Optimum Designs for Parameter Estimation in Linear Mixture Models with Synergistic Effects

In this article, we derive optimum designs for parameter estimation in a mixture experiment when the response function is linear in the mixing components with some synergistic effects. The D- and A-optimality criteria have been used for the purpose. The Equivalence Theorem has been used to check for the optimality of the proposed designs.     

Algorithmic construction of optimal block designs for two-colour cDNA microarray experiments using the linear mixed effects model

ABSTRACT

In this study, methods for efficient construction of A-, MV-, D- and E-optimal or near-optimal block designs for two-colour cDNA microarray experiments with array as the block effect are considered. Two algorithms, namely the array exchange and treatment exchange algorithms together with the complete enumeration technique are introduced. For large numbers of arrays or treatments or both, the complete enumeration method is highly computer intensive. The treatment exchange algorithm computes the optimal or near-optimal designs faster than the array exchange algorithm. The two methods however produce optimal or near-optimal designs with the same efficiency under the four optimality criteria.     

Optimal Block Designs With Interference Effects From Neighboring Units Under a Non Additive Model

Here, the optimality of block design with interference effect from neighboring unit under a general non additive model is investigated, which allows for the presence of interactions among the treatments applied in the adjacent plots. A non additive model with interference × direct effects of treatments is considered as these effects contribute significantly to the response. A class of complete block designs balanced for interference effects from left neighboring unit is shown to be universally optimal for the estimation of direct and interference effects of treatments and two such series of designs have been constructed. Furthermore, considering direct treatment × block non additivity with interference effects, the optimality is studied and the optimal designs are obtained.     

A bidimensional class of optimality criteria involving φp and characteristic criteria

A new biparametric class of criteria for Optimal Experimental Design generalizing the families of φ_p and Characteristic Criteria is presented. Some properties of the Characteristic Criteria are provided: in particular, differentiability, monotonicity and convexity. A statistical interpretation is also offered. Optimal designs with respect to Characteristic Criteria are obtained for polynomial and compartmental models and an extension of the Michaelis-Menten model. A thorough discussion for the trigonometric model is given. The computed optimal designs are shown to be quite efficient for A-, D- and D_s -optimality. Thus, some of them appear as a natural compromise between different optimality criteria. A simulation in multiple linear regression confirms the quality of Characteristic designs for discovering significant differences between parameters.     

CVX‐based algorithms for constructing various optimal regression designs

CVX‐based numerical algorithms are widely and freely available for solving convex optimization problems but their applications to solve optimal design problems are limited. Using the CVX programs in MATLAB, we demonstrate their utility and flexibility over traditional algorithms in statistics for finding different types of optimal approximate designs under a convex criterion for nonlinear models. They are generally fast and easy to implement for any model and any convex optimality criterion. We derive theoretical properties of the algorithms and use them to generate new A‐, c‐, D‐ and E‐optimal designs for various nonlinear models, including multi‐stage and multi‐objective optimal designs. We report properties of the optimal designs and provide sample CVX program codes for some of our examples that users can amend to find tailored optimal designs for their problems. The Canadian Journal of Statistics 47: 374–391; 2019 © 2019 Statistical Society of Canada     

Extrapolation designs and Φp-optimum designs for cubic regression on the q-ball

This paper continues earlier work of the authors in carrying out the program discussed in Kiefer (1975), of comparing the performance of designs under various optimality criteria. Designs for extrapolation problems are also obtained. The setting is that in which the controllable variable takes on values in the q-dimensional unit ball, and the regression is cubic. Thus, the ideas of comparison are tested for a model more complex than the quadratic models discussed previously. The E-optimum design performs well in terms of other criteria, as well as for extrapolation to larger balls. A method of simplifying the calculations to obtain approximately optimum designs, is illustrated.     

Tables of Cyclic Incomplete Block Designs: r=k

Initial blocks and efficiencies are given for cyclic incomplete block designs in the range t= 10(1)60 and r=k= 3(1)10. The designs have been selected by search procedures using optimality criteria which maximize the efficiency factor.     

Marginally restricted D-optimal designs for correlated observations

Two practical degrees of complexity may arise when designing an experiment for a model of a real life case. First, some explanatory variables may not be under the control of the practitioner. Secondly, the responses may be correlated. In this paper three real life cases in this situation are considered. Different covariance structures are studied and some designs are computed adapting the theory of marginally restricted designs for correlated observations. An exchange algorithm given by Brimkulov's algorithm is also adapted to marginally restricted D–optimality and it is applied to a complex situation.     

On optimality properties of simple block designs in the approximate design theory

Optimality properties of approximate block designs are studied under variations of (1) the class of competing designs, (2) the optimality criterion, (3) the parametric function of interest, and (4) the statistical model. The designs which are optimal turn out to be the product of their treatment and block marginals, and uniform designs when the support is specified in advance. Optimality here means uniform, universal, and simultaneous j_p-optimality. The classical balanced incomplete block designs are embedded into this approach, and shown to be simultaneously j_p-optimal for a maximal system of identifiable parameters. A geometric account of universal optimality is given which applies beyond the context of block designs.     

Optimizing the test power for a radiation retention model in the human body

The model that describes the retention in lungs of radioisotope particles is studied in this paper, considering the situation of an accident in facilities that handle radioactive materials. Optimal times to make the bioassays are computed for D‐ and c‐optimality, and efficiencies for the computed designs are provided and compared. Moreover, the test power is checked by means of simulations and replications. After that the inverse of the Fisher information matrix is compared to an estimation of the covariance matrix of the parameters. Finally, a study taking into consideration the randomness of the designs space is performed. Copyright © 2009 John Wiley & Sons, Ltd.     

