More Precise Estimation of Lower-Level Interaction Effects in Multilevel Models

Abstract

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.

Catching Up on Multilevel Modeling

“…In so-called variable-centering (Hoffman, 2019), level-1 predictors are usually centered using their level-2 means (as shown below), but other options are possible, such as using the baseline variable of time-level-1 predictors in longitudinal designs (Algina & Swaminathan, 2011), or centering level-1 predictors using more than one level-2 variable simultaneously (e.g., double decomposition; O' Keefe & Rodgers, 2017). When used for clustered data, variable-centering using the level-2 cluster mean has many names, such as cluster-mean-centering (e.g., Antonakis et al, 2021;Brinks et al, 2017;Loeys et al, 2018;Rights & Sterba, 2019) group-mean-centering (e.g., Algina & Swaminathan, 2011;Hofmann & Gavin, 1998;Raudenbush & Bryk, 2002, ch. 5;Snijders & Bosker, 2012, ch.…”

Acceptance and commitment improve the work–caregiving interface among dementia family caregivers.

“…Specifically, we used interaction terms between work/caregiving demands at Level 1 and acceptance/commitment to personal values at Level 2 to examine Hypothesis 2, and those between work/caregiving demands at Level 1 and work resources/caregiving support at Level 2 were used to examine Hypothesis 4. Following Loeys et al (2018), both Level 1-to-Level 1 and Level 2-to-Level 2 interaction terms of the variables included in the cross-level interaction terms were also simultaneously entered into the HLM model with the sociodemographic variables at Wave 1 (specific variables as described above) and time as covariates (see Online Supplemental A, for specific variables).…”

Catching Up on Multilevel Modeling (in press, Annual Review of Psychology)

“…Following the clustered example from Season 1, the time-level-1 compliance predictor can be constant-centered or variable-centered-the latter when using the level-2 person mean in longitudinal studies is called person-mean-centering (e.g., Algina & Swaminathan, 2001;Curran & Bauer, 2011;Hoffman, 2015, ch. 8;Wang & Maxwell, 2015) or subject-mean-centering (e.g., Loeys et al, 2018). Here, we center time-level-1 compliance using its patient-level-2 mean, creating 1 = − ̅̅̅̅̅̅̅ .…”