Lüdtke Oliver, Marsh Herbert W, Robitzsch Alexander, Trautwein Ulrich, Asparouhov Tihomir, Muthén Bengt
Center for Educational Research, Max Planck Institute for Human Development, Berlin, Germany.
Psychol Methods. 2008 Sep;13(3):203-29. doi: 10.1037/a0012869.
In multilevel modeling (MLM), group-level (L2) characteristics are often measured by aggregating individual-level (L1) characteristics within each group so as to assess contextual effects (e.g., group-average effects of socioeconomic status, achievement, climate). Most previous applications have used a multilevel manifest covariate (MMC) approach, in which the observed (manifest) group mean is assumed to be perfectly reliable. This article demonstrates mathematically and with simulation results that this MMC approach can result in substantially biased estimates of contextual effects and can substantially underestimate the associated standard errors, depending on the number of L1 individuals per group, the number of groups, the intraclass correlation, the sampling ratio (the percentage of cases within each group sampled), and the nature of the data. To address this pervasive problem, the authors introduce a new multilevel latent covariate (MLC) approach that corrects for unreliability at L2 and results in unbiased estimates of L2 constructs under appropriate conditions. However, under some circumstances when the sampling ratio approaches 100%, the MMC approach provides more accurate estimates. Based on 3 simulations and 2 real-data applications, the authors evaluate the MMC and MLC approaches and suggest when researchers should most appropriately use one, the other, or a combination of both approaches.
在多层模型（MLM）中，组级（L2）特征通常通过汇总每个组内的个体级（L1）特征来衡量，以便评估情境效应（例如，社会经济地位、成就、氛围的组平均效应）。大多数先前的应用都采用了多层显变量协变量（MMC）方法，其中观测到的（显）组均值被假定为完全可靠。本文通过数学推导和模拟结果表明，这种MMC方法可能会导致情境效应估计出现大幅偏差，并且可能会大幅低估相关的标准误差，这取决于每组L1个体的数量、组的数量、组内相关系数、抽样比例（每组抽样的案例百分比）以及数据的性质。为了解决这个普遍存在的问题，作者引入了一种新的多层潜变量协变量（MLC）方法，该方法可以校正L2的不可靠性，并在适当条件下得到L2结构的无偏估计。然而，在某些情况下，当抽样比例接近100%时，MMC方法能提供更准确的估计。基于3次模拟和2个实际数据应用，作者评估了MMC和MLC方法，并建议研究人员在何时最适合使用其中一种、另一种或两种方法的组合。
