Design optimality for treatment comparison experiments has been intensively studied by numerous researchers, employing a variety of statistically sound criteria. Their general formulation is based on the idea that optimality functions of the treatment information matrix are invariant to treatment permutation. This implies equal interest in all treatments. In practice, however, there are many experiments where not all treatments are equally important. When selecting a design for such an experiment, it would be better to weight the information gathered on different treatments according to their relative importance and/or interest. This dissertation develops a general theory of weighted design optimality, with special attention to the block design problem. Among others, this study develops and justifies weighted versions of the popular A, E and MV optimality criteria. These are based on the weighted information matrix, also introduced here. Sufficient conditions are derived for block designs to be weighted A, E and MV-optimal for situations where treatments fall into two groups according to two distinct levels of interest, these being important special cases of the "2-weight optimality" problem. Particularly, optimal designs are developed for experiments where one of the treatments is a control. The concept of efficiency balance is also studied in this dissertation. One view of efficiency balance and its generalizations is that unequal treatment replications are chosen to reflect unequal treatment interest. It is revealed that efficiency balance is closely related to the weighted-E approach to design selection. Functions of the canonical efficiency factors may be interpreted as weighted optimality criteria for comparison of designs with the same replication numbers. / Ph. D.

在本篇論文中我們使用矩陣來建構可分組設計(GDD), 我們列出了兩種型 式的建構, 第一種 -- 起因於 W.H. Haemers -- A .crtimes. J + I .crtimes. D, 利用此種建構我們將所有符合 r - .lambda.1 = 1 的 (m,n,k,.lambda.1,.lambda.2) GDD 分成三類: (i) A=0 或 J-I, (ii) A 為 .mu. - .lambda. = 1 強則圖的鄰接矩陣, (iii) J-2A 為斜對稱 矩陣的核心。第二種型式為 A .crtimes. D + .Abar .crtimes. .Dbar ,此種方法可以建構出 b=4(r-.lambda.2) 的正規和半正規 GDD 。另外在 論文中, 我們研究在這些建構中出現的相關題目。 / In this thesis we use matrices to construct group divisible designs (GDDs). We list two type of constructions, the first type is -- due to W.H. Heamers -- A .crtimes. J + I .crtimes. D and use this construction we classify all the (m,n,k,. lambda.1, .lambda.2) GDD with r - .lambda.1 = 1 in three classes according to (i) A = 0 or J-I, (ii) A is the adjacency matrix of a strongly regular graph with .mu. - .lambda. = 1, (iii) J - 2A is the core of a skew-symmetric Hadamard matrix. The second type is A .crtimes. D + .Abar .crtimes. .Dbar , this type can construct many regular and semi-regular GDDs with b=4(r-.lambda.2). In the thesis we investigate related topics that occur in these constructions.