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.     

Alternative modeling techniques for the quantal response data in mixture experiments

Mixture experiments are commonly encountered in many fields including chemical, pharmaceutical and consumer product industries. Due to their wide applications, mixture experiments, a special study of response surface methodology, have been given greater attention in both model building and determination of designs compared with other experimental studies. In this paper, some new approaches are suggested on model building and selection for the analysis of the data in mixture experiments by using a special generalized linear models, logistic regression model, proposed by Chen et al. [7]. Generally, the special mixture models, which do not have a constant term, are highly affected by collinearity in modeling the mixture experiments. For this reason, in order to alleviate the undesired effects of collinearity in the analysis of mixture experiments with logistic regression, a new mixture model is defined with an alternative ratio variable. The deviance analysis table is given for standard mixture polynomial models defined by transformations and special mixture models used as linear predictors. The effects of components on the response in the restricted experimental region are given by using an alternative representation of Cox's direction approach. In addition, odds ratio and the confidence intervals of odds ratio are identified according to the chosen reference and control groups. To compare the suggested models, some model selection criteria, graphical odds ratio and the confidence intervals of the odds ratio are used. The advantage of the suggested approaches is illustrated on tumor incidence data set.     

Optimal designs for non-competitive enzyme inhibition kinetic models

In this paper, we present a new method for determining optimal designs for enzyme inhibition kinetic models, which are used to model the influence of the concentration of a substrate and an inhibition on the velocity of a reaction. The approach uses a nonlinear transformation of the vector of predictors such that the model in the new coordinates is given by an incomplete response surface model. Although there exist no explicit solutions of the optimal design problem for incomplete response surface models so far, the corresponding design problem in the new coordinates is substantially more transparent, such that explicit or numerical solutions can be determined more easily. The designs for the original problem can finally be found by an inverse transformation of the optimal designs determined for the response surface model. We illustrate the method determining explicit solutions for the D-optimal design and for the optimal design problem for estimating the individual coefficients in a non-competitive enzyme inhibition kinetic model.     

Cross-over Designs in the Presence of Higher Order Carry-overs

In cross-over experiments, where different treatments are applied successively to the same experimental unit over a number of time periods, it is often expected that a treatment has a carry-over effect in one or more periods following its period of application. The effect of interaction between the treatments in the successive periods may also affect the response. However, it seems that all systematic studies of the optimality properties of cross-over designs have been done under models where carry-over effects are assumed to persist for only one subsequent period. This paper proposes a model which allows for the possible presence of carry-over effects up to k subsequent periods, together with all the interactions between treatments applied at k + 1 successive periods. This model allows the practitioner to choose k for any experiment according to the requirements of that particular experiment. Under this model, the cross-over designs are studied and the class of optimal designs is obtained. A method of constructing these optimal designs is also given.     

A new model and class of designs for mixture experiments with process variables

Experiments that involve the blending of several components are known as mixture experiments. In some mixture experiments, the response depends not only on the proportion of the mixture components, but also on the processing conditions, A new combined model is proposed which is based on Taylor series approximation and is intended to be a compromise between standard mixture models and standard response surface models. Cost and/or time constraints often limit the size of industrial experiments. With this in mind, we present a new class of designs that will accommodate the fitting of the new combined model.     

ALL-BIAS DESIGNS FOR POLYNOMIAL SPLINE REGRESSION MODELS

Polynomial spline regression models of low degree have proved useful in modeling responses from designed experiments in science and engineering when simple polynomial models are inadequate. Where there is uncertainty in the number and location of the knots, or breakpoints, of the spline, then designs that minimize the systematic errors resulting from model misspecification may be appropriate. This paper gives a method for constructing such all‐bias designs for a single variable spline when the distinct knots in the assumed and true models come from some specified set. A class of designs is defined in terms of the inter‐knot intervals and sufficient conditions are obtained for a design within this class to be all‐bias under linear, quadratic and cubic spline models. An example of the construction of all‐bias designs is given.     

Analysis of a supersaturated design using Entropy Prior Complexity for binary responses via generalized linear models

A supersaturated design is a factorial design in which the number of effects to be estimated is greater than the available number of experimental runs. It is used in many experiments for screening purposes, i.e., for studying a large number of factors and then identifying the active ones. The goal with such a design is to identify just a few of the factors under consideration, that have dominant effects and to do this at minimum cost. While most of the literature on supersaturated designs has focused on the construction of designs and their optimality, the data analysis of such designs remains still at an early stage. In this paper, we incorporate the parameter model complexity into the supersaturated design analysis process, by assuming generalized linear models for a Bernoulli response, for analyzing main effects designs and discovering simultaneously the effects that are significant.     

