[100 Multilevel Questions]Biased within effect and group means

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Dear all,

Consider the two-level random intercept model for students nested in schools:

Y(ij) = b0(j) + b1*X(ij) + e(ij)         (model 1)
with
b0(j) = gamma00 + u0(j)

As far as I know, the estimate of b1 will be biased, if u0(j) correlates with X(ij). To avoid such bias, Tom Snijders e.g.
recommends to add the schoolmean of X as a predictor to the model:

Y(ij) = b0(j) + b1*[ X(ij) - schoolmean.of.X(j) ]  + b2*schoolmean.of.X(j)  +   e(ij)     (model 2)

The term [  X(ij) - schoolmean.of.X(j) ] points to the deviation of the X value of student i in school j from
the (school)mean of X for school j. In model 2, b1 is called the within effect and b2 is called the between effect.

My first question is about the bias in the estimate of b1 1 in model 1. Is the bias in the direction of the value of b2 in model 2?
Could you say that the estimate of b1 in model 1 is a 'compromise' between the two values of b1 and b2 in model 2, and hence lies somewhere in between?

Further, suppose there is a school-level attribute, say 'schoolsize', which correlates with schoolmean.of.X(j). However, since 'schoolsize' doesn't occur in model 2 as an explanatory variable, the u0(j) term in model 2 will correlate with schoolmean.of.X(j) and therefore, the effect b2 in model 2 will be biased. Is this conclusion correct?

Could you please point me to an 'accessible' publication about the (direction of the) bias in the effect b1?

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关键词：Multilevel questions question within Effect within

I would say that the answer to all of your questions is yes.

Hopefully this article will be useful to you (it is not yet published but hopefully will be relatively soon - it has now been through a couple of rounds of revisions).

https://www.researchgate.net/pub ... nel_Data?ev=prf_pub