To make specific recommendations for practice, it is important to consider the circumstances under which the true initial condition specifications used in this manuscript would arise. Therefore, I will next discuss circumstances and constructs in psychological research that would give rise to each of the true initial condition specifications considered and provide suggestions for fitted initial condition specifi- cations based on the results from this thesis.
true process is indeed stationary, choosing one of the stationary approaches is best. A true initial condition resulting from using a model-implied approach results in a stationary process that has always maintained its stationarity. This may be viable for psychological processes that ebb and flow with time, such as emotions or daily affect (see, for example, Diener, Fujita, & Smith, 1995; Shifren, Hooker, Wood, & Nesselroade, 1997) sampled on a daily basis. A model-implied approach in this case would appear to be the best choice, however this study did not provide supporting evidence as the model-implied approach proved to have severe convergence issues.
Generating a process using the free parameter approach is akin to generating a process where there is only a change in the first occasion, in line with the free- parameter approach discussed by Du Toit and Browne (2007). When the process is stationary, this means that the process before the first occasion was different than the overall process; however, from the first time point onward the process does not display mean trends or changes in variability. This type of process is more likely to arise in psychological data than the null initial condition, although the assumption of stationarity still holds across all time points other than the first. When this is the case, the fitted free-parameter initial condition performs best as it captures the mean and covariance of the process at the first time point, and also the stationary process onward. However, it is important to carefully consider whether this assumption is valid given the type of psychological construct being studied. Still, applying such a specification will provide information regarding the process before the first time point in the form of mean and variance-covariance parameter estimates. There is a cost to this, however, as more degrees of freedom are used. In line with the model selection literature, and a more parsimonious model may be preferred (Preacher, 2006, 2000).
structs estimated in psychological studies, such as self-assuredness and extroversion in the empirical example considered in this manuscript, this seems like an untenable assumption. Given that the fitted free-parameter method can estimate the value in the initial condition matrices, it seems wise to use this approach over a fitted null ap- proach unless a researcher is certain that the process starts at the first time point, in which case the null model would be a more parsimonious model. The fitted diffuse approaches performed especially poorly in this case, so it would be wise to avoid using them here.
If the process is nonstationary, this thesis considered three true initial condition specifications: 1. free-parameter, 2. null, and 3. diffuse. Like in the nonstationary case, the free-parameter condition indicates that there was indeed a change in pro- cess at the first time point, but in this case the process after the first time point is nonstationary. A process like this may occur in an experimental design after a treat- ment in enacted. Specifically, the first time point serves as a person’s process prior to receiving a treatment, and time points thereafter serve as a person’s process after receiving a treatment. While the fitted free-parameter approach works well when there are no convergence issues, it may not be the best choice if the process is highly nonstationary, as the variance may be consumed fully by the variance and covari- ance parameters of the initial condition matrices leading to potential convergence problems, boundary conditions, and parameter estimate outliers.
In the nonstationary case, a true initial condition consisting of null matrices would arise when the process starts exactly at the first time point, but displays a nonstationary process onward. Given psychological constructs, this seems like an untenable assumption. However, there may be cases in experimental designs where, prior to treatment, every person has no proficiency in a given construct. Still, being that the free-parameter approach estimates the process before the first time point, it may be safer to just apply this approach, especially since simulation results suggested
this approach performs well if the true initial condition is null. As in the stationary case, the fitted diffuse approaches did not perform very well when the true initial condition was null.
The final true initial condition considered was that of a diffuse process. In this case, the process began in the distant past, there is no prior information regarding the process before the first time point, and the process after the first time point re- mains nonstationary. This situation may occur often in psychological processes, as people are constantly changing in complicated ways with respect to many psycholog- ical constructs such as personality development, intelligence, and severity of clinical disorders. It is rarely known what a given person’s trajectory was before the first time point was collected. It is also unlikely that a person began the process right at the first time point. As the process started in the distant past and there is no prior information about the process, fitting a null initial condition is not a good idea, as confirmed by the simulation results. Fitting the free-parameter approach may work well as it is able to capture some aspects of the process prior to the first observation, but again there are issues of convergence, boundary conditions, and extreme outly- ing parameter estimates. The de Jong diffuse initial condition specification worked well in this case in terms of both recover point and standard error estimates and not displaying large convergence, boundary, or outlier problems. The other diffuse approaches did not work as well, mainly due to boundary cases and convergence issues. Thus, using the fitted de Jong approach would appear to be the best choice, followed by the free-parameter condition.