The performance of subset response surface designs for estimating third order terms

Response surface designs are widely used in industries like chemicals, foods, pharmaceuticals, bioprocessing, agrochemicals, biology, biomedicine, agriculture and medicine. One of the major objectives of these designs is to study the functional relationship between one or more responses and a number of quantitative input factors. However, biological materials have more run to run variation than in many other experiments, leading to the conclusion that smaller response surface designs are inappropriate. Thus designs to be used in these research areas should have greater replication. Gilmour (2006) introduced a wide class of designs called “subset designs” which are useful in situations in which run to run variation is high. These designs allow the experimenter to fit the second order response surface model. However, there are situations in which the second order model representation proves to be inadequate and unrealistic due to the presence of lack of fit caused by third or higher order terms in the true response surface model. In such situations it becomes necessary for the experimenter to estimate these higher order terms. In this study, the properties of subset designs, in the context of the third order response surface model, are explored.     

Optimal design for discrete choice experiments

In this paper we derive locally optimal designs for discrete choice experiments. As in Kanninen (2002) we consider a multinomial logistic model, which contains various qualitative attributes as well as a quantitative one, which may range over a sufficiently large interval. The derived optimal designs improve upon those given in the literature, but have the feature that every choice set contains alternatives, which coincide in all but the quantitative attributes. The multinomial logistic model will then lead to a response behavior, which is apparently unrealistic.     

Optimal designs for contingent response models with application to toxicity–efficacy studies

We describe a general family of contingent response models. These models have ternary outcomes constructed from two Bernoulli outcomes, where one outcome is only observed if the other outcome is positive. This family is represented in a canonical form which yields general results for its Fisher information. A bivariate extreme value distribution illustrates the model and optimal design results. To provide a motivating context, we call the two binary events that compose the contingent responses toxicity and efficacy. Efficacy or lack thereof is assumed only to be observable in the absence of toxicity, resulting in the ternary response (toxicity, efficacy without toxicity, neither efficacy nor toxicity). The rate of toxicity, and the rate of efficacy conditional on no toxicity, are assumed to increase with dose. While optimal designs for contingent response models are numerically found, limiting optimal designs can be expressed in closed forms. In particular, in the family of four parameter bivariate location-scale models we study, as the marginal probability functions of toxicity and no efficacy diverge, limiting D optimal designs are shown to consist of a mixture of the D optimal designs for each failure (toxicity and no efficacy) univariately. Limiting designs are also obtained for the case of equal scale parameters.     

Orthogonally Blocked Mixture Experiments in Ellipsoidal Restricted Regions

In practical situations involving mixtures formed from several ingredients, interest is sometimes centered on the response in an ellipsoidal neighborhood around a standard formulation. We show that standard, orthogonally blocked, response surface designs, defined on a q ? 1 dimensional unit sphere, may be transformed into similarly orthogonally blocked q-ingredient mixture designs defined within an ellipsoid centered at the standard formulation. The method is illustrated using several examples of mixture experiments with three, four, and five ingredients, arranged in two, three, or four orthogonal blocks, obtained by projecting standard central composite designs and Box–Behnken designs into the ellipsoidal mixture region. Rotations of the resulting designs within the ellipsoidal regions are also considered.     

THE CONSTRUCTION OF EQUILEVERAGE DESIGNS FOR MULTIPLE LINEAR REGRESSION

The hat matrix is widely used as a diagnostic tool in linear regression because it contains the leverages which the independent variables exert on the fitted values. In some experiments, cases with high leverage may be avoided by judicious choice of design for the independent variables. A variety of methods for constructing equileverage designs for linear regression are discussed. Such designs remove one of the factors, namely large leverage points, which can lead to nonrobust estimators and tests. In addition, a method is given for combining equileverage designs to test for lack of fit of the linear model.     

On Exact K-optimal Designs Minimizing the Condition Number

A new design criterion based on the condition number of an information matrix is proposed to construct optimal designs for linear models, and the resulting designs are called K-optimal designs. The relationship between exact and asymptotic K-optimal designs is derived. Since it is usually hard to find exact optimal designs analytically, we apply a simulated annealing algorithm to compute K-optimal design points on continuous design spaces. Specific issues are addressed to make the algorithm effective. Through exact designs, we can examine some properties of the K-optimal designs such as symmetry and the number of support points. Examples and results are given for polynomial regression models and linear models for fractional factorial experiments. In addition, K-optimal designs are compared with A-optimal and D-optimal designs for polynomial regression models, showing that K-optimal designs are quite similar to A-optimal designs.     

