I’m reading a presentation by Preacher, who gives the following level-1 and level-2 equations

Level 1

$y_{ij} = \beta_{0j} + \beta_{1j}x_{1ij} + \epsilon_{ij}$

Level 2

$\beta_{0j} = \gamma_{00} + \gamma_{01}w_{1j} + u_{0j}$
$\beta_{1j} = \gamma_{10} + \gamma_{11}w_{1j} + u_{1j}$

Preacher notes:

Even though it is not possible to use a level-1 variable as a predictor in a level-3 equation, it is possible for the level-3 intercept variance to be reduced by $x_{ij}$. We can explain level-3 variance with level-1 predictors. For example, differences among classes may be partially explained by considering a student-level variable.

I think the reference to level-3 may be a typo, and Preacher actually means level-2, because at this point in his presentation Preacher hasn’t yet introduced the concept of a third level.

Why can’t the level-1 predictor $x_{ij}$ be included in either of the Level 2 equations?


1 Answer 1


There's a very simple reason, which also immediately suggests the work-around.

A level-1 predictor, $x_{ij}$ can't be used in the level-2 equation because there are many different values of $x_{ij}$ for a single level-2 unit $j$. You can't use, say, individual income as predictor in a model for neighbourhood crime levels, because there are lots of different individual incomes for each neighbourhood.

What you can do is use a summary of the level-1 predictor. You could use average individual income or median individual income or median household income or income interquartile range or whatever else you want.

As an additional note: it's possible to expand your class of mixed models from the multilevel representation to the Laird-Ware representation. This gets rid of the distinction between levels from the computational/expressive viewpoint. It doesn't get rid of the fact that there are many ways a set of level-1 variables can be summarised as a level-2 variable -- that's a real problem (or opportunity), not a computational one. You still have to decide.


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