Abstract
Cumulative link models have been widely used for ordered categorical responses. Uniform allocation of experimental units is commonly used in practice, but often suffers from a lack of efficiency. We consider D-optimal designs with ordered categorical responses and cumulative link models. For a predetermined set of design points, we derive the necessary and sufficient conditions for an allocation to be locally D-optimal and develop efficient algorithms for obtaining approximate and exact designs. We prove that the number of support points in a minimally supported design only depends on the number of predictors, which can be much less than the number of parameters in the model. We show that a D-optimal minimally supported allocation in this case is usually not uniform on its support points. In addition, we provide EW D-optimal designs as a highly efficient surrogate to Bayesian D-optimal designs. Both of them can be much more robust than uniform designs.
Highlights
In this paper we determine optimal and efficient designs for factorial experiments with qualitative factors and ordered categorical responses, or ordinal data
In order to check the efficiency of a rounded design, we used the lift-one algorithm to find that the D-optimal approximate design contains 100 positive pi’s out of the 729 distinct experimental settings
We have two surprising findings that are different from the cases under univariate generalized linear models (Yang and Mandal, 2015): (1) the minimum number of experimental settings can be strictly less than the number of parameters, and (2) the allocation of experimental units on the support points of a minimally supported design is usually not uniform
Summary
In this paper we determine optimal and efficient designs for factorial experiments with qualitative factors and ordered categorical responses, or ordinal data. For optimal designs under generalized linear models, there is a growing body of literature (see Khuri et al (2006), Atkinson, Donev and Tobias (2007), Stufken and Yang (2012), and references therein) In this case, it is known that the minimum number of experimental settings required by a nondegenerate Fisher information matrix is d + 1, which equals the number of parameters (Fedorov (1972); Yang and Mandal (2015)). One type of experiment deals with quantitative or continuous factors only Such a design problem includes the identification of a set of design points {xi}i=1,...,m and the corresponding weights {pi}i=1,...,m (see, for example, Atkinson, Donev and Tobias (2007) and Stufken and Yang (2012)).
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