Nested Balanced Incomplete Block Designs

If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v; b1;r;k1) are each partitioned into sub-blocks of size k2, and the b2 =b1k1=k2 sub-blocks themselves constitute a BIBD with parameters (v; b2;r;k2), then the system of blocks, sub-blocks and treatments is, by de4nition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k1 =2k2= 4. Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, they are almost unknown in the combinatorial literature, except in the literature of tournaments, and detailed combinatorial studies of them have been lacking. The present paper therefore reviews and extends mathematical knowledge of NBIBDs. Isomorphism and automorphisms are defined for NBIBDs, and methods of construction are outlined. Some special types of NBIBD are de4ned and illustrated. A first-ever detailed table of NBIBDs with v⩽16, r⩽30 is provided; this table contains many newly discovered NBIBDs. © 2001 Elsevier Science B.V. All rights reserved

Computer construction of experimental plans

Experimental plans identify the treatment allocated to each unit and they are necessary for the supervision of most comparative experiments. Few computer programs have been written for constructing experimental plans but many for analysing data arising from designed experiments. In this thesis the construction of experimental plans is reviewed so as to determine requirements for a computer program. One program, DSIGNX, is described. Four main steps in the construction are identified: declaration, formation of the unrandomized plan (the design), randomization and output. The formation of the design is given most attention. The designs considered are those found to be important in agricultural experimentation and a basic objective is set that the 'proposed' program should construct most designs presented in standard texts (e.g. Cochran and Cox (1957)) together with important designs which have been developed recently. Topics discussed include block designs, factorial designs, orthogonal Latin squares and designs for experiments with non-independent observations. Some topics are discussed in extra detail; these include forming standard designs and selecting defining contrasts in symmetric factorial experiments, general procedures for orthogonal Latin squares and constructing serially balanced designs. Emphasis is placed on design generators, especially the design key and generalized cyclic generators, because of their versatility. These generators are shown to provide solutions to most balanced and partially balanced incomplete block designs and to provide efficient block designs and row and column designs. They are seen to be of fundamental importance in constructing factorial designs. Other versatile generators are described but no attempt is made to include all construction techniques. Methods for deriving one design from another or for combining two or more designs are shown to extend the usefulness of the generators. Optimal design procedures and the evaluation of designs are briefly discussed. Methods of randomization are described including automatic procedures based on defined block structures and some forms of restricted randomization for the levels of specified factors. Many procedures presented in the thesis have been included in a computer program DSIGNX. The facilities provided by the program and the language are described and illustrated by practical examples. Finally, the structure of the program and its method of working are described and simplified versions of the principal algorithms presented

Cell Means Formulation of Mixed Models in the Analysis of Variance

34 pages, 1 article*Cell Means Formulation of Mixed Models in the Analysis of Variance* (Searle, S. R.) 34 page

Computational-experimental approach to drug-target interaction mapping : A case study on kinase inhibitors

Peer reviewe

On Nearly Balanced Designs for Sensory Trials

In sensory experiments, often designs are used that are balanced for carryover effects. It is hoped that this controls for possible carryover effects, like, e.g., a lingering taste of the products. Proper randomization is essential to guarantee the usual model assumption of independent identically distributed (i.i.d.) errors. We consider a randomization procedure that permutes treatment labels and assessors. This restricted randomization leaves the neighbour structure unchanged and validates the assumption of i.i.d. errors if the design used is a Generalized Youden Design (GYD). However, the use of a neighbour balanced GYD may require too many assessors. The question arises, whether nearly balanced designs may be used without grossly violating the validity of the analysis. We therefore do a simulation study to assess the properties (under this restricted randomization) of nearly balanced designs like, e.g., the ones proposed by Périnel and Pag?s (2004, Food Quality and Preference 15, 439?446). We observe that, if there are no carryover effects, the variance estimates for treatment contrasts are not significantly biased whenever we use designs that are nearly GYD. Additionally, designs that are nearly carryover balanced still produce conservative variance estimates, even in the presence of large carryover effects. In all, ?nearly neighbour balanced nearly GYD? as proposed by Périnel and Pag?s (2004) appear to be useful in experimental situations where the use of GYD is too restrictive. It should be stressed, however, that these results are true only if randomization is used as a protection against effects unaccounted for in the statistical model. --carryover balance,nearly balanced designs,randomization,validity

Applications of finite geometries to designs and codes

This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures. A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples. We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs. Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes

