Centering Strategies

Andy Grogan-Kaylor

Notation

In these notes I use

\(x\) to denote an independent variable
\(\bar{x}_{..}\) to denote the grand mean
\(\bar{x}_{.j}\) to denote a group mean

Then

\(x - \bar{x..}\) is a grand mean centered variable.
\(x - \bar{x}_{.j}\) is a group mean centered variable.

All of the approaches listed below seem plausible to me. I think that they are simply different regression sentences, or substantive questions, that one could pose of the data.

Uncentered x

\[y_{ij} = \beta_0 + \beta x + e_{ij} + u_{0i}\]

Grand Mean Centered x

\[y_{ij} = \beta_0 + \beta (x - \bar{x..}) + e_{ij} + u_{0i}\]

Group Mean Centered x and Group Mean of x

\[y_{ij} = \beta_0 + \beta (x - \bar{x}_{.j}) + \beta \bar{x}_{.j} + e_{ij} + u_{0i}\]

x and Group Mean of x

\[y_{ij} = \beta_0 + \beta x + \beta \bar{x}_{.j} + e_{ij} + u_{0i}\]

Citation

For attribution, please cite this work as

Grogan-Kaylor (2022, Feb. 2). Centering Strategies. Retrieved from https://agrogan1.github.io

BibTeX citation

@misc{grogan-kaylor2022centering,
  author = {Grogan-Kaylor, Andy},
  title = {Centering Strategies},
  url = {https://agrogan1.github.io},
  year = {2022}
}