# When to include the average of a Level 1 predictor as a Level 2 predictor?

Reflecting on this, I came to the conclusion that the most obvious reason to do it was if you were interested in whether the average of the Level 1 variable in a particular Level 2 unit impacts the outcome variable, even once we've controlled for the Level 1 variable itself.

I've also seen some other reasons for doing it suggested in the literature.

• Raudenbush & Bryk (2002) p. 261 suggest that it is one option to fix problems with bias that would otherwise occur when an omitted Level 2 predictor is associated with a Level 1 predictor.

• Gelman & Hill (2006) p. 480 also recommend doing it in an example they give in which adding a predictor would otherwise increase the residual variance.

• While not offering any comment on its merits, Heck et al (2014) give an example of a random intercepts model where individual pupils are nested in schools and both individual SES and the average SES at a pupil's school are used to predict individual test scores. The model is as follows:

$$Y_{ij} = \gamma_{00} + \gamma_{01}\text{ses_mean} + \gamma_{02}\text{pro4yrc} + \gamma_{03}\text{public} + \gamma_{10}\text{ses} + u_{0j} + \epsilon_{ij}$$

I checked and in their book the Level 2 variable ses_mean is calculated just from averaging the SESes of other pupils in the sample who are from the same school, i.e. it can be calculated simply from the Level 1 variable SES and a knowledge of which pupils go to which schools.

My questions:

• Are all these three reasons valid? Are there other good reasons to include the average of a Level 1 predictor as a Level 2 predictor?
• What are some reasons to not include the average of a Level 1 predictor as a Level 2 predictor?
• Should the average of a Level 1 predictor be routinely used as a Level 2 predictor, or only in special circumstances?

Gelman, A., & Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.

Heck, R. H., Thomas, S. L., & Tabata, L. N. (2013). Multilevel and longitudinal modeling with IBM SPSS. Routledge.

Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (Vol. 1). Sage.

• If you are referring to "group mean centering" rather than grand mean centering, this is often done when we want to seperate the between-effect from the within-effect, when a variable varies both between and within a grouping variable, and it's sometimes called "contextual effects" Commented Jun 25, 2021 at 15:26
• Thanks - to the best of my knowledge the issue is separate from group mean centering, and seeking to explain the scenario a bit more concretely I've updated the post with an example from a textbook. Commented Jun 26, 2021 at 6:29