Model Robust Designs Research Articles

Model Robust Optimal Designs for Kronecker Model for Mixture Experiments

In comparison to Scheffè’s canonical polynomial models (S-models), the Kronecker models (K-models) for mixture experiments are symmetric, compact in notation, and based on the Kronecker algebra of vectors and matrices. Further, there is a corresponding transition from S-models to K-models in the form of model re-parameterization. In the literature, it has been recommended to use second-degree K-models in practice compared to the widely used second-degree S-models especially when the moment matrix is of an ill-conditioning type. The motivation of the present article is to discriminate between K-models and S-models in terms of the model-robust D- and A-optimality criteria. These optimality criteria are discussed when there is uncertainty in selecting an appropriate model out of two rival models for a mixture experiment. International Journal of Statistical Sciences, Vol.24(1), March, 2024, pp 31-48

Optimizing the design of a pharmacokinetic trial to evaluate the dosing scheme of a novel tuberculosis drug in children living with or without HIV.

Pharmacokinetic (PK) studies in children are usually small and have ethical constraints due to the medical complexities of drawing blood in this special population. Often, population PK models for the drug(s) of interest are available in adults, and these models can be extended to incorporate the expected deviations seen in children. As a consequence, there is increasing interest in the use of optimal design methodology to design PK sampling schemes in children that maximize information using a small sample size and limited number of sampling times per dosing period. As a case study, we use the novel tuberculosis drug delamanid, and show how applications of optimal design methodology can result in highly efficient and model-robust designs in children for estimating PK parameters using a limited number of sampling measurements. Using developed population PK models based on available data from adults living with and without HIV, and limited data on children without HIV, competing designs for children living with HIV were derived and assessed based on robustness to model uncertainty.

Model misspecification and robust analysis foroutcome-dependent sampling designs under generalizedlinear models.

Outcome-dependent sampling (ODS) is a commonly used class of sampling designs to increase estimation efficiency in settings where response information (and possibly adjuster covariates) is available, but the exposure is expensive and/or cumbersome to collect. We focus on ODS within the context of a two-phase study, where in Phase One the response and adjuster covariate information is collected on a large cohort that is representative of the target population, but the expensive exposure variable is not yet measured. In Phase Two, using response information from Phase One, we selectively oversample a subset of informative subjects in whom we collect expensive exposure information. Importantly, the Phase Two sample is no longer representative, and we must use ascertainment-correcting analysis procedures for valid inferences. In this paper, we focus on likelihood-based analysis procedures, particularly a conditional-likelihood approach and a full-likelihood approach. Whereas the full-likelihood retains incomplete Phase One data for subjects not selected into Phase Two, the conditional-likelihood explicitly conditions on Phase Two sample selection (ie,it is a "complete case" analysis procedure). These designs and analysis procedures are typically implemented assuming a known, parametric model for the response distribution. However, in this paper, we approach analyses implementing a novel semi-parametric extension to generalized linear models (SPGLM) to develop likelihood-based procedures with improved robustness to misspecification of distributional assumptions. We specifically focus on the common setting where standard GLM distributional assumptions are not satisfied (eg,misspecified mean/variance relationship). We aim to provide practical design guidance and flexible tools for practitioners in these settings.

Model-robust design of mixture experiments

Optimal designs are often used for constrained mixture experiments because of the irregular design spaces. For these experiments, the number of blends needed to fit standard linear models may be too large when considering second- or third-order terms. We present a computationally-tractable algorithm for generating model-robust mixture designs that exploits anticipated effect sparsity by using a set of models defined by a user-specified number of higher-order terms. We compared the model-robust designs with Bayesian-optimal designs, and the model-robust designs show an improved ability to either estimate realistic models or make predictions for mixture experiments.

Model-robust G-optimal Designs in the Presence of Block Effects

Model-robust G-optimal Designs in the Presence of Block Effects

Constructing model robust mixture designs via weighted G-optimality criterion

Constructing model robust mixture designs via weighted G-optimality criterion

Optimal designs for spline wavelet regression models

In this article we investigate the optimal design problem for some wavelet regression models. Wavelets are very flexible in modeling complex relations, and optimal designs are appealing as a means of increasing the experimental precision. In contrast to the designs for the Haar wavelet regression model (Herzberg and Traves 1994; Oyet and Wiens 2000), the I-optimal designs we construct are different from the D-optimal designs. We also obtain c-optimal designs. Optimal (D- and I-) quadratic spline wavelet designs are constructed, both analytically and numerically. A case study shows that a significant saving of resources may be realized by employing an optimal design. We also construct model robust designs, to address response misspecification arising from fitting an incomplete set of wavelets.

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.

Model-robust designs for split-plot experiments

Split-plot experiments are appropriate when some factors are more difficult and/or expensive to change than others. They require two levels of randomization resulting in a non-independent error structure. The design of such experiments has garnered much recent attention, including work on exact D-optimal split-plot designs. However, many of these procedures rely on the a priori assumption that the form of the regression function is known. We relax this assumption by allowing a set of model forms to be specified, and use a scaled product criterion along with an exchange algorithm to produce designs that account for all models in the set. We include also a generalization which allows weights to be assigned to each model, though they appear to have only a slight effect. We present two examples from the literature, and compare the scaled product designs with designs optimal for a single model. We also discuss a maximin alternative.

