Methods in Algebraic Statistics for the Design of Experiments

We present a brief review of classical experimental design in the spirit of algebraic statistics. Notions of identifiability, aliasing and estimability of linear parametric functions, and confounding are expressed in relation to a set of polynomials identified by the design, called the design ideal. An effort has been made to indicate the classical linear algebra counterpart of the objects of interest in the polynomial space, and to indicate how the algebraic statistic approach generalizes the classical theory. In the second part of this chapter we address new questions: a seeming limitation of the algebraic approach is discussed and resolved using the ideas of minimal and maximal fan designs, again generalizing classical notions; an algorithm is provided to switch between two major representations of a design, the first using Gröbner bases and the second using indicator functions. Finally, all the theory in the chapter is applied and extended to the class of mixture designs, which presents a challenging structure and questions.