Equally spaced design points in polynomial regression: A comparison of systematic sampling methods with the optimal design of experiments

In some situations an experimenter may desire to have equally spaced design points. Three methods of obtaining such points on the interval [—1,1]—namely systematic random sampling, centrally located systematic sampling, and a purposive systematic sampling method which includes the endpoints - 1 and 1 as two of the design points-are evaluated under the D-optimal and G-optimal criteria. These methods are also compared to the optimal designs in polynomial regression and to the limiting designs of Kiefer and Studden (1976).     

Optimal allocation of time points for the random effects model

In this article the problem of the optimal selection and allocation of time points in repeated measures experiments is considered. D‐ optimal designs for linear regression models with a random intercept and first order auto‐regressive serial correlations are computed numerically and compared with designs having equally spaced time points. When the order of the polynomial is known and the serial correlations are not too small, the comparison shows that for any fixed number of repeated measures, a design with equally spaced time points is almost as efficient as the D‐ optimal design. When, however, there is no prior knowledge about the order of the underlying polynomial, the best choice in terms of efficiency is a D‐ optimal design for the highest possible relevant order of the polynomial. A design with equally‐spaced time points is the second best choice     

Sequential design for nonparametric inference

The performance of nonparametric function estimates often depends on the choice of design points. Based on the mean integrated squared error criterion, we propose a sequential design procedure that updates the model knowledge and optimal design density sequentially. The methodology is developed under a general framework covering a wide range of nonparametric inference problems, such as conditional mean and variance functions, the conditional distribution function, the conditional quantile function in quantile regression, functional coefficients in varying coefficient models and semiparametric inferences. Based on our empirical studies, nonparametric inference based on the proposed sequential design is more efficient than the uniform design and its performance is close to the true but unknown optimal design. The Canadian Journal of Statistics 40: 362–377; 2012 © 2012 Statistical Society of Canada     

D s -optimal designs for Kozak's tree taper model

In this work, we study D _s -optimal design for Kozak's tree taper model. The approximate D _s -optimal designs are found invariant to tree size and hence create a ground to construct a general replication-free D _s -optimal design. Even though the designs are found not to be dependent on the parameter value p of the Kozak's model, they are sensitive to the s×1 subset parameter vector values of the model. The 12 points replication-free design (with 91% efficiency) suggested in this study is believed to reduce cost and time for data collection and more importantly to precisely estimate the subset parameters of interest.     

Experimental designs for detecting synergy and antagonism between two drugs in a pre‐clinical study

The identification of synergistic interactions between combinations of drugs is an important area within drug discovery and development. Pre‐clinically, large numbers of screening studies to identify synergistic pairs of compounds can often be ran, necessitating efficient and robust experimental designs. We consider experimental designs for detecting interaction between two drugs in a pre‐clinical in vitro assay in the presence of uncertainty of the monotherapy response. The monotherapies are assumed to follow the Hill equation with common lower and upper asymptotes, and a common variance. The optimality criterion used is the variance of the interaction parameter. We focus on ray designs and investigate two algorithms for selecting the optimum set of dose combinations. The first is a forward algorithm in which design points are added sequentially. This is found to give useful solutions in simple cases but can lack robustness when knowledge about the monotherapy parameters is insufficient. The second algorithm is a more pragmatic approach where the design points are constrained to be distributed log‐normally along the rays and monotherapy doses. We find that the pragmatic algorithm is more stable than the forward algorithm, and even when the forward algorithm has converged, the pragmatic algorithm can still out‐perform it. Practically, we find that good designs for detecting an interaction have equal numbers of points on monotherapies and combination therapies, with those points typically placed in positions where a 50% response is expected. More uncertainty in monotherapy parameters leads to an optimal design with design points that are more spread out. Copyright © 2015 John Wiley & Sons, Ltd.     

Some remarks on experimental design for estimating the expectation of a stationary random process

For estimation of the mean of a stationary random process the variance-optimal choice of the observation points (the so called experimental design) is considered. For this discrete and continuous designs are introduced, some known results of process statistics are interpreted to experimental design and a proposal for simplification of the minimization problem is offered, moreover it is proved, that for monotone decreasing eovarianee functions a design, for which the points near the ends of the observation interval are more dense than in the middle, is better than the equidistant design.     

