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

Complete enumeration of two-Level orthogonal arrays of strength dd with d+2d+2 constraints

Enumerating nonisomorphic orthogonal arrays is an important, yet very difficult, problem. Although orthogonal arrays with a specified set of parameters have been enumerated in a number of cases, general results are extremely rare. In this paper, we provide a complete solution to enumerating nonisomorphic two-level orthogonal arrays of strength $d$ with $d+2$ constraints for any $d$ and any run size $n=\lambda2^d$. Our results not only give the number of nonisomorphic orthogonal arrays for given $d$ and $n$, but also provide a systematic way of explicitly constructing these arrays. Our approach to the problem is to make use of the recently developed theory of $J$-characteristics for fractional factorial designs. Besides the general theoretical results, the paper presents some results from applications of the theory to orthogonal arrays of strength two, three and four.Comment: Published at http://dx.doi.org/10.1214/009053606000001325 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Saturated locally optimal designs under differentiable optimality criteria

We develop general theory for finding locally optimal designs in a class of single-covariate models under any differentiable optimality criterion. Yang and Stufken [Ann. Statist. 40 (2012) 1665-1681] and Dette and Schorning [Ann. Statist. 41 (2013) 1260-1267] gave complete class results for optimal designs under such models. Based on their results, saturated optimal designs exist; however, how to find such designs has not been addressed. We develop tools to find saturated optimal designs, and also prove their uniqueness under mild conditions.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1263 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Support points of locally optimal designs for nonlinear models with two parameters

We propose a new approach for identifying the support points of a locally optimal design when the model is a nonlinear model. In contrast to the commonly used geometric approach, we use an approach based on algebraic tools. Considerations are restricted to models with two parameters, and the general results are applied to often used special cases, including logistic, probit, double exponential and double reciprocal models for binary data, a loglinear Poisson regression model for count data, and the Michaelis--Menten model. The approach, which is also of value for multi-stage experiments, works both with constrained and unconstrained design regions and is relatively easy to implement.Comment: Published in at http://dx.doi.org/10.1214/07-AOS560 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Information-based Optimal Subdata Selection for Clusterwise Linear Regression

Mixture-of-Experts models are commonly used when there exist distinct clusters with different relationships between the independent and dependent variables. Fitting such models for large datasets, however, is computationally virtually impossible. An attractive alternative is to use a subdata selected by ``maximizing" the Fisher information matrix. A major challenge is that no closed-form expression for the Fisher information matrix is available for such models. Focusing on clusterwise linear regression models, a subclass of MoE models, we develop a framework that overcomes this challenge. We prove that the proposed subdata selection approach is asymptotically optimal, i.e., no other method is statistically more efficient than the proposed one when the full data size is large.Comment: 23 pages, 5 figure

ON OPTIMAL AND HIGHLY EFFICIENT BLOCK DESIGNS FOR COMPARING TEST TREATMENTS WITH A CONTROL.

ON OPTIMAL AND HIGHLY EFFICIENT BLOCK DESIGNS FOR COMPARING TEST TREATMENTS WITH A CONTROL