The BLUEs for both the two-level GLM and the single-level GLM can be calculated using the General Least Squares approach .
Initially consider the two-level GLM. The parameter estimates at the
first level are
Similarly, the estimates of the (second-level) group parameters are
In practice, however, the second-level model uses the estimates from the first level as input and not the true (but unobservable) parameters. That is, equation 2 is modified, becoming
Therefore the two-level model, as used in practice, is specified by
equations 1 and 6. This has significant
implications, as the two-level version is no longer precisely
equivalent to the single-level model in terms of estimation. In
particular, the estimation of the group parameters in the two-level
model now is
Now consider the single-level GLM (equation 4), where the BLUE is
This equation directly relates the group parameter estimates of interest, , to the full data vector and so requires the GLM to be solved for matrices of greatly increased size.
Thus, instead of solving the single-level model all at once, we wish to use
the two-level approach. However, substituting equation 5
into equation 7 gives the two-level group parameter estimates as:
If this estimation (equation 9) can be made equivalent to the single-level estimation (equation 8) by accounting for the covariances of the first-level estimates within the second-level (i.e. setting appropriately), then the two approaches become exactly equivalent. This turns out to be possible and the general equivalence result is presented in the next section.