Bayesian continuous-time Rasch models.

Abstract

Continuous-time modeling offers a flexible approach to analyze longitudinal data from designs with unequally spaced measurement occasions. Measurement models are popular tools in psychological research to control for measurement error. The objective of the present article is to introduce the continuous-time Rasch model, a combination of the Rasch model and a continuous-time dynamic model. In a series of simulations we demonstrate the performance of the proposed model and that ignoring individual unequal time interval lengths, choosing a wrong measurement model, and selecting a wrong analysis strategy results in poor parameter estimates. The newly proposed continuous-time Rasch model overcomes these problems and offers a powerful new approach to longitudinal analysis with dichotomous items. A step-by-step tutorial on how to run a continuous-time Rasch model with the R package ctsem and an illustrative empirical example is given.

Continuous-time modeling in prevention research: An illustration

“…Thus, studies in which discrete-time crosslagged models based on different time intervals were used would come to different results and conclusions, whereas this problem is resolved in continuous-time models. Further, discrete-time models rely on equal-interval nonindividualized spacings of measurement occasions and may perform poorly when this design feature is not given (De Haan-Rietdijk, Voelkle, Keijsers, & Hamaker, 2017;Hecht, Hardt, et al, 2019).…”

“…. , P j being a running index that denotes the discrete measurement occasion and P j being the personspecific number of measurement occasions (see Hecht, Hardt, et al, 2019, for details and illustrations). The manifest responses, y jpf , of person j at measurement occasion p on variable f ¼ 1, .…”

“…2, where A à D jðp À 1Þ is the square autoregression matrix of order F containing the autoregressive effects on the main diagonal and the cross-lagged effects on the off-diagonals; Q à D jðp À 1Þ is the symmetric process error covariance matrix of order F containing the process error variances on the main diagonal and the process error covariances on the off-diagonals; b à D jðp À 1Þ is a column vector containing F discrete-time intercepts; and b à jD jðp À 1Þ is a column vector containing person-specific deviations from the discrete-time intercepts. These discrete-time parameters all depend on person-specific and occasion-specific interval lengths D j(p À 1) ¼ t jp À t j(p À 1) and can be calculated from the continuous-time parameters, equations (5) to (11) in the Appendix (for examples and illustrations, see Hecht, Hardt, et al, 2019). In Table 1, we provide an overview of possible terms to distinguish corresponding discrete-time and continuous-time parameters.…”

A Computationally More Efficient Bayesian Approach for Estimating Continuous-Time Models

“…. ; P being a running index denoting the discrete measurement occasion and P being the number of measurement occasions (see, e.g., Hecht, Hardt, Driver, & Voelkle, 2019, for details and illustrations). Time intervals Δ pÀ1 between measurement occasions are given by Δ pÀ1 ¼ t p À t pÀ1 for all p !…”

Integrating Out Nuisance Parameters for Computationally More Efficient Bayesian Estimation – An Illustration and Tutorial

“…Although the advantages of Bayesian approaches are certainly well appreciated, a frequently encountered obstacle is the high run time that might prevent users from using Bayesian estimation. For instance, Hecht, Hardt, Driver, and Voelkle (2019) report run times of hours to days for rather small Bayesian longitudinal models. Similar problems were encountered by Lüdtke, Robitzsch, and Wagner (2018) who note that "very long chains (more than 2 Â 10 6 iterations) […] provided only poor approximations of the posterior distributions (e.g., small effective sample size)" (p. 577).…”