Optimal designs for mixed models in experiments based on ordered units

We consider experiments for comparing treatments using units that are ordered linearly over time or space within blocks. In addition to the block effect, we assume that a trend effect influences the response. The latter is modeled as a smooth component plus a random term that captures departures from the smooth trend. The model is flexible enough to cover a variety of situations; for instance, most of the effects may be either random or fixed. The information matrix for a design will be a function of several variance parameters. While data will shed light on the values of these parameters, at the design stage, they are unlikely to be known, so we suggest a maximin approach, in which a minimal information matrix is maximized. We derive maximin universally optimal designs and study their robustness. These designs are based on semibalanced arrays. Special cases correspond to results available in the literature.Comment: Published in at http://dx.doi.org/10.1214/07-AOS518 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal Designs for 2^k Factorial Experiments with Binary Response

We consider the problem of obtaining D-optimal designs for factorial experiments with a binary response and $k$ qualitative factors each at two levels. We obtain a characterization for a design to be locally D-optimal. Based on this characterization, we develop efficient numerical techniques to search for locally D-optimal designs. Using prior distributions on the parameters, we investigate EW D-optimal designs, which are designs that maximize the determinant of the expected information matrix. It turns out that these designs can be obtained very easily using our algorithm for locally D-optimal designs and are very good surrogates for Bayes D-optimal designs. We also investigate the properties of fractional factorial designs and study the robustness with respect to the assumed parameter values of locally D-optimal designs.Comment: 41 pages, 3 figures, 8 table

Optimal designs for two-level factorial experiments with binary response, Statistica Sinica 22

Abstract: We consider the problem of obtaining locally D-optimal designs for factorial experiments with qualitative factors at two levels each and with binary response. For the 2 2 factorial experiment with main-effects model, we obtain optimal designs analytically in special cases and demonstrate how to obtain a solution in the general case using cylindrical algebraic decomposition. The optimal designs are shown to be robust to the choice of the assumed values of the prior, and when there is no basis to make an informed choice of the assumed values we recommend the use of the uniform design that assigns equal number of observations to each of the four points. For the general 2 k case we show that the uniform design has a maximin property

Optimal designs for two-level factorial experiments with binary response, Statistica Sinica 22

Abstract: We consider the problem of obtaining locally D-optimal designs for factorial experiments with qualitative factors at two levels each and with binary response. For the 2 2 factorial experiment with main-effects model, we obtain optimal designs analytically in special cases and demonstrate how to obtain a solution in the general case using cylindrical algebraic decomposition. The optimal designs are shown to be robust to the choice of the assumed values of the prior, and when there is no basis to make an informed choice of the assumed values we recommend the use of the uniform design that assigns equal number of observations to each of the four points. For the general 2 k case we show that the uniform design has a maximin property