiple subjects while ensuring sparsity of the resulting solution. Additional approaches include a latent class extension of dynamic SEM (Keleva et al. Reference Kelava, Kilian, Glaesser, Merk and Brandt2022) and machine learning extensions of mixed-effects models (Nestler & Hamburg, Reference Nestler and Humberg2021)
One theme that arises from this collection is the importance of disentangling intra- and inter-individual sources of variation. While this general theme is certainly familiar, it leads to some interesting considerations in the context of forecasting and prediction as distinct from estimation and inference. Lafit et al. (Reference Lafit, Meers and Ceulemans2021) nicely characterize this tension in the context of multilevel VAR. More generally, while there is a consensus that models for ILD should accommodate individual-specific parameterizations (e.g., via random effects), there are open questions as to the conditions under which prediction will be more accurate using individual-specific versus aggregate model properties (e.g., the predicted random effects or the fixed effects). The comparison of the forecasting properties of aggregate and individual-specific model specifications, such as those provided by Nestler and Hamburg (Reference Nestler and Humberg2021) as well as Li et al. (Reference Li, Oravecz, Zhou, Bodovski, Barnett, Chi, Zhou, Friedman, Vrieze and Chow2022), begins to address this question.
A second theme concerns how to explicitly link forecasting quality to model parameters. Often the forecasting distribution is not available analytically. In such cases it is difficult to understand how model parameters influence forecasts, or how changing a parameter (e.g., intervening on a system) affects the forecast distribution. Chow et al. (Reference Chow, Lee, Hofman, van der Maas, Pearl and Molenaar2022) discuss this problem from the perspective of control theory. Lafit et al. (Reference Lafit, Meers and Ceulemans2021) provide insights about how both model and data characteristics affect prediction with multilevel VAR, while Nestler and Hamburg (Reference Nestler and Humberg2021) discuss some analytical results in the context of mixed-effects models. In many applications, accurate forecasts may lead to an appropriate course of action even if we lack a precise understanding of how those actions affect a system’s dynamics (e.g., in the case of alcohol and substance use). Nonetheless, learning about these effects through the model is certainly one way in which statistical research can inform substantive theories about ILD.
Other than the contribution by Hunter et al. (Reference Hunter, Fatimah and Bornovalova2022), the papers in this collection implemented “batch” estimation in the usual way (i.e., after the data were in). While online estimation is not required for forecasting, in its absence one must address the question of how estimation and forecasting can be integrated in applications that involve ongoing data collection. Perhaps the model is “pre-calibrated” using an initial batch of data and then used for forecasting without updating model parameters, which is an inefficient use of data. Alternatively, the model might be “naively” re-estimated after each data point is observed, which is an inefficient (and perhaps infeasible) use of computational resources. Integration of estimation and forecasting via online approaches whose properties are well understood remains a long-term goal for advancing the study of ILD.
