Optimal designs for exponential regression

This paper is concerned with the optimal design problem for the particular case of non-linear parametrisation:the parameters to be estimated are included in exponents.Some properties of locally optimal designs as functions of estimated parameters are investigated and a table of such designs in given.We consider also designs to be optimal in the sense of minimax approach.     

Optimal design for the Bradley–Terry paired comparison model

Paired comparisons are a popular tool for questionnaires in psychological marketing research. The quality of the statistical analysis of the responses heavily depends on the design, i.e. the choice of the alternatives in the comparisons. In this paper we show that the structure of locally optimal designs changes substantially with the parameters in the underlying utility. This fact is illustrated by elementary examples, where the optimal designs can be completely characterized. As an alternative maximin efficient designs are proposed which perform well for all parameter settings. Research supported by grant Ho 1286 of the German Research Council (Deutsche Forschungsgemeinschaft).     

Efficient experimental designs for sigmoidal growth models

For the Weibull- and Richards-regression model robust designs are determined by maximizing a minimum of D  - or _D1$D_1$-efficiencies, taken over a certain range of the non-linear parameters. It is demonstrated that the derived designs yield a satisfactory solution of the optimal design problem for this type of model in the sense that these designs are efficient and robust with respect to misspecification of the unknown parameters. Moreover, the designs can also be used for testing the postulated form of the regression model against a simplified sub-model.     

Robust designs for linear mixed effects models

Summary.  In health sciences, medicine and social sciences linear mixed effects models are often used to analyse time-structured data. The search for optimal designs for these models is often hampered by two problems. The first problem is that these designs are only locally optimal. The second problem is that an optimal design for one model may not be optimal for other models. In this paper the maximin principle is adopted to handle both problems, simultaneously. The maximin criterion is formulated by means of a relative efficiency measure, which gives an indication of how much efficiency is lost when the uncertainty about the models over a prior domain of parameters is taken into account. The procedure is illustrated by means of three growth studies. Results are presented for a vocabulary growth study from education, a bone gain study from medical research and an epidemiological decline in height study. It is shown that, for the mixed effects polynomial models that are applied to these studies, the maximin designs remain highly efficient for different sets of models and combinations of parameter values.     

Optimal designs for generalized linear models

Summary This paper solves some D-optimal design problems for certain Generalized Linear Models where the mean depends on two parameters and two explanatory variables. In all of the cases considered the support point of the optimal designs are found to be independent of the unknown parameters. While in some cases the optimal design measures are given by two points with equal weights, in others the support is given by three point with weights depending on the unknown parameters, hence the designs are locally optimal in general. Empirical results on the efficiency of the locally optimal designs are also given. Some of the designs found can also be used for planning D-optimal experiments for the normal linear model, where the mean must be positive. This research was carried out in part at University College, London as an M.Sc. project. Thanks are due to Prof. I. Ford (University of Glasgow) and Prof. A. Giovagnoli (University of Perugia) for their valuable suggestions and critical observations.     

On d-optimal designs for binary data

We use the criterion of D-optimality of the Fisher information matrix to derive optimal vectors for binary data. Some concepts of totally positive functions and Polya functions of order II are used to derive properties of the determinant of the Fisher information matrix arising in quantal response bioassay and attribute life testing models. As is often the case in non-linear models the D-optimal vectors are functions of the unknown parameters. By using the criterion of D-optimality, general optimal vectors are characterized which could be used for constructing Bayesian or locally D-optimal designs.     

An investigation of a quadratic design criterion for estimation of the hemodynamic response function

Optimal experimental design for estimation of the hemodynamic response function (HRF) is investigated using a nonlinear model with a quadratic mean squared error design criterion. This criterion is used, along with a genetic algorithm, to select locally optimal designs that are shown to be, in most cases, more efficient than designs selected with the more commonly used linear expansion criterion. These designs are also shown to result in lower overall asymptotic estimator variance and bias. The investigation focuses on a single stimulus type, but the criterion can also be used with multiple stimulus types.     

Optimum Mixture Design for the Detection of Synergistic Effects

In mixture experiments, optimal designs for the estimation of parameters, both linear and non-linear, have been discussed by several authors. Optimal designs for the estimation of a subset of parameters have also been investigated. However, designs for testing the effects of certain factors and interactions have been studied only in the context of response surface models. In this article, we attempt to find the optimum design for testing the presence of synergistic effects in a mixture model. The classical F-test has been considered and the optimum design has been obtained so as to maximize the power of the test. It is observed that the barycenters are necessarily the support points of the trace-optimal design.     

Sample reuse simulation in optimal design for Tmax in pharmacokinetic experiments

T _max and C _max are important pharmacokinetic parameters in drug development processes. Often a nonparametric procedure is needed to estimate them when model independence is required. This paper proposes a simulation-based optimal design procedure for finding optimal sampling times for nonparametric estimates of T _max and C _max for each subject, assuming that the drug concentration follows a non-linear mixed model. The main difficulty of using standard optimal design procedures is that the property of the nonparametric estimate is very complicated. This procedure uses a sample reuse simulation to calculate the design criterion, which is an integral of multiple dimension, so that effective optimization procedures such as Newton-type procedures can be used directly to find optimal designs. This procedure is used to construct optimal designs for an open one-compartment model. An approximation based on the Taylor expansion is also derived and showed results that were consistent with those based on the sample reuse simulation.     

