Tuesday 16th July
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Continuous Time Modeling in Panel Research (N large) and Time-Series Analysis (N = 1 or small) 2 |
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| Convenor | Dr Johan Oud (Radboud University Nijmegen) |
| Coordinator 1 | Dr Manuel Voelkle (Max Planck Institute for Human Development Berlin ) |
Although virtually all processes in social reality develop in continuous time, continuous time modeling of those processes by means of differential equations is extremely rare. Scattered early attempts have been taken by the well-known scientists Herbert Simon in 1952 and James Coleman in 1968 but did not lead to much follow-up. Time-ordered causal modeling is almost always done in discrete time with the cross-lagged panel design being its most popular representative. The preference for discrete-time modeling is likely motivated by the inherently discrete-time nature of our measurements. It can be shown, however, that failing to properly account for the continuous time intervals between measurement occasions may lead to quite paradoxical and even contradictory conclusions. The session is open for all continuous-time modeling approaches in social science. Data sets analyzed may range from N = 1 and T large to N large and T small, from observation time points and intervals that are equal for all N subjects to individually varying observation intervals within and between subjects. Also welcome are contributions that discuss the different approximate and exact estimation procedures in continuous-time modeling. The trajectories analyzed may take arbitrary forms: oscillating and nonoscillating, with and without random subject effects. We especially welcome papers on substantive topics that apply continuous time modeling in their analyses.
The past decade has evidenced the increased prevalence of irregularly spaced longitudinal data in
social sciences. Clearly lacking, however, are modeling tools that allow researchers to fit dynamic
models to irregularly spaced data, particularly data that show nonlinearity and heterogeneity in
dynamical structures. We consider the issue of fitting multivariate nonlinear differential equation
models with random effects to irregularly spaced data. A stochastic approximation expectation-maximization
(SAEM) algorithm is proposed and its performance is evaluated using a benchmark nonlinear dynamical systems model, namely, the Van der Pol oscillator equations. The empirical utility of the proposed technique is illustrated using a set of 24-hour ambulatory cardiovascular data from 168 men and women. Pertinent methodological challenges and unresolved issues are discussed.
Many modeling efforts in the behavioral sciences aim at testing causal theories about reality. After presenting and estimating the model, drawing the causal picture in accordance with or inspired by the model, and giving recommendations for the construction of new and better models, most studies conclude. In longitudinal research, however, when the model is written in the form of the dynamic state space model, the Kalman filter and smoother open up the possibility to go one step further and to make the model practically useful by estimating optimally latent developmental curves for individual sample units. The approach has recently been extended with random subject effects and continuous time modeling. The extended procedure offers the following beneficial features:
- Not only at but also between measurements optimal estimates of fluent curves for individual subjects are made by optimally interpolating and predicting.
- This implies that there are no requirements with regard to data: data may be missing or observed at arbitrary time points for each individual.
- The individual developmental curves may be compared with the population of developmental curves and confidence intervals allow to test the differences.
- Individuals may also be compared with themselves (subject specific expected curves).
- Individual developmental curves may be compared with predicted individual curves and so, for example, the effect of improvement measures tested.
Psychodiagnostics and test theory are extremely cross-sectionally oriented. Test scoring techniques aim almost exclusively at comparing the testee with the population, based on data collected at a single point in time. Validity as well as reliability studies also are typically performed on test data collected at a single point in time. One exception is the test-retest method to assess the reliability of test scores. However, in this case the unrealistic assumption is made that the testees did not change in the time interval between test and retest, so that the retest can be considered to be simultaneous with the test and the interval to be virtually zero. By introducing a longitudinal model, which accounts for the change in the testees over time and for causal effects between latent variables, validity and reliability can be assessed across time without reliance on the no-change assumption. In addition, longitudinal data enable to get much more reliable estimates of the latent scores than cross-sectional data and to compare the testee with him- or herself across time instead of only with the population. It is shown how continuous time modeling is able to account for individually varying intervals and what difference in results this may make. Continuous is also able to combine cross-sectional and longitudinal data in the analysis, for example, when part of the total sample has only one measurement, for another part test-retest data are available and perhaps still another part has been measured three or more
The primary goal of this presentation is to demonstrate the close relationship between two classes of dynamic models in psychological research: Latent change score models and continuous time models. The secondary goal is to point out some differences. We begin with a quick review of both approaches, before demonstrating how the two methods are mathematically and conceptually related. It will be shown that most commonly used latent change score models are related to continuous time models based on the difference equation approximation. One way in which the two approaches differ, is the treatment of time. While there are theoretical and practical limits to the number of unequal time intervals in latent change score models, continuous time models do not suffer from such limitations. We illustrate our arguments by simulated data and a reanalysis of the Bradway-McArdle longitudinal study on intellectual abilities (McArdle & Hamagami, 2004) by means of the proportional change score model and the dual change score model in discrete and continuous time.