Three-stage designs for monitoring clinical trials

Optimal three-stage designs with equal sample sizes at each stage are presented and compared to fixed sample designs, fully sequential designs, designs restricted to use the fixed sample critical value at the final stage, and to modifications of other group sequential designs previously proposed in the literature. Typically, the greatest savings realized with interim analyses are obtained by the first interim look. More than 50% of the savings possible with a fully sequential design can be realized with a simple two-stage design. Three-stage designs can realize as much as 75% of the possible savings. Without much loss in efficiency, the designs can be modified so that the critical value at the final stage equals the usual fixed sample value while maintaining the overall level of significance, alleviating some potential confusion should a final stage be necessary. Some common group sequential designs, modified to allow early acceptance of the null hypothesis, are shown to be nearly optimal in some settings while performing poorly in others. An example is given to illustrate the use of several three-stage plans in the design of clinical trials.     

On the Construction of Nearest–Neighbour Balanced Row–Column Designs

Nearest–neighbour balance is considered a desirable property for an experiment to possess in situations where experimental units are influenced by their neighbours. This paper introduces a measure of the degree of nearest–neighbour balance of a design. The measure is used in an algorithm which generates nearest–neighbour balanced designs and is readily modified to obtain designs with various types of nearest–neighbour balance. Nearest–neighbour balanced designs are produced for a wide class of parameter settings, and in particular for those settings for which such designs cannot be found by existing direct combinatorial methods. In addition, designs with unequal row and column sizes, and designs with border plots are constructed using the approach presented here.     

Multilevel Augmented Pairs Second-Order Response Surface Designs and Their Robustness to Missing Data

Missing observations can occur even in a well-planned experiment. The effect of missing observations can be much more serious when the design is saturated or near saturated. The levels of factor settings that make a design more robust to missing observations are of great importance in the sense that the loss for missing observations becomes minimum. In this study, new augmented pairs minimax loss designs are constructed, which are more robust to one missing design point than the augmented pairs designs presented by Morris (2000 Morris , M. D. ( 2000 ). A class of three-level experimental designs for response surface modeling . Technometrics 42 : 111 – 121 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). New designs are compared with augmented pairs designs, central composite designs, and small composite designs under generalized scaled standard deviations. The model used is also studied for the regression coefficient estimates.     

Comparison of designs equivalent under one or two criteria

Two designs equivalent under one or two criteria may be compared under other criteria. For certain configurations of eigenvalues of the information matrices, we decide which design is the better of the two for many other such criteria. The relationship to universal optimality (in the case of equivalence under one criterion) is indicated. For two criteria, applications are given to weighing and treatment-with-covariate settings.     

Circular Binary Block Second- and Higher-Order Neighbor Designs

Some new neighbor designs are presented here. Second-order neighbor designs for different configurations are generated in circular binary blocks. Third-order and fourth-order neighbor designs for some cases are also constructed. In all cases, circular blocks are well separated and these designs are obtained through initial block/s. At the end of the study, some models for analysis of these designs are also presented.     

Fourth order rotatability

Fourth order rotatable designs are discussed. A general k, design moment inequality is given. The variance function for two-factor designs is derived, and plotted for a specific design. A minimum point set requirement for two-factor designs is established, thus enabling one to form an infinity of such designs. Some difficulties in obtaining deLigns for k>2 are described. Some questions are posed for future work.     

Balanced Treatment incomplete block (BTIB) designs for comparing treatments with a control: minimal complete sets of generator designs for k = 3, p = 3(1)10

Bechhofer and Tamhane (1981) proposed a new class of incomplete block designs called BTIB designs for comparing p ≥ 2 test treatments with a control treatment in blocks of equal size k < p + 1. All BTIB designs for given (p,k) can be constructed by forming unions of replications of a set of elementary BTIB designs called generator designs for that (p,k). In general, there are many generator designs for given (p,k) but only a small subset (called the minimal complete set) of these suffices to obtain all admissible BTIB designs (except possibly any equivalent ones). Determination of the minimal complete set of generator designs for given (p,k) was stated as an open problem in Bechhofer and Tamhane (1981). In this paper we solve this problem for k = 3. More specifically, we give the minimal complete sets of generator designs for k = 3, p = 3(1)10; the relevant proofs are given only for the cases p = 3(1)6. Some additional combinatorial results concerning BTIB designs are also given.     

On The Optimal Choice of Cube and Star Replications in Restricted Second-Order Designs

This article examines the central composite design in which some of the experimental runs are replicated. Three different classes of N-point designs are compared using the criterion of Schur's ordering under orthogonality, rotatablity, and slope- rotatablity conditions. The response surface designs with the star portion replicated seem to have more potential than others under orthogonality condition, while the designs with the cube portion replicated is preferable to the designs with their star portion or only the center point replicated under rotatable and slope-rotatable conditions.     

On balanced incomplete Latin square designs

Recently, balanced incomplete Latin square designs are introduced in the literature. We propose three methods of constructions of balanced incomplete Latin square designs. Particular classes of Latin squares namely Knut Vik designs, semi Knut Vik designs, and crisscross Latin squares play a key role in the construction.     

