Development and evaluation of R-squared effect size measures and methods for mixed effects models

分层数据结构在许多科学研究领域中无处不在，在心理学中尤其普遍。典型的例子包括嵌套在学校里的学生，或者嵌套在个人里的重复观察。这些数据通常用混合效应模型进行分析，混合效应模型扩展了标准回归模型，允许系数随聚类（例如，学校）而变化。尽管混合效应模型在心理学和其他学科中越来越受欢迎，但使用混合效应模型的一个主要限制是，研究人员很少报告效应大小（效应大小/变量之间关系的定量指示）的测量结果，而通常主要用p值来总结结果。这与广泛的建议形成鲜明对比，即传达实际意义，而不是完全依赖统计意义。特别是，标准回归r平方（R2）中经常报道的一个效应大小测量，反映了混合效应模型中很少报道的解释方差的比例。在混合效应模型中计算R2明显比在标准回归中要复杂得多，而且对于如何最好地做到这一点一直存在困惑。

Hierarchical data structures are ubiquitous in many areas of scientific research, and are particularly common in psychology. Typical examples include students nested within schools, or repeated observations nested within individuals. Such data are often analyzed with mixed effects models, which extend standard regression models by allowing coefficients to vary by cluster (e.g., school). Despite their increasing popularity in psychology and other disciplines, one major limitation in the use of mixed effects models is that researchers rarely report measures of effect size (a quantitative indication of the magnitude of effects/ relationships among variables) and instead typically summarize results primarily with p-values. This stands in sharp contrast to widespread recommendations to convey practical significance rather than relying exclusively on statistical significance. In particular, one effect size measure routinely reported in standard regressionR-squared (R2), reflecting the proportion of variance explainedis rarely reported for mixed effects models. Computing R2 in mixed effects models is markedly more complex than in standard regression, and there has been confusion about how best to do so. In my previous work, I explained how the methodological literature on mixed effect model R2 has historically suffered from many shortcomings, leading to researchers either not reporting such measures at all or, perhaps more troublingly, using and interpreting them incorrectly. To address these shortcomings, I provided a unifying framework to defining, computing, and interpreting R2 for mixed effects models (Rights & Sterba, 2019, in press). Several key shortcomings persist, however, in that (a) this framework does not apply to every possible model specification, (b) little work has been done to assess the statistical properties of the available mixed effect R2 measures, and (c) there is a lack of available software to compute measures. In this proposal, I aim to address these shortcomings. Objective 1 is to extend the Rights & Sterba R2 framework to a wider variety of modeling contexts, such as generalized linear mixed effects models, cross-classified models, longitudinal models with complex residual variance structures, and latent variable models. Objective 2 is to formally evaluate both preexisting and newly developed R2 methods in terms of bias, efficiency, and coverage via simulation. Different methods of bias adjustment and confidence interval construction will be considered and developed. Objective 3 is to ease the computational burden on researchers by providing open-source software to compute these measures and implement these methods. Overall, the long-term aim of this research program is to enable researchers to better compute effect size for mixed effects models. This responds to widespread recommendations to consider practical significance of statistical results, and will be useful in many research disciplines in psychology and beyond.