On the study of balanced incomplete block designs with repeated blocks and other incomplete block designs

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BIB designs with repeated blocks: review and perspectives

Experimental Design plays an important role on establishing an interface between Applied Mathematics and statistical applications in several fields, like Agriculture, Industry, Genetics,Biology and Education Sciences. The goal of any Experimental Design is to obtain the maximum amount of information for a given experimental effort, to allow comparisons between varieties and to control for sources of random variability. Randomized block designs are used to control for these sources. A Balanced Incomplete Block Design (BIB Design) is a randomized block design with number of varieties greater than the block size and with all pairs of varieties occurring equally often along the blocks. The Fisher related information of a balanced block design will remain invariant whether or not the design has repeated blocks. This fact can be used theoretically to build a large number of non-isomorphic designs for the same set of design parameters, which could be used for many different purposes both in experimentations and surveys from finite populations. The original and most important method on the construction of BIB Designs with repeated blocks (BIBDR) is due to Hedayat and Li (1979): the trade-off method. Since then, many authors and researchers have been paying particular attention to the construction of BIBDR, but still some unsolved problems remain. This issue will be briefly reviewed and new results on the existence and construction of BIBDR, as well as several unsolved problems for further research will be presented.Centro de Estatística e Aplicações da Universidade de Lisbo

Optimal and efficient crossover designs for comparing test treatments with a control treatment

This paper deals exclusively with crossover designs for the purpose of comparing t test treatments with a control treatment when the number of periods is no larger than t+1. Among other results it specifies sufficient conditions for a crossover design to be simultaneously A-optimal and MV-optimal in a very large and appealing class of crossover designs. It is expected that these optimal designs are highly efficient in the entire class of crossover designs. Some computationally useful tools are given and used to build assorted small optimal and efficient crossover designs. The model robustness of these newly discovered crossover designs is discussed.Comment: Published at http://dx.doi.org/10.1214/009053604000000887 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Planeamento de experiências : modelos e desafios dos planos em blocos incompletos.

O principal objetivo de um Planeamento de Experiências reside essencialmente na procura de relações entre variáveis e na comparação de níveis de fatores, recorrendo ao tratamento estatístico dos dados recolhidos. A utilização de blocos no Planeamento de Experiências é fundamental, pois permite reduzir ou eliminar a variabilidade introduzida por fatores que podem influenciar a experiência mas que não interessam e/ou não foram explicitamente incluídos durante o planeamento. Neste trabalho apresentamos os resultados do estudo e investigação dos Planos em Blocos Incompletos Equilibrados (BIBD), Planos em Blocos Incompletos Equilibrados com repetição de blocos (BIBDR) e Planos em Blocos Incompletos com blocos de diferentes dimensões (VBBD). Exploramos algumas propriedades e métodos de construção destes planos e ilustramos, sempre que possível, com exemplos. Tendo como base o planeamento em blocos, apresentamos uma aplicação dos BIBDR na área da Educação com o objetivo de comparar cinco domínios do pensamento algébrico de uma amostra de alunos do 1º ano do ensino superior em Cabo Verde. Para a análise dos dados da amostra foi utilizado o software R, versão 2.12.1. Pudemos constatar que existem diferenças significativas entre alguns dos domínios do pensamento algébrico, nomeadamente entre os domínios da Generalização da Aritmética e Tecnicismo Algébrico com os restantes domínios. Recomendamos a escolha de uma amostra mais representativa constituída por alunos de todas as instituições superiores de Cabo VerdeThe main purpose of an Experimental Design resides mainly in the search for relationships between variables and in comparing levels of factors, using statistical treatment of collected data. The use of blocks in Experimental Design is essential because it allows reducing or eliminating the variability introduced by factors that can influence the experience but are not of main interest and/or were not explicitly included during experiments. In this work we present the results of the study and research of Balanced Incomplete Block Designs (BIBD), Balanced Incomplete Block Designs with repeated blocks (BIBDR) and the Incomplete Blocks Designs with blocks with different dimensions (VBBD). We explore some properties and construction methods of such designs and illustrate, when possible, with examples. Based on Block Designs, we present an application of BIBDR in Education, with the aim of comparing five domains of algebraic thinking in a sample of 1st year students of higher education in Cape Verde. For the analysis of sample data, the software R was used, version 2.12.1. We observed that significant differences exist between some of the domains of algebraic thinking, especially among the domains of Generalization of Arithmetic and Algebraic Technicality with the remaining areas. For a more representative sample, we recommend a bigger sample consisting of students from all higher institutions of Cape Verde

Resistant And Susceptible BIB Designs

Statistic

Design Lines

The two basic equations satisfied by the parameters of a block design define a three-dimensional affine variety $\mathcal{D}$ in $\mathbb{R}^{5}$. A point of $\mathcal{D}$ that is not in some sense trivial lies on four lines lying in $\mathcal{D}$. These lines provide a degree of organization for certain general classes of designs, and the paper is devoted to exploring properties of the lines. Several examples of families of designs that seem naturally to follow the lines are presented.Comment: 16 page