Translation and Rotation Invariant Multiscale Image Registration

The most recent research involved registering images in the presence of translations and rotations using one iteration of the redundant discrete wavelet transform. We extend this work by creating a new multiscale transform to register two images with translation or rotation differences, independent of scale differences between the images. Our two-dimensional multiscale transform uses an innovative combination of lowpass filtering and the continuous wavelet transform to mimic the two-dimensional redundant discrete wavelet transform. This allows us to obtain multiple subbands at various scales while maintaining the desirable properties of the redundant discrete wavelet transform. Whereas the discrete wavelet transform produces results only at dyadic scales, our new multiscale transform produces data at all integer scales. This added flexibility improves registration accuracy without greatly increasing computational complexity and permits accurate registration even in the presence of scale differences

Design of sensory multi-session trials with preparation constraints

Design of sensory multi-session trials with preparation constraints Designs for sensory studies must satisfy several requirements. Usually a given number of products are to be evaluated and there is an upper limit to the number of assessors available. Due to variation in sensory perception, inter-assessor product comparisons are preferred. For large product numbers, trials are split into sessions to avoid sensory fatigue and the sequential presentation of products can cause order and carry-over effects. Thus, resolvable row-column or cross-over designs are required, which ensure that each assessor tastes all products the same number of times. In this thesis a three-step procedure is proposed to generate designs for trials where the number of products prepared for or served in each session is limited. First, an incomplete block design with a special column structure, the preparation design, is created, assigning products to sessions. Secondly, a cross-over design is constructed, assigning the columns of the preparation design to assessors. In the third step the two designs are combined by identifying the column-order of the preparation design that results in the highest average efficiency of the complete cross-over design. Search algorithms for incomplete block and cross-over designs are modified to produce preparation and panel designs with a special structure to guarantee resolvability of the complete sensory design. This procedure has been enhanced to produce designs for trials involving a control and several test products, in which control-test comparisons are estimated with higher precision than test-test comparisons. Two distinct construction methods have been developed for this case. By using factorial preparation designs the three step procedure can also be adapted for creating factorial multi-session designs with or without a control product

Optimality and Construction of Designs with Generalized Group Divisible Structure

This thesis is an investigation of the optimality and construction problems attendant to the assignment of v treatments to experimental units in b blocks of size k, paying special attention to settings for which equal replication of the treatments is not possible. The model is that of one way elimination of heterogeneity, in which the expectation of an observation on treatment i in block j is Ti + βj (treatment effect + block effect), where Ti and βj are unknown constants, 1 ≤ i ≤ v and 1 ≤ j ≤ b. All observations are assumed to be uncorrelated with same variance. The generalized group divisible design with s groups, or GGDD(s), is defined in terms of the elements of the information matrix, instead of in terms of the elements of the concurrence matrix as done by Adhikary (1965) and extended by Jacroux (1982). This definition extends the class of designs to include non-binary members, and allows for broader optimality results. Some sufficient conditions are derived for GGDD(s) to be E- and MV-optimal. It is also shown how augmentation of addition blocks to certain GGDD(s)s produces other nonbinary, unequally replicated E- and MV-optimal block designs. Where nonbinary designs are found, they are generally preferable to binary designs in terms of interpretability, and often in terms of one or more formal optimality criteria as well. The class of generalized nearly balanced incomplete block designs with maximum concurrence range l, or NBBD(l), is defined. This class extends the nearly balance incomplete block designs as defined by Cheng & Wu (1981), and the semi-regular graph designs as defined by Jacroux (1985), to cases where off-diagonal entries of the concurrence matrix differ by at most the positive integer l. Sufficient conditions are derived for a NBBD(2) to be optimal under a given type-I criterion. The conditions are used to establish the A- and D-optimality of an infinite series of NBBD(2)s having unequal numbers of replicates. Also, a result from Jacroux (1985) is used to establish the A-optimality of a new series of NBBD(1)s. Several methods of construction of GGDD(s)s are developed from which many infinite series of designs are derived. Generally these designs satisfy the obtained sufficient conditions for E- and MV-optimality. Finally, in the nested row-column setting, the necessary conditions for existence of 2 x 2 balanced incomplete block designs with nested rows and columns (BIBRCs) are found to be sufficient. It is also shown that, sufficient for a BIBRC with p=q to generally balanced, is that the row and column classifications together form a balanced incomplete block design, as does the block classification. All of the 2 x 2 BIBRCs are constructed to have this property