A spectral form of the dispersion model in designs with arbitrarily unequal block sizes

Under a usual additive mixed model, a spectral decomposition is given to each observed vector Y from a class of block designs with arbitrarily unequal block sizes. As a result, an analysis of variance, in the sense defined by Graybill and Hultquist (1961) and Speed (1987), exist for such a design. The intrablock anova is discussed. Furthermore based on the dispersion structure, it is seen that the unbalancedness forces some useful information beyond the intrablock analysis, and even interblock comparisons. Let a group be a set of observations from the blocks of same sizes, a further decomposition of the spectral form provides an intergroup comparison which completes the anova of such designs.