Rotatability is a Sufficient Condition for A- and D-Rotatability

Designs for estimating axial slopes of a response surface are considered. A design is said to be A- (D-) rotatable if the trace (determinant) of variance–covariance matrix of the estimated slopes at a point in the factor space is a function of the distance of the point from the center of the design. It is shown that for any order d and any dimension k, a design which is rotatable in the Box and Hunter (1957 Box , G. E. P. , Hunter , J. S. ( 1957 ). Multifactor experimental designs for exploring response surfaces . Annals of Mathematical Statistics 28 : 195 – 241 .[Crossref] , [Google Scholar]) sense is automatically A- and D-rotatable. Some specific examples illustrating the validity of this general result are provided.     

Comparison of Operating Characteristics of Commonly Used Sample Size Re-Estimation Procedures in a Two-Stage Design

In group sequential clinical trials, there are several sample size re-estimation methods proposed in the literature that allow for change of sample size at the interim analysis. Most of these methods are based on either the conditional error function or the interim effect size. Our simulation studies compared the operating characteristics of three commonly used sample size re-estimation methods, Chen et al. (2004), Cui et al. (1999), and Muller and Schafer (2001). Gao et al. (2008) extended the CDL method and provided an analytical expression of lower and upper threshold of conditional power where the type I error is preserved. Recently, Mehta and Pocock (2010) extensively discussed that the real benefit of the adaptive approach is to invest the sample size resources in stages and increasing the sample size only if the interim results are in the so called “promising zone” which they define in their article. We incorporated this concept in our simulations while comparing the three methods. To test the robustness of these methods, we explored the impact of incorrect variance assumption on the operating characteristics. We found that the operating characteristics of the three methods are very comparable. In addition, the concept of promising zone, as suggested by MP, gives the desired power and smaller average sample size, and thus increases the efficiency of the trial design.     

Computer-Generated Neighbor Designs

Neighbor designs are useful to remove the neighbor effects. In this article, an algorithm is developed and is coded in Visual C + +to generate the initial block for possible first, second,…, and all order neighbor designs. To get the required design, a block (0, 1, 2,…, k ? 1) is then augmented with (v ? 1) blocks obtained by developing the initial block cyclically mod (v ? 1).     

Some New Series of Partially Balanced Residual Treatment Effects Designs

Blaisdell and Raghavarao (1980 Blaisdell , E. A. , Raghavarao , D. ( 1980 ). Partially balanced change-over designs based on m-associate class PBIB designs . JRSS B 42 : 334 – 338 . [Google Scholar] 1985 Blaisdell , E. A. , Raghavarao , D. ( 1985 ). Efficiency bounds for partially balanced change-over designs based on m-associate class PBIB designs . JRSS B 47 : 132 – 135 . [Google Scholar]) introduced partially balanced residual treatment effects designs and gave methods of construction of these designs. They also gave efficiencies for the estimated direct and residual elementary treatment contrasts of these designs. In this paper, we construct some new series of partially balanced residual treatment effects designs and also give their efficiencies for the estimated direct and residual elementary treatment contrasts.     

The Mean Squared Error Optimum Design Criterion for Parameter Estimation in Nonlinear Regression Models

In the context of nonlinear regression models, we propose an optimal experimental design criterion for estimating the parameters that account for the intrinsic and parameter-effects nonlinearity. The optimal design criterion proposed in this article minimizes the determinant of the mean squared error matrix of the parameter estimator that is quadratically approximated using the curvature array. The design criterion reduces to the D-optimal design criterion if there are no intrinsic and parameter-effects nonlinearity in the model, and depends on the scale parameter estimator and on the reparameterization used. Some examples, using a well known nonlinear kinetics model, demonstrate the application of the proposed criterion to nonsequential design of experiments as compared with the D-optimal criterion.     

A-Optimality Standardized Through the Coefficient of Variation

Many of the usual criteria for optimal experimental design do not take into account the different scale of the variance of the parameters. Dette (1997 Dette , H. ( 1997 ). Designing experiments with respect to “standardized” optimality criteria . J. J. Roy. Statist. Soc. Ser. B Statist. Methodol. 59 ( 1 ): 97 – 110 .[Crossref] , [Google Scholar]) provided a standardization which leads to designs with similar efficiencies for all of the parameters.

In this article, a new way of standardization through the coefficient of variation is given. This leads to designs useful when one of the parameters is expected to be very small. Thus, this criterion may be used if there is special interest in maximizing the power of some parameter tests. Locally standardized through the coefficient of variation A-optimal designs are computed for simple linear and quadratic regression.     

Minimax Design for the Difference Between Estimated Responses for the Quadratic Model Over Hypercubic Regions

Abstract

A second-order model involving the intercept and only the pure quadratic terms is considered for regression over hypercubes. Minimization of the variance of the difference between estimated responses at two points, maximized over all pairs of points in the region of interest in factor space, is taken as the design criterion. Optimal design under the minimax criterion is derived and found to be the one which is also simultaneously A-, D-, and E-optimal for the parameters excluding the intercept. The minimax design is compared with other standard designs and is found to perform extremely well.     

Circular Strongly Balanced Repeated Measurements Designs

Magda (1980 Magda , C. G. ( 1980 ). Circular balanced repeated measurements designs . Commun. Statist. Theor. Meth. 9 : 1901 – 1918 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Hedayat (1981 Hedayat , A. S. ( 1981 ). Repeated measurements designs-IV: recent advances (with discussion) . Bull. Int. Statist. Inst. 1 : 591 – 610 . [Google Scholar]) first considered the construction of circular strongly balanced repeated measurements designs. Sen and Mukerjee (1987 Sen , M. , Mukerjee , R. ( 1987 ). Optimal repeated measurements designs under interaction . J. Statist. Plann. Infer. 17 : 81 – 91 .[Crossref], [Web of Science ®] , [Google Scholar]) and Roy (1988 Roy , B. K. ( 1988 ). Construction of strongly balanced uniform repeated measurements designs . J. Statist. Plann. Infer. 19 : 341 – 348 .[Crossref], [Web of Science ®] , [Google Scholar]) considered the optimality and existence of circular strongly balanced repeated measurements designs based on the method of differences and Hamiltonian decomposition of lexicographic product of two graphs. In this article, we consider the construction of circular strongly balanced repeated measurements designs using the newly proposed method called cyclic shifts, and propose some new designs for p < v.     

Analytical Characterization of the Information Matrix for Split-Plot Designs

This article provides the analytical characterization of the inverse of the information matrix for second-order SPD. A particular feature of these explicit expressions is that they are functions of the design parameters enabling the development of analytical functions to efficiently compute exact design optimality criteria. The application of these analytical expressions is demonstrated using the generalized variance of the parameter estimates for second-order SPD. An example illustrating the use of these expressions is also presented.