On the Common but Problematic Specification of Conflated Random Slopes in Multilevel Models

• Published in: Multivariate behavioral research 2023-11, Vol.58 (6), p.1106-1133
• Main Authors: Rights, Jason D., Sterba, Sonya K.
• Format: Article
• Language: English
• Subjects: Clusters; contextual effect models; Estimation; group mean centering; Heterogeneity; Multilevel modeling; random slopes; Slopes; Specifications; Standard error
• ISSN: 0027-3171; 1532-7906
• DOI: 10.1080/00273171.2023.2174490

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, we 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 we explain that a model commonly used to disaggregate the fixed component-the contextual effect model with random slopes-troublingly still yields a conflated random component. Negative consequences of such random conflation have not been demonstrated. Here we 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, we show that this random conflation can yield inappropriate standard errors for fixed effects. To aid researchers in practice, we delineate which types of random slope specifications yield an unconflated random component. We 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.