A characterization of saturated designs for factorial experiments

•Algebraic Statistics is used to study saturated fractions of a factorial design.•We exploit connections between DoE and contingency tables analysis.•We define a criterion to check whether a fraction is saturated or not.•We present a connection with optimization.

In this paper we study saturated fractions of factorial designs under the perspective of Algebraic Statistics. Exploiting the identification of a fraction with a binary contingency table, we define a criterion to check whether a fraction is saturated or not with respect to a given model. The proposed criterion is based on combinatorial algebraic objects, namely the circuit basis of the toric ideal associated to the design matrix of the model.