Fitting Multilevel Models When Predictors and Group Effects Correlate

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Compared to what? Variation in the impacts of early childhood education by alternative care type (2016, methodology): “We include the proportion assigned to treatment, z̄j , as a site-level predictor in order to account for differing proportions randomized to treatment by site [see Bafumi and Gelman (2006), Raudenbush (2015)].”
Multilevel Modeling and Ordinary Least Squares Regression: How Comparable Are They? (2018, background): “The solution involves including the aggregated level-one variable (or including-the-group-mean; IGM) as a predictor at level two (Bafumi & Gelman, 2006; Hanchane & Mostafa, 2012; Mundlak, 1978).”
But You Don’t Look Like A Scientist!: Women Scientists with Feminine Appearance are Deemed Less Likely to be Scientists (2016, unknown): “Importantly, decomposing the femininity predictor in this way also avoids the problem of biased parameter estimates that can result from pooling together effects from different levels of analysis (Bafumi and Gelman 2006; Bell and Jones 2015).”