We have shown how multi-level hierarchical GLM inference can be split into different levels with the summary statistics of a multivariate non-central t-distribution being passed between the levels. This was achieved by formulating the model in a fully Bayesian framework and using reference analysis to drive our crucial choice of priors (sections 3.3 and 3.4). Using this framework we have proposed two approaches to inferring at the top-level. A fast approximation to the marginal posterior, and a slower approach utilising Markov Chain Monte Carlo (MCMC) followed by a multivariate non-central t-distribution fit to the MCMC chains. These inference approaches are applicable whether we are attempting to infer using the all-in-one approach or the summary statistic split-model approach. We have validated the crucial assumption of the marginal distribution of the GLM regressions parameters being a multivariate non-central t-distribution at levels higher than the first using artificial data. The artificial data also demonstrates the difference between a standard OLS approach and the approach proposed in this paper. We have also shown results on FMRI data.