Multiple group variances

We can use the framework we have described to work with multiple group variances at any level after the first-level. An example of when this would be useful is when we might expect different between-subject variances for a patient group and a control group. We can easily deal with such multiple group variances if we limit ourselves to design matrices which are ``separable'' with respect to the variance groupings.

We define a sub-design matrix as the part of the design matrix belonging to a group of observations for which we want to have a separate variance group. A design matrix would be ``separable'' with respect to the variance groupings if the sub-design matrices could be inferred upon using separate GLMs to give the same result as inferring on one GLM using the full design matrix.

We define a ``group-regressor'' as that part of an regressor that belongs to a particular group variance:

Variance Group	regressor 1	regressor 2
1	1	0
1	1	0
1	1	0
2	0	1
2	0	1
2	0	1

The group-regressor for regressor 1, and for group 1 is $[1,1,1]^T$. The group-regressor for regressor 1, group 2 is $[0,0,0]^T$.

We can check if our design matrix is ``separable'' by checking that within each regressor only one group-regressor has non-zero values in it. An example of a design matrix which violates this is:

Variance Group	regressor 1	regressor 2
1	1	1
1	1	1
1	1	1
2	1	-1
2	1	-1
2	1	-1

Simulations have shown that if this constraint is not met then the resulting $\beta _g$ vector is not generally multivariate t-distributed. Whilst MCMC could deal with it, this violation prohibits the use of BIDET. This would require the use of longer MCMC chains and would also prohibit carrying the output to higher levels as the output from a level with these properties could not be summarised as a multivariate t-distribution. Hence, we need in practice to ensure that our designs are ``separable'' with respect to the variance groupings.

These ``separable'' multiple group variance designs can then be implemented by inferring on separate GLMs using the fast approximation or MCMC plus BIDET. The results for different variance groups are pooled into one multivariate t-distribution for $\beta _g$. We can then proceed to the contrast stage and ask questions within- or across-variance groupings.