Developmental researchers commonly utilize multilevel models (MLMs) to describe and predict individual differences in change over time. In such growth model applications, researchers have been widely encouraged to supplement reporting of statistical significance with measures of effect size, such as R‐squareds (R²) that convey variance explained by terms in the model. An integrative framework for computing R‐squareds in MLMs with random intercepts and/or slopes was recently introduced by Rights and Sterba and it subsumed pre‐existing MLM R‐squareds as special cases. However, this work focused on cross‐sectional applications, and hence did not address how the computation and interpretation of MLM R‐squareds are affected by modeling considerations typically arising in longitudinal settings: (a) alternative centering choices for time (e.g., centering‐at‐a‐constant vs. person‐mean‐centering), (b) nonlinear effects of predictors such as time, (c) heteroscedastic level‐1 errors and/or (d) autocorrelated level‐1 errors. This paper addresses these gaps by extending the Rights and Sterba R‐squared framework to longitudinal contexts. We: (a) provide a full framework of total and level‐specific R‐squared measures for MLMs that utilize any type of centering, and contrast these with Rights and Sterba's measures assuming cluster‐mean‐centering, (b) explain and derive which measures are applicable for MLMs with nonlinear terms, and extend the R‐squared computation to accommodate (c) heteroscedastic and/or (d) autocorrelated errors. Additionally, we show how to use differences in R‐squared (ΔR²) measures between growth models (adding, for instance, time‐varying covariates as level‐1 predictors or time‐invariant covariates as level‐2 predictors) to obtain effects sizes for individual terms. We provide R software (r2MLMlong) and a running pedagogical example analyzing growth in adolescent self‐efficacy to illustrate these methodological developments. With these developments, researchers will have greater ability to consider effect size when analyzing and predicting change using MLMs.

中文翻译：

---

纵向增长分析的效应大小测量：扩展多级模型 R 平方的框架以适应异方差、自相关、非线性和替代中心策略

发育研究人员通常利用多层次模型 (MLM) 来描述和预测随时间变化的个体差异。在这种增长模型应用中，研究人员被广泛鼓励用效应大小的度量来补充统计显着性的报告，例如 R 平方 ( R ²) 表示由模型中的术语解释的方差。最近，Rights 和 Sterba 引入了一个用于在具有随机截距和/或斜率的 MLM 中计算 R 平方的综合框架，它将预先存在的 MLM R 平方归为特殊情况。然而，这项工作侧重于横截面应用，因此没有解决 MLM R 平方的计算和解释如何受到纵向设置中通常出现的建模考虑因素的影响：（a）时间的替代居中选择（例如，居中 - at-a-constant vs. person-mean-centering), (b) 预测变量的非线性效应，如时间, (c) 异方差 1 级误差和/或 (d) 自相关 1 级误差。本文通过将 Rights 和 Sterba R 平方框架扩展到纵向背景来解决这些差距。我们：(a) 为使用任何类型居中的 MLM 提供了一个完整的和特定级别 R 平方度量的完整框架，并将这些与 Rights 和 Sterba 假设集群均值居中的度量进行对比，(b) 解释并推导出哪些度量适用于具有非线性项的 MLM，并扩展 R 平方计算以适应 (c) 异方差和/或 (d) 自相关误差。此外，我们展示了如何使用 R 平方 (Δ R ²) 增长模型之间的度量（例如，添加时变协变量作为 1 级预测变量或时变协变量作为 2 级预测变量）以获得单个项的效应大小。我们提供了 R 软件 ( r2MLMlong ) 和一个正在运行的教学示例，用于分析青少年自我效能感的增长，以说明这些方法论的发展。随着这些发展，研究人员在使用 MLM 分析和预测变化时将有更大的能力考虑效应大小。