Stefanie Biedermann

Robust and Efficient Designs for the Michaelis-Menten Model

For the Michaelis–Menten model, we determine designs that maximize the minimum of the D-efficiencies over a certain interval for the nonlinear parameter. The best two point designs can be found explicitly, and a characterization is given when these designs are optimal within the class of all designs. In most cases of practical interest, the determined designs are highly efficient and robust with respect to misspecification of the nonlinear parameter. The results are illustrated and applied in an example of a hormone receptor assay.

Optimal Designs for Dose–Response Models With Restricted Design Spaces

In dose–response studies, the dose range is often restricted because of concerns over drug toxicity and/or efficacy. We derive optimal designs for estimating the underlying dose–response curve for a restricted or unrestricted dose range with respect to a broad class of optimality criteria. The underlying curve belongs to a diversified set of link functions suitable for the dose–response studies and having a common canonical form. These include the fundamental binary response models—the logit and the probit, as well as the skewed versions of these models. Our methodology is based on a new geometric interpretation of optimal designs with respect to Kiefers Φp criteria in regression models with two parameters, which is of independent interest. It provides an intuitive illustration of the number and locations of the support points of Φp-optimal designs. Moreover, the geometric results generalize the classical characterization of D-optimal designs by the minimum covering ellipsoid to the class of Kiefers Φp criteria. The results are illustrated through the redesign of a dose ranging trial.

On Optimal Designs for Nonlinear Models: A General and Efficient Algorithm

Finding optimal designs for nonlinear models is challenging in general. Although some recent results allow us to focus on a simple subclass of designs for most problems, deriving a specific optimal design still mainly depends on numerical approaches. There is need for a general and efficient algorithm that is more broadly applicable than the current state-of-the-art methods. We present a new algorithm that can be used to find optimal designs with respect to a broad class of optimality criteria, when the model parameters or functions thereof are of interest, and for both locally optimal and multistage design strategies. We prove convergence to the optimal design, and show in various examples that the new algorithm outperforms the current state-of-the-art algorithms.

Testing linearity of regression models with dependent errors by kernel based methods

In a recent paper Gonzalez Manteiga and Vilar Fernandez (1995) considered the problem of testing linearity of a regression under MA structure of the errors using a weighted L1-distance between a parametric and a nonparametric fit. They established asymptotic normality of the corresponding test statistic under the hypothesis and under local alternatives. In the present paper we extend these results and establish asymptotic normality of the statistic under fixed alternatives. This result is then used to prove that the optimal (with respect to uniform maximization of power) weight function in the test of Gonzalez Manteiga and Vilar Fernandez (1995) is given by the Lebesgue measure independently of the design density_ The paper also discusses several extensions of tests proposed by Azzalini and Bow_ man (1993) Zheng (1996) and Dette (1999) to the case of non-independent errors and compares these methods with the method of Gonzalez Manteiga and Vilar Fernandez (1995). It is demonstrated that among the kernel based methods the approach of the latter authors is the most efficient from an asymptotic point of view.

Maximin optimal designs for the compartmental model

For the compartmental model we determine optimal designs, which are robust against misspecifications of the unknown model parameters. We propose a maximin approach based on D-efficiencies and provide designs that are optimal with respect to the particular choice of various parameter regions.

Minimax Optimal Designs for Nonparametric Regression — A Further Optimality Property of the Uniform Distribution

In the common nonparametric regression model y(i) = g(ti) + a (ti) ei , i=1….,n with i.i.d - noise and nonrepeatable design points ti we consider the problem of choosing an optimal design for the estimation of the regression function g. A minimax approach is adopted which searches for designs minimizing the maximum of the asymptotic integrated mean squared error_ where the maximum is taken over an appropriately bounded class of functions (g,a). The minimax designs are found explicitly and for certain special cases the optimality of the uniform distribution can be established.

Tests in a Case–control Design Including Relatives

We present a new approach to handle dependencies within the general framework of case-control designs, illustrating our approach by a particular application from the field of genetic epidemiology. The method is derived for parent-offspring trios, which will later be relaxed to more general family structures. For applications in genetic epidemiology we consider tests on equality of allele frequencies among cases and controls utilizing well-known risk measures to test for independence of phenotype and genotype at the observed locus. These test statistics are derived as functions of the entries in the associated contingency table containing the numbers of the alleles under consideration in the case and the control group. We find the joint asymptotic distribution of these entries, which enables us to derive critical values for any test constructed on this basis. A simulation study reveals the finite sample behaviour of our test statistics. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..

A Note on Maximin and Bayesian D-optimal Designs in Weighted Polynomial Regression

We consider the problem of finding D-optimal designs for estimating the coefficients in a weighted polynominal regression model with a certain efficiency function depending on two unknown parameters, which models he heteroscedastic error structure. This problem is tackled by adopting a Bayesian and a maximin approach, and optimal designs supported on a minimal number of support points are determined explicitly.

Constrained optimal discriminating designs for Fourier regression models

In this article, the problem of constructing efficient discriminating designs in a Fourier regression model is considered. We propose designs which maximize the efficiency for the estimation of the coefficient corresponding to the highest frequency subject to the constraints that the coefficients of the lower frequencies are estimated with at least some given efficiency. A complete solution is presented using the theory of canonical moments, and for the special case of equal constraints the optimal designs can be found analytically.

Model robust designs for survival trials

The exponential-based proportional hazards model is often assumed in time-to-event experiments but may only approximately hold. Deviations in different neighbourhoods of this model are considered that include other widely used parametric proportional hazards models and the data are assumed to be subject to censoring. Minimax designs are then found explicitly, based on criteria corresponding to classical c- and D-optimality. Analytical characterisations of optimal designs are provided which, unlike optimal designs for related problems in the literature, have finite support and thus avoid the issues of implementing a density-based design in practice. Finally, the proposed designs are compared with the balanced design that is traditionally used in practice, and recommendations for practitioners are given.