On the problem of augmented fractional factorial designs

Let D be a saturated fractional factorial design of the general K₁ x K₂ ...x K_t factorial such that it consists of m distinct treatment combinations and it is capable of providing an unbiased estimator of a subvector of m factorial parameters under the assumption that the remaining k-m,t (k = H it ) factorial parameters are negligible. Such a design will not provide an unbiased estimator of the varianceσ² Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatmentcombinations with the aim to estimate ² Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatment combinations with the aim to estimate σ² unbiasedly. The problem then is how to select the c treatment combinations such that the augmented design D retains its optimality property. This problem, in all its generality is extremely complex. The objective of this paper is to provide some insight in the problem by providing a partial answer in the case of the 2^tfactorial, using the d-optimality criterion.     

Economic Design of Cumulative Sum Control Charts for Monitoring a Process with Correlated Samples

The underlying assumption for the design of control charts is the measurements within a sample are independently distributed. However, there are many situations where the uncorrelation assumption may be unacceptable in practice. In this paper, the economic design of cumulative sum (CUSUM) control chart for correlated data within a sample is developed. The genetic algorithm is applied to find the optimal design parameters of the CUSUM control chart by minimizing the cost function. An illustrative example is given. A sensitivity analysis is then conducted to evaluate the effects of cost parameters, process parameters, and correlation coefficient on the economic design.     

A measure of aliasing and applications to two-level fractional factorials

This paper examines some properties of a measure of aliasing proposed by Hedayat, Raktoe, and Federer (1974). It is shown that in the case of balanced orthogonal designs with no repeated treatments, minimizing the alias measure is equivalent to minimizing tr Yar(ψ). Lower bounds are found for fixed eigenvalues of the design matrix. These results are applied to two-level fractional factorials to show that in certain cases classical fractional-factorial designs yield minimal solutions for the alias measure.     

Optimal designs for the development of personalized treatment rules

We study the design of multi-armed parallel group clinical trials to estimate personalized treatment rules that identify the best treatment for a given patient with given covariates. Assuming that the outcomes in each treatment arm are given by a homoscedastic linear model, with possibly different variances between treatment arms, and that the trial subjects form a random sample from an unselected overall population, we optimize the (possibly randomized) treatment allocation allowing the allocation rates to depend on the covariates. We find that, for the case of two treatments, the approximately optimal allocation rule does not depend on the value of the covariates but only on the variances of the responses. In contrast, for the case of three treatments or more, the optimal treatment allocation does depend on the values of the covariates as well as the true regression coefficients. The methods are illustrated with a recently published dietary clinical trial.     

Economic design of non parametric sign control chart

An economic design of sign chart to control the median is proposed. Since the sign chart is distribution free, it can easily be applied to any process without prior knowledge of process quality distribution. The effect on loss cost observed for different shifts in location shows that the sign chart performs better for large shifts. The economic statistical performance study reveals that statistical performance of sign chart can be improved sufficiently for moderate shifts in the process. Sensitivity study shows that design is more sensitive for change in values of penalty loss cost and time required for search and repair of an assignable cause.     

A general method to determine sampling windows for nonlinear mixed effects models with an application to population pharmacokinetic studies

Optimal design methods have been proposed to determine the best sampling times when sparse blood sampling is required in clinical pharmacokinetic studies. However, the optimal blood sampling time points may not be feasible in clinical practice. Sampling windows, a time interval for blood sample collection, have been proposed to provide flexibility in blood sampling times while preserving efficient parameter estimation. Because of the complexity of the population pharmacokinetic models, which are generally nonlinear mixed effects models, there is no analytical solution available to determine sampling windows. We propose a method for determination of sampling windows based on MCMC sampling techniques. The proposed method attains a stationary distribution rapidly and provides time-sensitive windows around the optimal design points. The proposed method is applicable to determine sampling windows for any nonlinear mixed effects model although our work focuses on an application to population pharmacokinetic models.     

A clustering-based coordinate exchange algorithm for generating G-optimal experimental designs

In the optimal experimental design literature, the G-optimality is defined as minimizing the maximum prediction variance over the entire experimental design space. Although the G-optimality is a highly desirable property in many applications, there are few computer algorithms developed for constructing G-optimal designs. Some existing methods employ an exhaustive search over all candidate designs, which is time-consuming and inefficient. In this paper, a new algorithm for constructing G-optimal experimental designs is developed for both linear and generalized linear models. The new algorithm is made based on the clustering of candidate or evaluation points over the design space and it is a combination of point exchange algorithm and coordinate exchange algorithm. In addition, a robust design algorithm is proposed for generalized linear models with modification of an existing method. The proposed algorithm are compared with the methods proposed by Rodriguez et al. [Generating and assessing exact G-optimal designs. J. Qual. Technol. 2010;42(1):3–20] and Borkowski [Using a genetic algorithm to generate small exact response surface designs. J. Prob. Stat. Sci. 2003;1(1):65–88] for linear models and with the simulated annealing method and the genetic algorithm for generalized linear models through several examples in terms of the G-efficiency and computation time. The result shows that the proposed algorithm can obtain a design with higher G-efficiency in a much shorter time. Moreover, the computation time of the proposed algorithm only increases polynomially when the size of model increases.     