Mixture designs for constrained components in orthogonal blocks

It is often the case in mixture experiments that some of the ingredients, such as additives or flavourings, are included with proportions constrained to lie in a restricted interval, while the majority of the mixture is made up of a particular ingredient used as a filler. The experimental region in such cases is restricted to a parallelepiped in or near one corner of the full simplex region. In this paper, orthogonally blocked designs with two experimental blends on each edge of the constrained region are considered for mixture experiments with three and four ingredients. The optimal symmetric orthogonally blocked designs within this class are determined and it is shown that even better designs are obtained for the asymmetric situation, in which some experimental blends are taken at the vertices of the experimental region. Some examples are given to show how these ideas may be extended to identify good designs in three and four blocks. Finally, an example is included to illustrate how to overcome the problems of collinearity that sometimes occur when fitting quadratic models to experimental data from mixture experiments in which some of the ingredient proportions are restricted to small values.     

Design optimality in multi-factor generalized linear models in the presence of an unrestricted quantitative factor

In this paper we show that product type designs are optimal in partially heteroscedastic multi-factor linear models. This result is applied to obtain locally D-optimal designs in multi-factor generalized linear models by means of a canonical transformation. As a consequence we can construct optimal designs for direct logistic response as well as for Bradley–Terry type paired comparison experiments.     

E-optimal designs for regression models with quantitative factors— a reasonable choice?

For regression models with quantitative factors it is illustrated that the E-optimal design can be extremely inefficient in the sense that it degenerates to a design which takes all observations at only one point. This phenomenon is caused by the different size of the elements in the covariance matrix of the least-squares estimator for the unknown parameters. For these reasons we propose to replace the E-criterion by a corresponding standardized version. The advantage of this approach is demonstrated for the polynomial regression on a nonnegative interval, where the classical and standardized E-optimal designs can be found explicitly. The described phenomena are not restricted to the E-criterion but appear for nearly all optimality criteria proposed in the literature. Therefore standardization is recommended for optimal experimental design in regression models with quantitative factors. The optimal designs with respect to the new standardized criteria satisfy a similar invariance property as the famous D-optimal designs, which allows an easy calculation of standardized optimal designs on many linearly transformed design spaces.     

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.     

Construction and analysis of edge designs from skew-symmetric supplementary difference sets

The purpose of screening experiments is to identify the dominant variables from a set of many potentially active variables which may affect some characteristic y. Edge designs were recently introduced in the literature and are constructed by using conferences matrices and were proved to be robust. We introduce a new class of edge designs which are constructed from skew-symmetric supplementary difference sets. These designs are particularly useful since they can be applied for experiments with an even number of factors and they may exist for orders where conference matrices do not exist. Using this methodology, examples of new edge designs for 6, 14, 22, 26, 38, 42, 46, 58, and 62 factors are constructed. Of special interest are the new edge designs for studying 22 and 58 factors because edge designs with these parameters have not been constructed in the literature since conference matrices of the corresponding order do not exist. The suggested new edge designs achieve the same model-robustness as the traditional edge designs. We also suggest the use of a mirror edge method as a test for the linearity of the true underlying model. We give the details of the methodology and provide some illustrating examples for this new approach. We also show that the new designs have good D-efficiencies when applied to first order models.     

Scaled density models for binary response data and their D- and Ds-optimal designs

We introduce scaled density models for binary response data which can be much more reasonable than the traditional binary response models for particular types of binary response data. We show the maximum-likelihood estimates for the new models and it seems that the model works well with some sets of data. We also considered optimum designs for parameter estimation for the models and found that the D- and D_s-optimum designs are independent of parameters corresponding to the linear function of dose level, but the optimum designs are simple functions of a scale parameter only.     

Adaptive sequential design for regression on multi-resolution bases

This work investigates the problem of construction of designs for estimation and discrimination between competing linear models. In our framework, the unknown signal is observed with the addition of a noise and only a few evaluations of the noisy signal are available. The model selection is performed in a multi-resolution setting. In this setting, the locations of discrete sequential D and A designs are precisely constraint in a small number of explicit points. Hence, an efficient stochastic algorithm can be constructed that alternately improves the design and the model. Several numerical experiments illustrate the efficiency of our method for regression. One can also use this algorithm as a preliminary step to build response surfaces for sensitivity analysis.