Panel studies, in which the same subjects are repeatedly observed at multiple time points, are among the most popular longitudinal designs in psychology. Meanwhile, there exists a wide range of different methods to analyze such data, with autoregressive and cross-lagged models being 2 of the most well known representatives. Unfortunately, in these models time is only considered implicitly, making it difficult to account for unequally spaced measurement occasions or to compare parameter estimates across studies that are based on different time intervals. Stochastic differential equations offer a solution to this problem by relating the discrete time model to its underlying model in continuous time. It is the goal of the present article to introduce this approach to a broader psychological audience. A step-by-step review of the relationship between discrete and continuous time modeling is provided, and we demonstrate how continuous time parameters can be obtained via structural equation modeling. An empirical example on the relationship between authoritarianism and anomia is used to illustrate the approach. Person to ContactCorrespondence concerning this article should be addressed to Manuel C. Voelkle, anomia is used to illustrate the approach.

This is the first paper in a series of two that synthesizes, compares, and extends methods for causal inference with longitudinal panel data in a structural equation modeling (SEM) framework. Starting with a cross-lagged approach, this paper builds a general cross-lagged panel model (GCLM) with parameters to account for stable factors while increasing the range of dynamic processes that can be modeled. We illustrate the GCLM by examining the relationship between national income and subjective well-being (SWB), showing how to examine hypotheses about short-run (via Granger-Sims tests) versus long-run effects (via impulse responses). When controlling for stable factors, we find no short-run or long-run effects among these variables, showing national SWB to be relatively stable, whereas income is less so. Our second paper addresses the differences between the GCLM and other methods. Online Supplementary Materials offer an Excel file automating GCLM input for Mplus (with an example also for Lavaan in R) and analyses using additional data sets and all program input/output. We also offer an introductory GCLM presentation at https://youtu.be/tHnnaRNPbXs . We conclude with a discussion of issues surrounding causal inference.

We introduce ctsem, an R package for continuous time structural equation modeling of panel (N > 1) and time series (N = 1) data, using full information maximum likelihood. Most dynamic models (e.g., cross-lagged panel models) in the social and behavioural sciences are discrete time models. An assumption of discrete time models is that time intervals between measurements are equal, and that all subjects were assessed at the same intervals. Violations of this assumption are often ignored due to the difficulty of accounting for varying time intervals, therefore parameter estimates can be biased and the time course of effects becomes ambiguous. By using stochastic differential equations to estimate an underlying continuous process, continuous time models allow for any pattern of measurement occasions. By interfacing to OpenMx, ctsem combines the flexible specification of structural equation models with the enhanced data gathering opportunities and improved estimation of continuous time models. ctsem can estimate relationships over time for multiple latent processes, measured by multiple noisy indicators with varying time intervals between observations. Within and between effects are estimated simultaneously by modeling both observed covariates and unobserved heterogeneity. Exogenous shocks with different shapes, group differences, higher order diffusion effects and oscillating processes can all be simply modeled. We first introduce and define continuous time models, then show how to specify and estimate a range of continuous time models using ctsem.

Perceptions of age influence how we evaluate, approach, and interact with other people. Based on a paramorphic human judgment model, the present study investigates possible determinants of accuracy and bias in age estimation across the adult life span. For this purpose, 154 young, middle-aged, and older participants of both genders estimated the age of 171 faces of young, middle-aged, and older men and women, portrayed on a total of 2,052 photographs. Each face displayed either an angry, fearful, disgusted, happy, sad, or neutral expression (FACES database;Ebner, Riediger, & Lindenberger, 2010). We found that age estimation ability decreased with age. Older and young adults, however, were more accurate and less biased in estimating the age of members of their own as compared with those of the other age group. In contrast, no reliable own-gender advantage was observed. Generally, the age of older faces was more difficult to estimate than the age of younger faces. Furthermore, facial expressions had a substantial impact on accuracy and bias of age estimation. Relative to other facial expressions, the age of neutral faces was estimated most accurately, while the age of faces displaying happy expressions was most likely underestimated. Results are discussed in terms of methodological and practical implications for research on age estimation.

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When designing longitudinal studies, researchers often aim at equal intervals. In practice, however, this goal is hardly ever met, with different time intervals between assessment waves and different time intervals between individuals being more the rule than the exception. One of the reasons for the introduction of continuous time models by means of structural equation modelling has been to deal with irregularly spaced assessment waves (e.g., Oud & Delsing, 2010). In the present paper we extend the approach to individually varying time intervals for oscillating and non-oscillating processes. In addition, we show not only that equal intervals are unnecessary but also that it can be advantageous to use unequal sampling intervals, in particular when the sampling rate is low. Two examples are provided to support our arguments. In the first example we compare a continuous time model of a bivariate coupled process with varying time intervals to a standard discrete time model to illustrate the importance of accounting for the exact time intervals. In the second example the effect of different sampling intervals on estimating a damped linear oscillator is investigated by means of a Monte Carlo simulation. We conclude that it is important to account for individually varying time intervals, and encourage researchers to conceive of longitudinal studies with different time intervals within and between individuals as an opportunity rather than a problem.

Continuous time dynamic models are similar to popular discrete time models such as autoregressive cross-lagged models, but through use of stochastic differential equations can accurately account for differences in time intervals between measurements, and more parsimoniously specify complex dynamics. As such they offer powerful and flexible approaches to understand ongoing psychological processes and interventions, and allow for measurements to be taken a variable number of times, and at irregular intervals. However, limited developments have taken place regarding the use of continuous time models in a fully hierarchical context, in which all model parameters are allowed to vary over individuals. This has meant that questions regarding individual differences in parameters have had to rely on single-subject time series approaches, which require far more measurement occasions per individual. We present a hierarchical Bayesian approach to estimating continuous time dynamic models, allowing for individual variation in all model parameters. We also describe an extension to the ctsem package for R, which interfaces to the Stan software and allows simple specification and fitting of such models. To demonstrate the approach, we use a subsample from the German socioeconomic panel and relate overall life satisfaction and satisfaction with health. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

This study tested whether the structure of affect observed on the basis of between-person (BP) differences is equivalent to the affect structures that organize the variability of affective states within persons (WP) over time. Further aims were to identify individual differences in the degree of divergence between the WP and BP structure and examine its association to dispositional and contextual variables (neuroticism, extraversion, well-being and stress). In 100 daily sessions, 101 younger adults rated their mood on the Positive and Negative Affect Schedule. Variability of five negative affect items across time was so low that they were excluded from the analyses. We thus worked with a modified negative affect subscale. WP affect structures diverged reliably from the BP structure, with individual differences in the degree of divergence. Differences in the WP structural characteristics and the degree of divergence could be predicted by well-being and stress. We conclude that BP and WP structures of affect are not equivalent and that BP and WP variation should be considered as distinct phenomena. It would be wrong, for example, to conceive of positive and negative affect as independent at the WP level, as suggested by BP findings. Yet, individual differences in WP structural characteristics are related to stable BP differences, and the degree to which individuals' affect structures diverge from the BP structure can provide important insights into intraindividual functioning.