a Department of Data Analysis , Ghent University.
b Department of Clinical Psychological Science , Maastricht University.
Multivariate Behav Res. 2018 May-Jun;53(3):335-347. doi: 10.1080/00273171.2018.1444975. Epub 2018 Mar 20.
In hierarchical data, the effect of a lower-level predictor on a lower-level outcome may often be confounded by an (un)measured upper-level factor. When such confounding is left unaddressed, the effect of the lower-level predictor is estimated with bias. Separating this effect into a within- and between-component removes such bias in a linear random intercept model under a specific set of assumptions for the confounder. When the effect of the lower-level predictor is additionally moderated by another lower-level predictor, an interaction between both lower-level predictors is included into the model. To address unmeasured upper-level confounding, this interaction term ought to be decomposed into a within- and between-component as well. This can be achieved by first multiplying both predictors and centering that product term next, or vice versa. We show that while both approaches, on average, yield the same estimates of the interaction effect in linear models, the former decomposition is much more precise and robust against misspecification of the effects of cross-level and upper-level terms, compared to the latter.
在层次数据中，较低层次的预测因子对较低层次结果的影响往往可能受到未测量的较高层次因素的干扰。当未解决这种混杂时，较低层次预测因子的效果会产生偏差。在线性随机截距模型中，在混杂因素的特定假设下，将该效果分离为组内和组间成分，可以消除这种偏差。当较低层次预测因子的效果还受到另一个较低层次预测因子的调节时，会将两个较低层次预测因子之间的交互作用项纳入模型中。为了解决未测量的上层次混杂问题，也需要将这个交互项分解为组内和组间成分。可以通过先对两个预测因子进行相乘，然后对乘积项进行中心化，或者反过来进行中心化，来实现这种分解。我们表明，虽然这两种方法平均都会产生线性模型中交互作用效果的相同估计值，但与后者相比，前者的分解在交叉层次和上层次项的效应的指定方面更加精确和稳健，不会出现偏差。
