An increasing number of studies have three levels, in particular: a within session level, a session level, and a subject level. With multiple sessions for multiple subjects it becomes possible to model the between-session variance separately from the between-subject variance, and hence one can benefit from the improvements in sensitivity (due to heterogeneity of variance) this produces.
In section 3.4 we showed that we could infer on the
full two-level model using just the summary result of the first
level without using the data 
. We can use a similar argument to
show that we can infer on a full three-level model using the
summary result of the two-level model (given by
equation 16) without using the data . The
resulting distribution is similar to that in
equation 14. Hence, we similarly assume that
the marginal posterior is a multivariate non-central
t-distribution equivalent to equation 16, and
again we can use the fast posterior approximation or MCMC
approaches to get the distribution parameters.
Higher-level models can be considered using exactly the same argument. This is because after the first-level, outputs and inputs, for subsequent levels can be summarised as a multivariate non-central t-distribution. Hence, the assumption that the marginal distribution in equation 14 is a multivariate non-central t-distribution is integral to the idea of being able to split inference on multiple-level models into inference on the different levels. We shall test the validity of this assumption later.