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## Estimation

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      (5)

Similarly, the estimates of the (second-level) group parameters are given by      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 (6)

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      (7)

where represents the potentially different covariance in this new two-level model.

Now consider the single-level GLM (equation 4), where the BLUE is      (8)

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:      (9)

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.   Next: Model Equivalence Up: Models Previous: Single-level GLM
Christian Beckmann 2003-07-16