This paper generalizes the results of Paik and Federer (1970), Ehrenfeld and Zacks (1961) and Zacks (1963, 1964) regarding invariance and randomization in fractional replication. It is shown that (i) the characteristic roots of the information matrix of a design in the general factorial relative to an admissible vector of effects remain invariant under a permutation of levels; (ii) the unbiased estimation of a linear function of an admissible vector of effects can be obtained under equal probability randomization. In addition some applications of the results are indicated.
"On Invariance and Randomization in Fractional Replication." Ann. Statist. 4 (2) 423 - 430, March, 1976. https://doi.org/10.1214/aos/1176343421