The basics of continuous time modelling by means of SEM can be summarized in five steps. A more detailed description is provided by Oud and Delsing (2010). A less technical introduction to the approach for a broader psychological readership is provided by Voelkle, Oud, Davidov, and Schmidt (in press), who also provide a more detailed description of the five steps summarized below. Technical aspects and model identification are discussed by Oud and Jansen (2000). For an illustrative empirical example on the relationship between decoding skill and reading comprehension, see Oud (2007b). Other examples include the analysis of the relationship between externalizing and internalizing problem behaviour, and the analysis of the relationships between individualism, nationalism, and ethnocentrism in Flanders (Oud & Delsing, 2010). For an investigation of the reciprocal influence between antisocial behaviour and depressive symptoms by means of a closely related model (the so-called continuous time autoregressive latent trajectory model), see Delsing and Oud (2008).
2.1. Continuous time modelling in five steps
Second, the stochastic differential equation corresponding to equation (1), and from which equation (1) is derived as a solution, can be expressed as
In contrast to equation (1), the coefficient matrix A– the so-called drift matrix – and the intercept vector b are no longer dependent on the time interval. W (t) denotes the random walk process in continuous time (the so-called Wiener process). In order to model arbitrary variances and covariances among the error terms, W (t) is premultiplied by the Cholesky triangle so that the resulting error covariance matrix in continuous time – the so-called diffusion matrix – is Q=GGT, with T denoting the transposition operator.
Third, for initial value and any time interval , the solution of the stochastic differential equation (3) is
for Q=GGT and . The row operator puts the elements of the Q matrix row-wise into a column vector, while irow represents the inverse operation (for details, see Arnold, 1974; Ruymgaart & Soong, 1985; Oud & Jansen, 2000).
Fourth, for the observed time intervals in a given study the discrete time coefficients in equation (1) are constrained to the underlying continuous time parameters as shown in equation (4), so that
with discrete time error covariance matrix
Fifth, the model given in equation (1), but with the discrete time parameters constrained to the underlying continuous time parameters as defined above (i.e., the continuous time model), is translated into an SEM framework – as demonstrated in the next paragraph – with
Given equations (2) and (5), it is easy to derive the model implied covariance matrix (e.g., Bollen, 1989, p. 325). Parameter estimates are then obtained by minimizing the raw maximum likelihood function
with y representing the vector of observed variables as defined above, and the corresponding mean vector. Note that missing values are easily handled by allowing the number of variables (m=p · T), to differ across all individuals (i.e., full information maximum likelihood estimation; cf. Arbuckle, 1996). However, before returning to the definition of the matrices in equation (5), we will first extend the approach to individually varying time intervals.
2.2. Individually varying time intervals
Individually varying time intervals are easily introduced into the model by allowing t and to be different for each individual i and time point j, that is, ti,j and . As a consequence, all discrete time parameters in equation (1) may also differ across individuals (i.e., ). Only the underlying continuous time parameters remain the same. In terms of the SEM equation (5), for j= 0, 1, …, T− 1 discrete time points (t) and i= 1,…, N independent individuals, the final continuous time model with individually varying time intervals can thus be defined as
with being a (p · T + 1) × 1 vector, which contains the p-variate outcome vectors that have been stacked above each other across all T time points, plus the constant 1. Likewise, B is defined as