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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 [16].

Initially consider the two-level GLM. The parameter estimates at the first level are

$\displaystyle \widehat{\beta}$ $\displaystyle =$ $\displaystyle (X^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\script...
...size\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}Y,$  
$\displaystyle \textrm{Cov}(\widehat{\beta})$ $\displaystyle =$ $\displaystyle (X^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}X)^{\mbox{\scriptsize\textit{\sffamily {-1}}}}.$ (5)

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

$\displaystyle \widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}}$ $\displaystyle =$ $\displaystyle (X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsize\te...
...y\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsize\textit{\sffamily {-1}}}}\beta,$  
$\displaystyle \textrm{Cov}(\widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}})$ $\displaystyle =$ $\displaystyle (X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsize\te...
...it{\sffamily {$\!$G}}}}^{\mbox{}})^{\mbox{\scriptsize\textit{\sffamily {-1}}}}.$  

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

$\displaystyle \widehat{\beta} = X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}+ \eta'.$ (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

$\displaystyle \widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}}$ $\displaystyle =$ $\displaystyle (X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsize\te...
...ffamily {$\!$G2}}}}^{\mbox{\scriptsize\textit{\sffamily {-1}}}}\widehat{\beta},$  
$\displaystyle \textrm{Cov}(\widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}})$ $\displaystyle =$ $\displaystyle (X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsize\te...
...it{\sffamily {$\!$G}}}}^{\mbox{}})^{\mbox{\scriptsize\textit{\sffamily {-1}}}},$ (7)

where $ V_{\mbox{\tiny\textit{\sffamily {$\!$G2}}}}^{\mbox{}}= \textrm{Cov}(\eta')$ represents the potentially different covariance in this new two-level model.

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

$\displaystyle \widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}}$ $\displaystyle =$ $\displaystyle (X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsize\te...
...size\textit{\sffamily {$\!$T}}}}W^{\mbox{\scriptsize\textit{\sffamily {-1}}}}Y,$  
$\displaystyle \textrm{Cov}(\widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}})$ $\displaystyle =$ $\displaystyle (X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsize\te...
...it{\sffamily {$\!$G}}}}^{\mbox{}})^{\mbox{\scriptsize\textit{\sffamily {-1}}}}.$ (8)

This equation directly relates the group parameter estimates of interest, $ \widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}}$, to the full data vector $ Y$ 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:

$\displaystyle \widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}}$ $\displaystyle =$ $\displaystyle (X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsize\te...
...size\textit{\sffamily {$\!$T}}}}V^{\mbox{\scriptsize\textit{\sffamily {-1}}}}Y,$  
$\displaystyle \textrm{Cov}(\widehat{\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}})$ $\displaystyle =$ $\displaystyle (X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{\scriptsize\te...
...it{\sffamily {$\!$G}}}}^{\mbox{}})^{\mbox{\scriptsize\textit{\sffamily {-1}}}}.$ (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 $ V_{\mbox{\tiny\textit{\sffamily {$\!$G2}}}}^{\mbox{}}$ 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 up previous
Next: Model Equivalence Up: Models Previous: Single-level GLM
Christian Beckmann 2003-07-16