A three-level piecewise growth model (3L-PGM) can be used to break up nonlinear growth into multiple components, providing the opportunity to examine potential sources of variation in individual and contextual growth within different segments of the model. The conventional 3L-PGM assumes that the data are strictly hierarchical in nature, where measurement occasions (level 1) are nested within individuals (level 2) who are members of a single cluster (level 3). However, in longitudinal research, it is sometimes difficult for data structures to remain purely clustered during a study, such as when some students change classrooms or schools over time. One resulting data structure in this situation is known as a multiple membership structure, where some lower-level units are members of more than one higher-level unit. The new multiple membership PGM (MM-PGM) extends the 3L-PGM to handle multiple membership data structures frequently found in the social sciences. This study sought to examine the consequences of ignoring individual mobility across clusters when estimating a 3L-PGM in comparison to estimating a MM-PGM. MM-PGM estimates were less biased (especially in the cluster-level coefficient estimates), although we found substantial bias in cluster-level variance components across some conditions for both models.

中文翻译：

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在估计分段增长模型时忽略跨集群的个体流动的影响

三级分段增长模型 (3L-PGM) 可用于将非线性增长分解为多个组成部分，从而有机会检查模型不同部分内个体和环境增长的潜在变异来源。传统的 3L-PGM 假设数据本质上是严格分层的，其中测量时机（级别 1）嵌套在作为单个集群（级别 3）成员的个人（级别 2）中。然而，在纵向研究中，数据结构有时很难在研究过程中保持纯粹的集群，例如当一些学生随着时间的推移改变教室或学校时。在这种情况下产生的一种数据结构称为多成员结构，其中一些较低级别的单元是多个较高级别单元的成员。新的多成员 PGM (MM-PGM) 扩展了 3L-PGM 以处理社会科学中常见的多成员数据结构。本研究试图检查在估计 3L-PGM 时忽略集群间个体移动性与估计 MM-PGM 相比的后果。MM-PGM 估计偏差较小（尤其是在集群级别系数估计中），尽管我们发现两种模型在某些条件下的集群级别方差分量存在很大偏差。