Bioassay case study applying the maximin D‐optimal design algorithm to the four‐parameter logistic model

Cell‐based potency assays play an important role in the characterization of biopharmaceuticals but they can be challenging to develop in part because of greater inherent variability than other analytical methods. Our objective is to select concentrations on a dose–response curve that will enhance assay robustness. We apply the maximin D‐optimal design concept to the four‐parameter logistic (4PL) model and then derive and compute the maximin D‐optimal design for a challenging bioassay using curves representative of assay variation. The selected concentration points from this ‘best worst case’ design adequately fit a variety of 4PL shapes and demonstrate improved robustness. Copyright © 2015 John Wiley & Sons, Ltd.     

The robustness of chain block designs and coat-of-mail designs

Chain block designs are relatively vulnerable to loss of information when missing values or outliers may occur. An alternative class of designs, coat-of-mail designs, are proposed and the relative robustness of the two types of design are compared.     

Bayesian two-stage optimal design for mixture models

In this paper, a Bayesian two-stage D–D optimal design for mixture experimental models under model uncertainty is developed. A Bayesian D-optimality criterion is used in the first stage to minimize the determinant of the posterior variances of the parameters. The second stage design is then generated according to an optimalityprocedure that collaborates with the improved model from the first stage data. The results show that a Bayesian two-stage D–D-optimal design for mixture experiments under model uncertainty is more efficient than both the Bayesian one-stage D-optimal design and the non-Bayesian one-stage D-optimal design in most situations. Furthermore, simulations are used to obtain a reasonable ratio of the sample sizes between the two stages.     

Statistical methodology for binary data from a set of partially structured two-way random cross-classifications

A complete two-way cross-classification design is not practical in many settings. For example, in a toxicological study where 30 male rats are mated with 30 female rats and each mating outcome (successful or unsuccessful)is observed, time and resource considerations can make the use of the complete design prohibitively costly. Partially structured variations of this design are, therefore, of interest (e.g., the balanced disjoint rectangle design, the fully diagonal design, and the "S"-design). Methodology for analyzing binary data from such incomplete designs is illustrated with an example. This methodology, which is based on infinite population sampling arguments, allows the estimation of the mean response, among-row correlation coefficient, among-column correlation coefficient, and the within-cell correlation coefficient as well as their standard errors.     

A relaxation procedure for calculating (G–)minimax optimal designs

Summary: In nonlinear statistical models, standard optimality functions for experimental designs depend on the unknown parameters of the model. An appealing and robust concept for choosing a design is the minimax criterion. However, so far, minimax optimal designs have been calculated efficiently under various restrictive conditions only. We extend an iterative relaxation scheme originally proposed by Shimizu and Aiyoshi (1980) and prove its convergence under very general assumptions which cover a variety of situations considered in experimental design. Application to different specific design criteria is discussed and issues of practical implementation are addressed. First numerical results suggest that the method may be very efficient with respect to the number of iterations required.*Supported by a grant from the Deutsche Forschungsgemeinschaft. We are grateful to a referee for his constructive suggestions.     

An introduction to designh optimality with an overview of the literature

A brief introduction to oprimal design theorv is given for those who are not familiar with the subject. A list of 312 selected articles on the theory of optimal design is provided. The bibliography should be sufficiently thorough to be of use to researchers in the field.     

Optimal designs for experiments with mixtures: a survey

This is a survey article on known results about analytic solutions and numerical solutions of optimal designs for various regression models for experiments with mixtures. The regression models include polynomial models, models containing homogeneous functions, models containing inverse terms and ratios, log contrast models, models with quantitative variables, and mod els containing the amount of mixture, Optimality criteria considered include D-, A-, E-,φ_p- and I_λ-Optimalities. Uniform design and uniform optimal design for mixture components, and efficiencies of the {q,2} simplex-controid design are briefly discussed.