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### Geometry of Optimal Designs for nonlinear models in statistics

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 β_{1} , β_{2} , β_{3} and and the identifiability condition β_{4} = −β_{1} − β_{2} − β_{3} , the design that takes an equal number of observations on the pairs (1, 2), (2, 3) and
(3, 4) has the following optimality region:

Figure 1: Optimality region for the design with an equal number of observations
on the pairs (1, 2), (2, 3) and (3, 4).

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.

Last Modification: 2018-06-11 - Contact Person: Sebastian Sager -

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