D-Optimum Designs for Optimum Mixture in a Quadratic Log Contrast Model

Optimal designs for estimating the optimum mixing proportions in a quadratic mixture model was first investigated by Pal and Mandal (2006). In this article, similar investigation is carried out when mean response in a mixture experiment is described by a quadratic log contrast model. It is found that in a symmetric subspace of the finite dimensional simplex, there exists a D-optimal design that puts weights at the centroid of the sub-space and the vertices of the experimental domain. The optimality is checked by numerical computation using Equivalence Theorem.     

Developing a Wide Easy-to-Generate Class of Bivariate Copulas

As of late, copulas have drawn great attention in stochastic simulation, financial engineering, and risk management. Their power lies under their ability of modeling dependent random variables. Using a known theorem in probability which proves that the fractional part of the sum of a uniform and an arbitrary independent continuous random variable follows a uniform distribution, we construct a wide class of bivariate copulas in which bivariate random vector generation can be performed easily. Some important members of this new class and their properties together with two invariant correlation measures and some insights in their application are presented.     

Statistical Inference of Adaptive Designs with Binary Responses

Adaptive designs of clinical trials are ethical alternatives when the traditional randomization becomes ethically infeasible in desperate medical situations. However, such a design creates a dependency among trial data and its statistical analysis becomes more complex than the analysis for traditional randomized clinical trials. In this article, we examine adaptive designs with dichotomous responses from two treatments and extend some commonly used statistical methods for independent data. Under a regularity condition, the estimated odds ratio and its logarithm are shown to follow asymptotically normal distributions. Moreover, the ordinary goodness-of-fit test statistic for two-by-two contingency tables with dependent data is shown to be asymptotically chi-square distributed. We also discuss the consistency of maximum likelihood estimators of the unknown parameters for a wide class of adaptive designs.     

Bayes designs for multiple linear regression on the unit sphere

We deal with experimental designs minimizing the mean square error of the linear BAYES estimator for the parameter vector of a multiple linear regression model where the experimental region is the k-dimensional unit sphere. After computing the uniquely determined optimum information matrix, we construct, separately for the homogeneous and the inhomogeneous model, both approximate and exact designs having such an information matrix.     

The Parshvanath yantram yields a constant-sum partially balanced incomplete block design

Yantrams have been used to generate mixture designs in the interior of a simplex. In this note, we show a connection between Parshvanath yantram and a particular partially balanced incomplete block design. This block design is rather special and somewhat unexpected due to the feature that sum of the treatment symbols in any block is constant.     

D- and V-optimal population designs for the quadratic regression model with a random intercept term

In this paper D- and V-optimal population designs for the quadratic regression model with a random intercept term and with values of the explanatory variable taken from a set of equally spaced, non-repeated time points are considered. D-optimal population designs based on single-point individual designs were readily found but the derivation of explicit expressions for designs based on two-point individual designs was not straightforward and was complicated by the fact that the designs now depend on ratio of the variance components. Further algebraic results pertaining to d-point D-optimal population designs where d≥3 and to V-optimal population designs proved elusive. The requisite designs can be calculated by careful programming and this is illustrated by means of a simple example.     

A new lower bound to A2-optimality measure for multi-level and mixed-level column balanced designs and its applications

In this paper, a new lower bound to A₂-optimality measure is derived and is applied to multi-level and mixed-level column balanced designs. A₂-optimal multi-level and mixed-level designs are obtained by the application of the new lower bound.     

Optimal designs for tumor regrowth models

Most growth curves can only be used to model the tumor growth under no intervention. To model the growth curves for treated tumor, both the growth delay due to the treatment and the regrowth of the tumor after the treatment need to be taken into account. In this paper, we consider two tumor regrowth models and determine the locally D- and c-optimal designs for these models. We then show that the locally D- and c-optimal designs are minimally supported. We also consider two equally spaced designs as alternative designs and evaluate their efficiencies.