Meaningful aspects of change as novel random coefficients: a general method for reparameterizing longitudinal models

A fundamental goal of longitudinal modeling is to obtain estimates of model parameters that reflect meaningful aspects of change over time. Often, a linear or nonlinear model may be sensible from a theoretical perspective, yet may have parameters that are difficult to interpret in a way that sheds light on substantive hypotheses. Fortunately, such models may be reparameterized to yield more easily interpretable parameters. This article has 3 goals. First, we provide theoretical background and elaboration on Preacher and Hancock's (2012) 4-step method for reparameterizing growth curve models. Second, we extend this method by providing a user-friendly modification of the structured latent curve model in the third step that enables fitting models that are not estimable with the original method. This modification also allows researchers to specify the mean structure without having to determine which parameters enter nonlinearly and without needing to solve complex matrix expressions. Third, we illustrate how this general reparameterization method allows researchers to treat the average rate of change, half-life, and knot (transition point) as random coefficients; these aspects of change have not before been treated as random coefficients in structural equation modeling. We supply Mplus code for illustrative examples in an online supplement. Our core message is that growth curve models are considerably more flexible than most researchers may suspect. Virtually any parameter can be treated as a random coefficient that varies across individuals. Alternative parameterizations of a given model may yield unique insights that are not available with traditional parameterizations.