A-Optimal Block Designs with Additional Singly Replicated Treatments

Block designs to which have been added a number of singly-replicated treatments, known as secondary treatments, are particularly useful for experiments where only small amounts of material are available for some treatments, for example new plant varieties. The designs are of particular use in the microarray situation. Such designs are known as 'augmented designs'. This paper obtains the properties of these designs and shows that, with an equal number of secondary treatments in each block, the A-optimal design is obtained by using the A-optimal design for the original block design. It develops formulae for the variance of treatment comparisons, for both the primary and the secondary treatments. A number of examples are used to illustrate the results.     

Optimal block designs for triallel cross experiments

Optimal block designs for a certain type of triallel cross experiments are investigated. Nested balanced block designs are introduced and it is shown how these designs give rise to optimal designs for triallel crosses. Several .series of nested balanced block designs, leading to optimal designs for triallel crosses are reported.     

Partially efficiency-balanced designs-II

efficiency balanced (£R) and partially efficiency balanced (PEB) designs were studied hy Calinski (1971) and Puri and Higam (1971, 1977a). In tais paper we have studied the dual desinas of two classes of PEB designs. It is shown that the dual of any symmetrical PCS design with HN'=N'H is again a PEB design. Also the dual of a PEB(S) with μ₁ = 0 μ₂ = 0and is shown to the again a PEB(S). The relations between PEB designs and linked and partially linked blocks have heen established. As a conseguence, the analysis of linked Mock and partially linkei block designs can he sinrlififid. Sore useful classes of supplencntcd block designs having simple analysis have also been discussed.     

On d- and e- minimax optimal designs for estimating the axial slopes of a second-order response surface over hypercubic regions

Design of experiments for estimating the slopes of a response surface is considered. Design criteria analogous to the traditional ones but based upon the variance-covariance matrix of the estimated slopes along factor axes are proposed. Optimal designs under the proposed criteria are derived for second-order polynomial regression over hypercubic regions. Best de¬signs within some commonly used classes of designs are also obtained and their efficiencies are investigated.     

The interplay of optimality and combinatorics in experimental design

The design of statistical experiments, as developed by R. A. Fisher and his followers, often used combinatorial structures that yielded simple calculation of estimates and/or symmetric variances and covariances. Examples are block designs with balance, regression experiments with equally spaced observations, etc. More recently, considerations of optimality (choosing a design that achieves most accurate inference in some sense) have sometimes justified the traditional designs, but have sometimes led to new combinatorial investigations. Illustrations are given.     

Families of proper generalized neighbor designs

A generalized neighbor design relaxes the equality condition on the number of times two treatments occur as neighbors in the design. In this article we have constructed a new series of generalized neighbor designs with equal block sizes, a series of neighbor designs of Rees [1967. Some designs of use in serology. Biometrics 23, 779–791] and a series of neighbor designs with two distinct block sizes. Two more new series of GN₂ designs are also constructed for even number of treatments. It has been shown that quasi neighbor designs introduced by Preece [1994. Balanced Ouchterlony neighbor designs. J. Combin. Math. Combin. Comput. 15, 197–219] are special cases of generalized neighbor designs with t=2$t=2$. All the designs given here are binary. A new definition—partially balanced circuit design is introduced which is a special case of generalized neighbor designs with binary blocks.     

On Some Two- and Three-dimensional D-minimax Designs for Estimating Slopes of a Third-order Response Surface

Design of experiments is considered for the situation where estimation of the slopes of a response surface is the main interest. Under the D-minimax criterion, the objective is to minimize the generalized variance of the estimated axial slopes at a point maximized over all points in the region of interest in the factor space. For the third-order model over spherical regions, the D-minimax designs are derived in two and three dimensions. The efficiencies of some two- and three-dimensional designs available in the literature are also investigated.     

Minimax designs for estimating slopes in a trigonometric regression model

Abstract

Designs for the first order trigonometric regression model over an interval on the real line are considered for the situation where estimation of the slope of the response surface at various points in the factor space is of primary interest. Minimization of the variance of the estimated slope at a point maximized over all points in the region of interest is taken as the design criterion. Optimal designs under the minimax criterion are derived for the situation where the design region and the region of interest are identical and a symmetric “partial cycle”. Some comparisons of the minimax designs with the traditional D- and A-optimal designs are provided. Efficiencies of some exact designs under the minimax criterion are also investigated.     

Example of Block Designs for Plant and Tree Breeding Trials

Experimental designs which use extensive blocking and which are particularly useful in plant and tree breeding trials are discussed. They can be constructed either to accommodate field restrictions or take advantage of favourable plot layouts. Computer software is available to generate these design types for use in practice. Examples cover latinized row-column designs, t -latinized and partially-latinized designs and designs with unequal block sizes.     

UPPER BOUNDS FOR LATINIZED DESIGNS

Upper bounds axe derived for the efficiency factor of a class of resolvable incomplete block designs known as latinized designs. These designs are particularly useful in glasshouse and field trials, and can be readily extended to two-dimensional blocking structures. Existing bounds for resolvable designs axe also reviewed and a comparison is made between the third moment bounds discussed by Jarrett (1989) and the second moment bounds of Tjur (1990).     

Interruptible supersaturated two-level designs

Interruptible designs possess a robustness against possible premature termination of an experiment. We consider such two-level designs for a first-order model and present interruptible sequences which lead to the D-optimal saturated design for four to nine factors if not interrupted. Premature termination of the experiment at any stage results in a supersaturated design with minimum loss of information about the factors. The loss for these designs, which is measured by the pairwise orthogonality between columns, is compared with that of the worst case f o r randomly ordered sequences.