Some robust designs for polynomial regression models

The author identifies static optimal designs for polynomial regression models with or without intercept. His optimality criterion is an average between the D‐optimality criterion for the estimation of low‐degree terms and the D₈‐optimality criterion for testing the significance of higher degree terms. His work relies on classical results concerning canonical moments and the theory of continued fractions.     

On the choice of optimality criteria in comparing statistical designs

We formulate in a reasonable sense a class of optimality functionals for comparing feasible statistical designs available in a given setup. It is desired that the optimality functionals reflect symmetric measures of the lack of information contained in the designs being compared. In view of this, Kiefer's (1975) universal optimality criterion is seen to rest on stringent conditions, some of which can be relaxed while preserving optimality (in an extended sense) of the so-called balanced designs.     

Optimal block designs with non additive mixed effects interference

This paper deals with optimality aspects of block designs balanced for interference effects from neighboring units on both sides under a general non additive model along with random block effects. Here, a class of complete, circular block designs strongly balanced for interference effects has been shown to be universally optimal for the estimation of direct effects and interference effects (left and right) of treatments under a non additive mixed effects model.     

OPTIMAL EXPERIMENTAL DESIGNS FOR MULTILEVEL MODELS WITH COVARIATES

In this paper optimal experimental designs for multilevel models with covariates and two levels of nesting are considered. Multilevel models are used to describe the relationship between an outcome variable and a treatment condition and covariate. It is assumed that the outcome variable is measured on a continuous scale. As optimality criteria D-optimality, and L-optimality are chosen. It is shown that pre-stratification on the covariate leads to a more efficient design and that the person level is the optimal level of randomization. Furthermore, optimal sample sizes are given and it is shown that these do not depend on the optimality criterion when randomization is done at the group level.     

Response surface design evaluation and comparison

Designing an experiment to fit a response surface model typically involves selecting among several candidate designs. There are often many competing criteria that could be considered in selecting the design, and practitioners are typically forced to make trade-offs between these objectives when choosing the final design. Traditional alphabetic optimality criteria are often used in evaluating and comparing competing designs. These optimality criteria are single-number summaries for quality properties of the design such as the precision with which the model parameters are estimated or the uncertainty associated with prediction. Other important considerations include the robustness of the design to model misspecification and potential problems arising from spurious or missing data. Several qualitative and quantitative properties of good response surface designs are discussed, and some of their important trade-offs are considered. Graphical methods for evaluating design performance for several important response surface problems are discussed and we show how these techniques can be used to compare competing designs. These graphical methods are generally superior to the simplistic summaries of alphabetic optimality criteria. Several special cases are considered, including robust parameter designs, split-plot designs, mixture experiment designs, and designs for generalized linear models.     

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.     

An extension of Banerjee and Rahim model in economic and economic statistical designs for multivariate quality characteristics under Burr XII distribution

The design parameters of the economic and economic statistical designs of control charts depend on the distribution of process failure mechanism or shock model. So far, only a small number of failure distributions, such as exponential, gamma, and Weibull with fixed or increasing hazard rates, have been used as a shock model in the economic and economic statistical designs of the Hotelling T² control charts. Due to both theoretical and practical aspects, the lifetime of the process under study may not follow a distribution with fixed or increasing hazard rate. A proper alternative for this situation may be the Burr distribution, in which the hazard rate can be fixed, increasing, decreasing, single mode, or even U-shaped. In this research article, economic and economic statistical designs of the Hotelling T² control charts under the Burr XII shock models under two uniform and non uniform sampling schemes were proposed, constructed, and compared. The obtained design models were implemented by a numerical example, and a sensitivity analysis was conducted to evaluate the effect of changing parameters of shock model distribution on the optimum values of the proposed design models. The results showed that first the proposed designs under non uniform sampling scheme perform better and second the optimum values of the designs are not significantly sensitive to changing of the Burr XII distribution parameters. We showed that the obtained design models are also true for the beta Burr XII shock model.     

Optimal design in average for inference in generalized linear models

This paper considers the problem of optimal design for inference in Generalized Linear Models, when prior information about the parameters is available. The general theory of optimum design usually requires knowledge of the parameter values. These are usually unknown and optimal design can, therefore, not be used in practice. However, one way to circumvent this problem is through so-called “optimal design in average”, or shortly, “ave optimal”. The ave optimal design is chosen to minimize the expected value of some criterion function over a prior distribution. We focus our interest on the aveD _A-optimality, including aveD- and avec-optimality and show the appropriate equivalence theorems for these optimality criterions, which give necessary conditions for an optimal design. Ave optimal designs are of interest when e.g. a factorial experiment with a binary or a Poisson response in to be conducted. The results are applied to factorial experiments, including a control group experiment and a 2×2 experiment.     

Optimality of circular neighbor-balanced designs for total effects with autoregressive correlated observations

This paper studies the optimality of circular neighbor-balanced designs (CNBDs) for total effects when the one-sided or two-sided neighbor effects are present in the model and the observation errors are correlated according to a first-order circular autoregressive (AR(1,C$C$)) process. Some optimality results under some specified conditions are provided and the efficiency of a CNBD relative to the optimal block design is investigated. In order to discuss the efficiency of a CNBD among all possible block designs with the same size, the optimal equivalence classes of sequences under the one-sided neighbor effects model are characterized and the efficiencies of CNBDs with blocks of small size are illustrated.     

Minimization of model inadequacy when observations subject to shift in means and variances

In this article optimality of experimental design for fitting a lower-order polynomial to a higher order response function for the situation in which observations may be subject to shift in means as well as in variances is considered. It is found that Karson, Manson and Hader‘s (1969) optimum designs provide pro-tection, in some sense, against model inadequacies even when observations are subject to shift in means and variances.