Exchange Algorithms for Constructing Model-Robust Experimental Designs

Optimal experimental design procedures, utilizing criteria such as D-optimality, are useful for producing designs for quantitative responses, often under nonstandard conditions such as constrained design spaces. However, these methods require a priori knowledge of the exact form of the response function, an often unrealistic assumption. Model-robust designs are those that, from our perspective, are efficient with respect to a set of possible models. In this paper, we develop a model-robust technique motivated by a connection to multiresponse D-optimal design. This link spawns a generalization of the modified Fedorov exchange algorithm, which is then used to construct exact model-robust designs. We also study the effectiveness of designs robust for a small set of models compared with designs that account for much larger sets. We give several examples and compare our designs with two model-robust procedures in the literature.

Computing optimal designs of multiresponse experiments reduces to second-order cone programming

Elfving's theorem is a major result in the theory of optimal experimental design, which gives a geometrical characterization of c-optimality. In this paper, we extend this theorem to the case of multiresponse experiments, and we show that when the number of experiments is finite, the c‐,A‐,T‐ and D-optimal design of multiresponse experiments can be computed by second-order cone programming (SOCP). Moreover, the present SOCP approach can deal with design problems in which the variable is subject to several linear constraints.We give two proofs of this generalization of Elfving's theorem. One is based on Lagrangian dualization techniques and relies on the fact that the semidefinite programming (SDP) formulation of the multiresponse c-optimal design always has a solution which is a matrix of rank 1. Therefore, the complexity of this problem fades.We also investigate a model robust generalization of c-optimality, for which an Elfving-type theorem was established by Dette (1993). We show with the same Lagrangian approach that these model robust designs can be computed efficiently by minimizing a geometric mean under some norm constraints. Moreover, we show that the optimality conditions of this geometric programming problem yield an extension of Dette's theorem to the case of multiresponse experiments.When the goal is to identify a small number of linear functions of the unknown parameter (typically for c-optimality), we show by numerical examples that the present approach can be between 10 and 1000 times faster than the classic, state-of-the-art algorithms.

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 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.

Model discrimination—another perspective on model-robust designs

Recent progress in model-robust designs has focused on maximizing estimation capacities. However, for a given design, two competing models may be both estimable and yet difficult or impossible to discriminate in the model selection procedure. In this paper, we propose several criteria for gauging the capability of a design for model discrimination. The criteria are then used to evaluate a class of 18-run orthogonal designs in terms of their model-discriminating capabilities. We demonstrate that designs having the same estimation capacity may differ considerably with respect to model-discrimination capabilities. The best designs according to the proposed model-discrimination criteria are obtained and tabulated for practical use.

Model-robust designs in multiresponse situations

The multiresponse model E(y α| x )=∑ l=1 p α θ αlf αl( x )+h α( x ) , α=1,…, r, is considered, where h α( x ) is an unknown bias or contamination function from some class H α with a probability measure. Optimal designs are studied in terms of generalized least squares estimation and the average expected quadratic loss. The performance of the uniform design is also explored.

A Class of Optimal Robust Parameter Designs

In robust designs, the control-by-noise interactions are usually considered to be more important than other two-factor interactions. We extend the work on model-robust factorial designs of Li and Nachtsheim (2000) to robust designs, where not all factors are treated equally. A new criterion is proposed to maximize a design's ability to estimate models with at least one control-by-noise interaction. Optimal designs are chosen from the following three groups: regular orthogonal arrays (OAs), non-regular OAs, and balanced designs. Designs with economic run-sizes are constructed algorithmically and are tabulated for practical use.

The design of model-robust experiments

Model-robust designs are generated using a combination of algorithms based on several optimality criteria. The content of the candidate list and the sequencing of the point-selection algorithms has a major impact on overall design quality. When properly constructed, model-robust designs enable nonlinear variable effects to be visualized and estimated due to the sampling of the interior of the design space accomplished by these designs. There are also many cases in which the assumption of a normally and independently distributed (NID) experimental error is not valid. Model-robust designs are also shown to support these experiment circumstances, among others. Additionally, the model independence of these designs aids outlier and residuals diagnostics, and response data transformation analysis. Thus the many benefits of these designs more than offset the cost of the small number of additional runs required by these designs. Copyright © 2000 John Wiley & Sons, Ltd.

The Theory of Canonical Moments with Applications in Statistics, Probability and Analysis

Canonical Moments. Orthogonal Polynomials. Continued Fractions and the Stieltjes Transform. Special Sequences of Canonical Moments. Canonical Moments and Optimal DesignFirst Applications. Discrimination and Model Robust Designs. Applications in Approximation Theory. Canonical Moments and Random Walks. The Circle and Trigonometric Functions. Further Applications. Bibliography. Indexes.

Optimal discrimination designs for multifactor experiments

In this paper efficient designs are determined when Anderson's procedure is applied in order to identify the degree of a multivariate polynomial regression model. It is shown that the optimal designs are very closely related to model robust designs which maximize a weighted p-mean of D-efficiencies. As a consequence we obtain designs with high efficiency for model discrimination and for the statistical analysis in the identified model.

A Geometric Approach to Optimal Design for One-Parameter Non-Linear Models

SUMMARY A geometric framework for constructing optimal Bayesian designs and maximin designs for non-linear models with a single unknown parameter and a prior distribution on that parameter, which is restricted in that it comprises exactly two points of support, is presented. The approach is illustrated by means of selected examples involving logistic regression and the simple exponential model, and its applicability to the construction of optimal designs for models with uncontrolled variation and to model robust designs is also demonstrated. In addition, the method is shown to provide some valuable insights into the general properties of optimal Bayesian designs for non-linear models.

Addendum to the Paper, Bayesian D-Optimal and Model Robust Designs in Linear Regression Models

In the recent paper of Dette (1993) there are two figures missing which will be given in this addendum.