2
$\begingroup$

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?

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.