- [[intraclass correlation]] - [[mixed-effects models centering simulations]] - [[220412_174201 MLM centering effects summary]] # Idea - [[Bell 2018 understanding and misunderstanding group mean centering]] ![[s20220413_143438.png|600]] ## Grand-mean centering (GMC, CGM) [[Enders 2007 centering predictor variables]] > The presence of between-cluster variation in the centered scores means that the CGM regression slope is also a mixture of the within- and between-cluster association between workload and well-being. Raudenbush and Bryk (2002) made this point, stating that, “the hierarchical estimator under grand-mean centering is an inappropriate estimator of the person-level [i.e., Level 1] effect. It too is an uninterpretable blend: neither $\beta_w$ nor $\beta_b$ ” (p. 139). The slope variance is potentially biased because of a dependency on intercepts that were subject to shrinkage. The same is not true of cluster/group-mean centering, so the resulting estimate of the slope variation is generally more accurate (Raudenbush & Bryk, 2002). A conflated slope estimate should be pulled closer to $\beta_{between}$ as the intraclass correlation of the predictor ($ICC_x$) increases, and closer to $\beta_{within}$ was $ICC_x$ decreases, all else being equal. ## Cluster/group-mean centering (CMC, WCC) The estimates are now expressed relative to other cases belong to the same cluster/group. The intercept should be interpreted as the unadjusted cluster mean. ## Contextual effects and the Mundlak model Also known as compositional effect—the same level 1 unit moving to another level-2 cluster. If student A is in school X and moves to school Y, what is this effect on their outcome? It's the difference in between-cluster versus within-cluster effects: $\beta_{contextual} = \beta_{between} - \beta_{within}$ [[Raudenbush 2002 hierarchical linear models]] p140 ![[s20220412_224141.png]] ![[s20220412_224311.png]] What is the effect of an individual (level 1) moving from one level-2 cluster to another level-2 cluster? ![[s20220412_221642.png]] # References - [My advisor told me I should group-mean center my predictors in my multilevel model because it might "make my effects significant" but this doesn't seem right to me. What exactly is involved in centering predictors within the multilevel model? - CenterStat](https://centerstat.org/centering/) - [[Enders 2007 centering predictor variables]] - [[Raudenbush 2002 hierarchical linear models]] - [How to center in multilevel models – Philipp K. Masur](https://philippmasur.de/2018/05/23/how-to-center-in-multilevel-models/) - [[Yaremych 2021 centering categorical variables in multilevel models]] - [Fitting multilevel models when predictors and group effects correlate | Statistical Modeling, Causal Inference, and Social Science](https://statmodeling.stat.columbia.edu/2017/11/12/fitting-multilevel-models-predictors-group-effects-correlate/) - [[Raudenbush 2009 adaptive centering with random effects]]