Latent growth curve models with spline functions are flexible and accessible statistical tools for investigating nonlinear change patterns that exhibit distinct phases of development in manifested variables. Among such models, the bilinear spline growth model (BLSGM) is the most straightforward and intuitive but useful. An existing study has demonstrated that the BLSGM allows the knot (or change-point), at which two linear segments join together, to be an additional growth factor other than the intercept and slopes so that researchers can estimate the knot and its variability in the framework of individual measurement occasions. However, developmental processes usually unfold in a joint development where two or more outcomes and their change patterns are correlated over time. As an extension of the existing BLSGM with an unknown knot, this study considers a parallel BLSGM (PBLSGM) for investigating multiple nonlinear growth processes and estimating the knot with its variability of each process as well as the knot-knot association in the framework of individual measurement occasions. We present the proposed model by simulation studies and a real-world data analysis. Our simulation studies demonstrate that the proposed PBLSGM generally estimate the parameters of interest unbiasedly, precisely and exhibit appropriate confidence interval coverage. An empirical example using longitudinal reading scores, mathematics scores, and science scores shows that the model can estimate the knot with its variance for each growth curve and the covariance between two knots. We also provide the corresponding code for the proposed model. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

中文翻译：

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在各个测量场合的框架中，在平行双线性样条曲线增长曲线模型中估计结及其关联。

具有样条函数的潜伏增长曲线模型是灵活且易于访问的统计工具，用于研究非线性变化模式，这些变化模式在明显的变量中表现出不同的发展阶段。在这些模型中，双线性样条增长模型（BLSGM）是最直接，最直观但最有用的模型。现有研究表明，BLSGM允许将两个线性段连接在一起的结（或变化点）作为除截距和斜率之外的其他增长因子，以便研究人员可以估算结中的结及其变化性。个别测量场合的框架。但是，发展过程通常是在联合发展中展开的，在联合发展中，两个或多个结果及其变化模式随时间而相互关联。作为现有BLSGM的扩展，具有未知的结，这项研究考虑了并行BLSGM（PBLSGM），用于研究多个非线性增长过程，并根据每个过程的可变性以及各个测量场合的结节关联来估计结节。我们通过仿真研究和现实世界的数据分析提出了提出的模型。我们的仿真研究表明，提出的PBLSGM通常无偏，准确地估计感兴趣的参数，并展现出适当的置信区间覆盖范围。一个使用纵向阅读分数，数学分数和科学分数的经验示例表明，该模型可以估计出每个结曲线的方差和两个结之间的协方差所带来的结。我们还为建议的模型提供了相应的代码。（PsycInfo数据库记录（c）2021 APA，保留所有权利）。