Preacher Kristopher J, Hancock Gregory R
Department of Psychology and Human Development, Vanderbilt University.
Department of Human Development and Quantitative Methodology, University of Maryland.
Psychol Methods. 2015 Mar;20(1):84-101. doi: 10.1037/met0000028.
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.
纵向建模的一个基本目标是获得能够反映随时间变化的有意义方面的模型参数估计值。通常，从理论角度来看，线性或非线性模型可能是合理的，但可能具有难以以阐明实质性假设的方式进行解释的参数。幸运的是，可以对这类模型进行重新参数化，以产生更易于解释的参数。本文有三个目标。首先，我们提供理论背景并详细阐述普雷彻和汉考克（2012年）用于对增长曲线模型进行重新参数化的四步方法。其次，我们在第三步中通过对结构化潜在曲线模型进行用户友好的修改来扩展此方法，这使得能够拟合原始方法无法估计的模型。这种修改还允许研究人员指定均值结构，而无需确定哪些参数以非线性方式进入，也无需求解复杂的矩阵表达式。第三，我们说明这种一般的重新参数化方法如何使研究人员能够将平均变化率、半衰期和节点（转折点）视为随机系数；在结构方程建模中，这些变化方面以前尚未被视为随机系数。我们在在线补充材料中提供了用于说明性示例的Mplus代码。我们的核心信息是，增长曲线模型比大多数研究人员可能认为的要灵活得多。实际上，任何参数都可以被视为个体间变化的随机系数。给定模型的替代参数化可能会产生传统参数化无法获得的独特见解。