On representing maximin efficient designs by Taylor series

This paper considers exponential and rational regression models that are nonlinear in some parameters. Recently, locally D-optimal designs for such models were investigated in [Melas, V. B., 2005. On the functional approach to optimal designs for nonlinear models. J. Statist. Plann. Inference 132, 93–116] based upon a functional approach. In this article a similar method is applied to construct maximin efficient D-optimal designs. This approach allows one to represent the support points of the designs by Taylor series, which gives us the opportunity to construct the designs by hand using tables of the coefficients of the series. Such tables are provided here for models with two nonlinear parameters. Furthermore, the recurrent formulas for constructing the tables for arbitrary numbers of parameters are introduced.     

Locally optimal designs for some dose–response models with continuous endpoints

We consider the problem of constructing static (or non sequential), approximate optimal designs for a class of dose–response models with continuous outcomes. We obtain conditions for a design being D-optimal or c-optimal. The designs are locally optimal in that they depend on the model parameters. The efficiency studies show that these designs have high efficiency when the mis-specification of the initial values of model parameters is not severe. A case study indicates that using an optimal design may result in a significant saving of resources.     

Non-linear design problem in a chemical kinetic model with non-constant error variance

For a locally optimum non-linear design problem for a chemical kinetic model, we investigate the influence of the dispersion structure of the random observation errors on the design and its efficiency. We find that there are two kinds of design determined by the model parameters and the error variance function: “interior” designs, and “boundary” designs that depend also on the design range. We give an exact criterion for determining which kind of design will arise and we illustrate the qualitative difference between the two kinds of design in terms of the design locus and the equivalence theorem. We tabulate quantitative details of the designs for a range of parameter values.     

Generalized case–cohort sampling

A class of cohort sampling designs, including nested case–control, case–cohort and classical case–control designs involving survival data, is studied through a unified approach using Cox's proportional hazards model. By finding an optimal sample reuse method via local averaging, a closed form estimating function is obtained, leading directly to the estimators of the regression parameters that are relatively easy to compute and are more efficient than some commonly used estimators in case–cohort and nested case–control studies. A semiparametric efficient estimator can also be found with some further computation. In addition, the class of sampling designs in this study provides a variety of sampling options and relaxes the restrictions of sampling schemes that are currently available.     

A new characterization of Elfving's method for high dimensional computation

We give a new characterization of Elfving's (1952) method for computing c-optimal designs in k dimensions which gives explicit formulae for the k unknown optimal weights and k unknown signs in Elfving's characterization. This eliminates the need to search over these parameters to compute c-optimal designs, and thus reduces the computational burden from solving a family of optimization problems to solving a single optimization problem for the optimal finite support set. We give two illustrative examples: a high dimensional polynomial regression model and a logistic regression model, the latter showing that the method can be used for locally optimal designs in nonlinear models as well.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

Optimal designs for the power logistic model

We investigate non-sequential designs for estimating model parameters in a power logistic model when the power is assumed to be approximately known and only the ranges for the other two parameters are available. The sensitivity of these designs to nominal values of all the three parameters are studied and our proposed optimal designs are shown to be reasonably robust under moderate deviation from the assumed model. An application to a toxicity experiment involving adult beetles is discussed, including the benefits of using an optimal design.     

A candidate-set-free algorithm for generating D-optimal split-plot designs

Summary.  We introduce a new method for generating optimal split-plot designs. These designs are optimal in the sense that they are efficient for estimating the fixed effects of the statistical model that is appropriate given the split-plot design structure. One advantage of the method is that it does not require the prior specification of a candidate set. This makes the production of split-plot designs computationally feasible in situations where the candidate set is too large to be tractable. The method allows for flexible choice of the sample size and supports inclusion of both continuous and categorical factors. The model can be any linear regression model and may include arbitrary polynomial terms in the continuous factors and interaction terms of any order. We demonstrate the usefulness of this flexibility with a 100-run polypropylene experiment involving 11 factors where we found a design that is substantially more efficient than designs that are produced by using other approaches.     

Optimal discrimination designs for exponential regression models

We investigate optimal designs for discriminating between exponential regression models of different complexity, which are widely used in the biological sciences; see, e.g., Landaw [1995. Robust sampling designs for compartmental models under large prior eigenvalue uncertainties. Math. Comput. Biomed. Appl. 181–187] or Gibaldi and Perrier [1982. Pharmacokinetics. Marcel Dekker, New York]. We discuss different approaches for the construction of appropriate optimality criteria, and find sharper upper bounds on the number of support points of locally optimal discrimination designs than those given by Caratheodory's Theorem. These results greatly facilitate the numerical construction of optimal designs. Various examples of optimal designs are then presented and compared to different other designs. Moreover, to protect the experiment against misspecifications of the nonlinear model parameters, we adapt the design criteria such that the resulting designs are robust with respect to such misspecifications and, again, provide several examples, which demonstrate the advantages of our approach.     

New adaptive designs for delayed response models

Adaptive designs are effective mechanisms for flexibly allocating experimental resources. In clinical trials particularly, such designs allow researchers to balance short- and long-term goals. Unfortunately, fully sequential strategies require outcomes from all previous allocations prior to the next allocation. This can prolong an experiment unduly. As a result, we seek designs for models that specifically incorporate delays.We utilize a delay model in which patients arrive according to a Poisson process and their response times are exponential. We examine three designs with an eye towards minimizing patient losses: a delayed two-armed bandit rule which is optimal for the model and objective of interest; a newly proposed hyperopic rule; and a randomized play-the-winner rule. The results show that, except when the delay rate is several orders of magnitude different than the patient arrival rate, the delayed response bandit is nearly as efficient as the immediate response bandit. The delayed hyperopic design also performs extremely well throughout the range of delays, despite the fact that the rate of delay is not one of its design parameters. The delayed randomized play-the-winner rule is far less efficient than either of the other methods.