On the common but problematic specification of conflated random slopes in multilevel models

Abstract

For multilevel models (MLMs) with fixed slopes, it has been widely recognized that a level-1 variable can have distinct between-cluster and within-cluster fixed effects, and that failing to disaggregate these effects yields a conflated, uninterpretable fixed effect. For MLMs with random slopes, however, I clarify that two different types of slope conflation can occur: that of the fixed component (termed fixed conflation) and that of the random component (termed random conflation). The latter is rarely recognized and not well understood. Here I explain that a model commonly used to disaggregate the fixed component-the contextual effect model with random slopes-troublingly yields a conflated random component. Negative consequences of such random conflation have not been demonstrated. Here I show that they include erroneous interpretation and inferences about the substantively important extent of between-cluster differences in slopes, including either underestimating or overestimating such slope heterogeneity. Furthermore, I show that this random conflation can yield inappropriate standard errors for fixed effects. To aid researchers in practice, I delineate which types of random slope specifications yield an unconflated random component (namely, group-mean-centered models or a proposed unconflated contextual effect model that adds a random component for the cluster mean). I demonstrate the advantages of these unconflated models in terms of estimating and testing random slope variance (i.e., improved power, type I error, and bias) and in terms of standard error estimation for fixed effects (i.e., more accurate standard errors), and make recommendations for which specifications to use for particular research purposes.