Testing Hypotheses in Randomized Factorial Experiments

lem in performing ANOVA tests in the non-randomized designs. The problem is due to the effects of the nuisance parameters, which may be excessive and yet not under our control. To be more specific, as will be shown in the sequel, the conditional distribution of the F-like ratio test statistics, given the fractional replicate chosen, is like that of a double non-central F[vi , V2; X, X*1. Here, vi and V2 are the appropriate degrees of freedom, X and X* the parameters of non-centrality, being functions of the fractional replicate chosen, and of the vector of unknown parameters. Even under the null hypotheses, that the pre-assigned parameters are zero, X and X* might be quite large due to the effects of the nuisance parameters. In the classical fractional replication model the assumptions imply that, under the null hypotheses,