  We explore the effect of finite population sampling in design problems with
many variables cross-classified in many ways. In particular, we investigate
designs where we wish to sample individuals belonging to different groups for
which the underlying covariance matrices are separable between groups and
variables. We exploit the generalised conditional independence structure of the
model to show how the analysis of the full model can be reduced to an
interpretable series of lower dimensional problems. The types of information we
gain by sampling are identified with the orthogonal canonical directions. We
first solve a variable problem, which utilises the powerful properties of the
adjustment of second-order exchangeable vectors, which has the same qualitative
features, represented by the underlying canonical variable directions,
irrespective of chosen group, population size or sample size. We then solve a
series of group problems which in a balanced design reduce to the sampling of
second-order exchangeable vectors. If the population sizes are finite then the
qualitative and quantitative features of each group problem will depend upon
the sampling fractions in each group, mimicking the infinite problem when the
sampling fractions in each group are the same.
