Orthogonal Balanced Block Designs

Orthogonal balanced (or nearly balanced) block designs ( or generalized orthogonal arrays) are new ones which are similar to the orthogonal Latin squares (or orthogonal arrays). Not only the run size of orthogonal balanced (or nearly balanced) block designs (or generalized orthogonal arrays) is small, but there is also the orthogonality in their data analysis. Similarly to the orthogonal decomposition of projection matrices for orthogonal arrays, a concept of matrix images based on orthogonal balanced (or nearly balanced) block designs (or generalized orthogonal arrays) has been demonstrated in this paper. By using the matrix images and properties of projection matrices, it is proved that the orthogonality of orthogonal balanced (or nearly balanced) block designs ( or generalized orthogonal arrays) is equivalent to that of projection matrices. The theory of orthogonal balanced incomplete block designs is further explained. In this paper, we try to expound the theory of the new designs. This paper has six chapters as the following:The first chapter is an introduction, which is a retrospect on the process of the experiment design. The importance of orthogonal balanced block designs is also discussed here.Chapter two presents the definition, property and example of orthogonal balanced block designs.Chapter three introduces matrix images, which is the primary tool in studying orthogonality of orthogonal balanced block designs. It is necessary to give a brief of the theory of matrix images.Chapter four aims at elaborating the statistical model of orthogonal balanced block designs. The methodology of designing the test are firstly provided, and then the statistical model and the new model after transformation, including vector format, is given. The calculation of quadratic form and parameter evaluation are presented. Some examples are also included for illustration.Chapter five emphasizes on expounding the method of data analysis. The quadratic form from the new model can well meet the law of Cochran. Therefore, this section focuses on the variance analysis of unsaturated model and zero-composition selecting of saturated model. There also give the statistic to check the grand mean and block column after the primary model.Chapter six principally investigates the simulation analysis of orthogonal balanced block designs. According to different conditions, several simulations are done to analysis the parameter evaluation, analysis of variance, zero-composition selecting, variance estimation, modified analysis of variance and the two-kinds-errors probability of orthogonal balanced block designs. Besides, simulations based on generalized orthogonal arrays GL₁₂(3¹;2³3³) and L₃₆(2³3⁴) are given out. It is illustrated that orthogonal balanced block designs is a recommendable designs because not only the run size is small but also keep the orthogonality between columns.The appendix provides some constructed generalized orthogonal arrays for reference.