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We explore the effect of finite population sampling in design problems withmany variables cross-classified in many ways. In particular, we investigatedesigns where we wish to sample individuals belonging to different groups forwhich the underlying covariance matrices are separable between groups andvariables. We exploit the generalised conditional independence structure of themodel to show how the analysis of the full model can be reduced to aninterpretable series of lower dimensional problems. The types of information wegain by sampling are identified with the orthogonal canonical directions. Wefirst solve a variable problem, which utilises the powerful properties of theadjustment of second-order exchangeable vectors, which has the same qualitativefeatures, represented by the underlying canonical variable directions,irrespective of chosen group, population size or sample size. We then solve aseries of group problems which in a balanced design reduce to the sampling ofsecond-order exchangeable vectors. If the population sizes are finite then thequalitative and quantitative features of each group problem will depend uponthe sampling fractions in each group, mimicking the infinite problem when thesampling fractions in each group are the same.

Bayesian analysis of finite population sampling in multivariate co-exchangeable structures with separable covariance matric