We are searching for optimality regions of experimental designs for statistical models, especially for generalized linear models with Poisson or logistic response. These regions are described by systems of polynomial inequalities in the parameter space, which means that they are nothing else than semialgebraic sets. Hence we can use algebraic geometry to study the properties of these optimality regions. For example, in the Bradley-Terry paired comparison model, which is a statistical model for comparisons of alternatives depending on logistic parameters, we are interested in the optimality regions of so called saturated designs. E.g, for a model with 4 alternatives denoted by (1, 2, 3, 4) with parameters β1 , β2 , β3 and and the identifiability condition β4 = −β1 − β2 − β3 , the design that takes an equal number of observations on the pairs (1, 2), (2, 3) and (3, 4) has the following optimality region:
The optimality regions for the Bradley-Terry paired comparison model with more than 4 alternatives and more than two factors are still unknown. We hope to find optimality regions in higher dimensions through the combined forces of optimal design theory and algebraic statistics.