Individual differences in the timing of developmental processes are often of interest in longitudinal studies, yet common statistical approaches to modeling change cannot directly estimate the timing of when change occurs. The time-to-criterion framework was recently developed to incorporate the timing of a prespecified criterion value; however, this framework has difficulty accommodating contexts where the criterion value differs across people or when the criterion value is not known a priori, such as when the interest is in individual differences in when change starts or stops. This article combines aspects of reparameterized quadratic models and multiphase models to provide information on the timing of change. We first consider the more common situation of modeling decelerating change to an offset point, defined as the point in time at which change ceases. For increasing trajectories, the offset occurs when the criterion attains its maximum (“inverted J-shaped” trajectories). For decreasing trajectories, offset instead occurs at the minimum. Our model allows for individual differences in both the timing of offset and ultimate level of the outcome. The same model, reparameterized slightly, captures accelerating change from a point of onset (“J-shaped” trajectories). We then extend the framework to accommodate “S-shaped” curves where both the onset and offset of change are within the observation window. We provide demonstrations that span neuroscience, educational psychology, developmental psychology, and cognitive science, illustrating the applicability of the modeling framework to a variety of research questions about individual differences in the timing of change. (PsycInfo Database Record (c) 2021 APA, all rights reserved)

中文翻译：

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对变化开始和抵消时间的个体差异进行建模。


纵向研究通常对发育过程时间的个体差异感兴趣，但建模变化的常见统计方法无法直接估计变化发生的时间。最近开发了达到标准的时间框架，以纳入预先指定的标准值的时间；然而，该框架难以适应不同人的标准值不同或先验未知标准值的情况，例如当关注的是变化何时开始或停止的个体差异时。本文结合了重新参数化二次模型和多相模型的各个方面，以提供有关变化时间的信息。我们首先考虑对偏移点进行减速变化建模的更常见情况，偏移点定义为变化停止的时间点。对于增加的轨迹，当标准达到最大值时（“倒 J 形”轨迹），就会发生偏移。对于递减的轨迹，偏移量反而发生在最小值。我们的模型考虑到抵消时间和最终结果水平的个体差异。同一模型，稍微重新参数化，捕获从起始点开始的加速变化（“J 形”轨迹）。然后，我们扩展框架以适应“S 形”曲线，其中变化的开始和偏移都在观察窗口内。我们提供涵盖神经科学、教育心理学、发展心理学和认知科学的演示，说明建模框架对有关变化时间的个体差异的各种研究问题的适用性。 （PsycInfo 数据库记录 (c) 2021 APA，保留所有权利）