Theorem:
The twolevel model specified by equations 1 and 6 is fully equivalent to the singlelevel model specified by equation 4 in terms of both modelling and estimation when
That is, the secondlevel covariance is set as the sum of the groupcovariance from the singlelevel model and the firstlevel parameter covariance from the twolevel model.
Proof:
We employ the ShermanMorrisonWoodbury formula (equation 17 from the appendix)
to write as
and let
which is the covariance estimate for the s, from equation 5.
Then
Inserting this into equation 8 gives
which becomes equivalent to equation 9 if

For this choice of , the grouplevel parameter estimates can be written as
that is, they become a function of the firstlevel parameter estimates and their associated covariances only.
Note that by simply applying this theorem multiple times, these results extend to any multilevel GLM. For example, one can calculate parameter estimates for groups of groups using a multilevel approach by only keeping track of parameter estimates and associated covariances at each level. Note also that parts of this proof can simply be obtained by characterising the necessary conditions on sufficient statistics for [16].