Kirti R. Shah

Theory of optimal designs

1. Optimality Criteria In Design of Experiments.- 1. General Objectives.- 2. The Linear Model Set-up.- 3. Choice of Optimality Criteria.- References.- 2. Block Designs: General Optimality.- 1. Introduction.- 2. Universal Optimality of the BBDs.- 3. Optimality of Some Classes of Asymmetrical Designs w.r.t the Generalized Criteria.- References.- 3. Block Designs: Specific Optimality.- 1. Introduction.- 2. E-optimal Designs.- 3. Efficiency Factor and A-optimal Designs.- 4. MV-optimal Designs.- 5. D-optimal Designs.- 6. Regular Graph Designs and John-Mitchell Conjecture.- 7. Optimal Designs with Unequal Block Sizes.- References.- 4 Row-Column Designs.- 1. Introduction.- 2. Universal Optimality of the Regular GYDs.- 3. Nonregular GYDs: Specific Optimality Results.- 4. Optimality of Other Row-Column Designs.- References.- 5. Mixed Effects Models.- 1. Introduction.- 2. Optimality Aspects of Block Designs Under a Mixed Effects Model.- 3. Optimality of GYDs Under a Mixed Effects Model.- 4. Concluding Remarks.- References.- 6. Repeated Measurements Designs.- 1. Introduction.- 2. The Linear Model(s), Definitions and Notations.- 3. Universal Optimality of Strongly Balanced Uniform RMDs.- 4. Universal Optimality of Nearly Strongly Balanced Uniform RMDs.- 5. Universal Optimality of Balanced Uniform RMDs.- 6. Concluding Remarks.- References.- 7. Optimal Designs For Some Special Cases.- 1. Introduction.- 2. Models with Correlated Observations.- 3. Models with Covariates.- 4. Designs for Comparing Treatments vs. Control.- References.- 8. Weighing Designs.- 1. Introduction.- 2. A Study of Chemical Balance Weighing Designs.- 3. A Study of Spring Balance Weighing Designs.- 4. Optimal Estimation of Total Weight.- 5. Miscellaneous Topics in Weighing Designs.- References.- Author Index.

Repeated Measurements Designs

In the preceding Chapters, we dealt with optimality aspects of traditional block designs and/or row-column designs, assuming fixed/mixed effects models. In many fields of scientific investigations, experiments are to be designed in such a manner that each experimental unit (eu) receives some or all of the treatments, one at a time, over a certain period of time. Such designs have been discussed in the literature under various names, viz., cross-over or change-over designs, time series designs or before-after designs in some special cases. Following Hedayat and Afsarinejad (1975), we will call such designs as Repeated Measurement Designs (RMDs). In effect, an RMD can be viewed as a row-column design with a set of eu’s displayed across the columns and a set of periods (of time) displayed across the rows wherein the eu’s receive some or all of a given set of treatments, one at a time, over these periods. The peculiarity of such an experiment is that any treatment applied to a unit in a certain period influences the response of the unit not only in the current period but also leaves residual effects in the following periods. In practice, only the first order residual effect (carry-over effect) i.e., residual effect of any treatment up to just the next period is of importance. For a general review of such designs, including practical applications, reference is made to Hedayat and Afsarinejad (1975). An extreme form of an RMD is the one in which only one experimental unit is involved in the entire experiment. For such experiments, Finney and Outhwaite (1955, 1956) introduced the notions of serially balanced sequences of types 1 and 2.

Exact Variance of Combined Inter- and Intra-Block Estimates in Incomplete Block Designs

Abstract This article presents methods for evaluating exact variance of combined inter- and intra-block estimates of treatment effects in incomplete block designs. The results are valid for all incomplete block designs and hold for a wide class of combined estimates including most in the literature. Tables of exact variance for some estimates are presented for some balanced designs and for some partially balanced designs.

On some aspects of row-column designs

Abstract We consider row-column designs with unequal replication. However the results obtained hold for equal replication designs also. Some results are derived concerning the projection of treatment contrasts onto the mutually orthogonal subspaces viz., the row space, the column space and the “interaction” space. Efficiency bounds for row-column designs are derived which depend on a commutativity property of the incidence matrices. Canonical efficiencies are investigated also and finally some examples which illustrate the results are presented.

On the optimality of a class of row-column designs

Abstract We establish optimality of a class of row-column designs. This includes the designs given by Anderson and Eccleston. The optimality result is valid for any Schur criterion and thus holds for the commonly used criteria such as A-, D- or E-optimality.

Uncertain Resources and Optimal Designs : Problems and Perspectives

ABSTRACT: Considered is an experimental situation where an experimenter has a certain amount of guaranteed fund at the current period and a 100% chance of an enhanced fund to be made available at a later date. The objective is to decide on an optimal planned experiment. Ideally, the experimenter should start with an optimal experiment and extend it in an optimal fashion as and when the additional fund is made available. This, however, may not lead to an optimal strategy in the long run. We speciaiize to the set‐up of a block design and discuss variovs aspects of this prablcm, after properly formulating the same in proper perspectives. Severa 1 illustrative examples are presented to highlight the compiexities involved.

On the Determination and Construction of Optimal Block Designs in the Presence of Linear Trends

Abstract This article considers experimental situations in which v treatments are to be applied to experimental units arranged in b blocks of size k and where there may be unknown or uncontrollable linear trends (possibly different) within blocks. Methods are given for determining and constructing universally optimal designs for such situations.

Recovery of interblock information: an update

Abstract This paper presents an updated review of results on inference in block designs based on combined inter- and intra-block models. Detailed treatment for block designs is followed by a brief description of results for row-column designs. Our approach is to seek combined estimators which are uniformly better than those based on intra-block information only.

Optimality aspects of row—column designs with non-orthogonal structure

Abstract This paper deals with row-column designs in which the row-column incidence patterns are not necessarily orthogonal. A set of sufficient conditions are obtained for universally optimal designs (for compar- ing treatment effects) to exist in the usual fixed effects 3-factor additive model. It is shown that if these conditions on the row-column incidence pattern are satisfied, a design can always be constructed. Some methods of construction are given for specific row-column incidence patterns. Various other related issues are also discussed.

Block Designs: General Optimality

In this Chapter we intend to present the essential results known so far regarding optimality of certain classes of block designs w.r.t. some general optimality criteria. Unless otherwise stated, the competing class of designs, to be denoted by D(b,v,k), will comprise of all connected designs in which v treatments are compared in b blocks each of size k. While presenting the results on optimal block designs we will deliberately restrict ourselves to the experimental set-up of blocks of equal sizes. With blocks of unequal sizes, the assumption of homogeneity of error variances is itself questionable and, further, the optimality results are seen to depend on too many design parameters. In Chapter Three (Section 7), we have mentioned some current work on optimality results involving the latter set-up.