Department of Educational Psychology and Learning Systems, College of Education, Florida State University, Tallahassee, Florida, USA.
University of Notre Dame, Indiana, USA.
Br J Math Stat Psychol. 2019 May;72(2):334-354. doi: 10.1111/bmsp.12151. Epub 2018 Nov 25.
Moderation analysis is useful for addressing interesting research questions in social sciences and behavioural research. In practice, moderated multiple regression (MMR) models have been most widely used. However, missing data pose a challenge, mainly because the interaction term is a product of two or more variables and thus is a non-linear function of the involved variables. Normal-distribution-based maximum likelihood (NML) has been proposed and applied for estimating MMR models with incomplete data. When data are missing completely at random, moderation effect estimates are consistent. However, simulation results have found that when data in the predictor are missing at random (MAR), NML can yield inaccurate estimates of moderation effects when the moderation effects are non-null. Simulation studies are subject to the limitation of confounding systematic bias with sampling errors. Thus, the purpose of this paper is to analytically derive asymptotic bias of NML estimates of moderation effects with MAR data. Results show that when the moderation effect is zero, there is no asymptotic bias in moderation effect estimates with either normal or non-normal data. When the moderation effect is non-zero, however, asymptotic bias may exist and is determined by factors such as the moderation effect size, missing-data proportion, and type of missingness dependence. Our analytical results suggest that researchers should apply NML to MMR models with caution when missing data exist. Suggestions are given regarding moderation analysis with missing data.
moderation 分析在社会科学和行为研究中非常有用，可以帮助解决有趣的研究问题。在实践中，最广泛使用的是调节多元回归（MMR）模型。然而，缺失数据是一个挑战，主要是因为交互项是两个或更多变量的乘积，因此是涉及变量的非线性函数。基于正态分布的最大似然（NML）已被提出并应用于估计具有不完全数据的 MMR 模型。当数据完全随机缺失时，调节效应估计是一致的。然而，模拟结果发现，当预测变量中的数据随机缺失（MAR）时，当调节效应不为零时，NML 可能会对调节效应产生不准确的估计。模拟研究受到与抽样误差混淆的系统偏差的限制。因此，本文的目的是分析具有 MAR 数据的 NML 估计的渐近偏差。结果表明，当调节效应为零时，无论是正态数据还是非正态数据，调节效应估计都没有渐近偏差。然而，当调节效应不为零时，可能存在渐近偏差，其大小取决于调节效应大小、缺失数据比例以及缺失依赖类型等因素。我们的分析结果表明，当存在缺失数据时，研究人员应谨慎地将 NML 应用于 MMR 模型。针对缺失数据的调节分析提出了建议。
