The Lattice of N-Run Orthogonal Arrays

If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the ``expansive replacement'' construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investigate the height and the size of the lattice. It is shown that the height is at most [c(N-1)], where c= 1.4039... and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on ``mixed spreads'', all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.Comment: 28 pages, 4 figure

Optimal Design of Experiments for Nonlinear Response Surface Models

Many chemical and biological experiments involve multiple treatment factors and often it is convenient to fit a nonlinear model in these factors. This nonlinear model can be mechanistic, empirical or a hybrid of the two. Motivated by experiments in chemical engineering, we focus on D-optimal design for multifactor nonlinear response surfaces in general. In order to find and study optimal designs, we first implement conventional point and coordinate exchange algorithms. Next, we develop a novel multiphase optimisation method to construct D-optimal designs with improved properties. The benefits of this method are demonstrated by application to two experiments involving nonlinear regression models. The designs obtained are shown to be considerably more informative than designs obtained using traditional design optimality algorithms

Regression games

The solution of a TU cooperative game can be a distribution of the value of the grand coalition, i.e. it can be a distribution of the payo (utility) all the players together achieve. In a regression model, the evaluation of the explanatory variables can be a distribution of the overall t, i.e. the t of the model every regressor variable is involved. Furthermore, we can take regression models as TU cooperative games where the explanatory (regressor) variables are the players. In this paper we introduce the class of regression games, characterize it and apply the Shapley value to evaluating the explanatory variables in regression models. In order to support our approach we consider Young (1985)'s axiomatization of the Shapley value, and conclude that the Shapley value is a reasonable tool to evaluate the explanatory variables of regression models

Optimal Crossover Designs in a Model With Self and Mixed Carryover Effects

We consider a variant of the usual model for crossover designs with carryover effects. Instead of assuming that the carryover eect of a treatment is the same regardless of the treatment in the next period, the model assumes that the carryover effect of a treatment on itself is different from the carryover effect on other treatments. For the traditional model optimal designs tend to have pairs of consecutive identical treatments; for the model considered here they tend to avoid such pairs. Practitioners have long expressed reservations about designs that exhibit such pairs, resulting in reservations about the traditional model. Our results provide support for these reservations if the carryover effect of a treatment depends also on the treatment in the next period

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Overzicht van de eerste resultaten met betrekking tot het onderzoek in het kader van het Plan van Aanpak

Sampling plans excluding contiguous units

We consider fixed size sampling plans for which the second order inclusion probabilities are zero for pairs of contiguous units and constant for pairs of non-contiguous units. A practical motivation for the use of such plans is pointed out and a statistical condition is identified under which these plans are more efficient than the corresponding simple random sampling plans. Results on the existence and construction of these plans are obtained