Bikas K. Sinha

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.

Statistical methods in assessing agreement: Models, issues, and tools

Measurements of agreement are needed to assess the acceptability of a new or generic process, methodology, and formulation in areas of laboratory performance, instrument or assay validation, method comparisons, statistical process control, goodness of fit, and individual bioequivalence. In all of these areas, one needs measurements that capture a large proportion of data that are within a meaningful boundary from target values. Target values can be considered random (measured with error) or fixed (known), depending on the situation. Various meaningful measures to cope with such diverse and complex situations have become available only in the last decade. These measures often assume that the target values are random. This article reviews the literature and presents methodologies in terms of “coverage probability.” In addition, analytical expressions are introduced for all of the aforementioned measurements when the target values are fixed and when the error structure is homogenous or heterogeneous (proportional to target values). This article compares the asymptotic power of accepting the agreement across all competing methods and discusses the pros and cons of each. Data when the target values are random or fixed are used for illustration. A SAS macro program to compute all of the proposed methods is available for download at http://www.uic.edu/~hedayat/.

Design Issues for Generalized Linear Models: A Review

Generalized linear models (GLMs) have been used quite effectively in the modeling of a mean response under nonstandard conditions, where discrete as well as continuous data distributions can be accommodated. The choice of design for a GLM is a very important task in the development and building of an adequate model. However, one major problem that handicaps the construction of a GLM design is its dependence on the unknown parameters of the fitted model. Several approaches have been proposed in the past 25 years to solve this problem. These approaches, however, have provided only partial solutions that apply in only some special cases, and the problem, in general, remains largely unresolved. The purpose of this article is to focus attention on the aforementioned dependence problem. We provide a survey of various existing techniques dealing with the dependence problem. This survey includes discussions concerning locally optimal designs, sequential designs, Bayesian designs and the quantile dispersion graph approach for comparing designs for GLMs.

Optimal designs for binary data under logistic regression

A unified approach is presented for the derivation of D- and A-optimal designs for binary data under the two-parameter logistic regression model. The optimal design is constructed for the estimation of several pairs of parameters. The E-optimal design is also obtained in some cases.

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.

Surveillance Strategies for Detecting Changepoint in Incidence Rate Based on Exponentially Weighted Moving Average Methods

Surveillance is a major issue in the todays world. The need to develop adequate surveillance strategies is genuine in many spheres of human activity. In this article we focus on a specific problem of surveillance and discuss some related statistical issues. This specific problem deals with a possible change in the incidence rate of an event (to a higher value) when a system is studied at discrete time points. We apply the exponentially weighted moving average methods for detecting an increased incidence rate per exposure unit of an event. Different measures of evaluation, suitable in different types of applications, such as the expected delay, out-of-control average run length, and probability of successful detection, are studied. Analytical bounds are provided for those measures of evaluation. Results from intensive simulations indicate that the analytical bounds perform fairly well when the weight parameter is large.

Optimal experimental designs for models with covariates

Abstract The model refers to a treatment design or to a block-treatment design in the presence of non-stochastic covariates, attached to each experimental unit. The problem is that of most efficient estimation of covariates parameters on one side and the treatment contrasts and/or block contrasts on the other side. If the components in the two sides are “orthogonal”, then we can invoke optimality [in some sense] separately in each side and thereby characterize optimal designs in such a set-up. Following Lopes Troya (J. Statist. Plann. Inference 6 (1982a) 373, J. Statist. Plann. Inference 7 (1982b) 49), we investigate the underlying combinatorial problems in the context of CRD, RBD and BIBD in order to accommodate maximum number of covariates. Hadamard matrices and mutually orthogonal Latin squares play a central role in this study.

Optimal designs in growth curve models - II Correlated model for quadratic growth : optimal designs for parameter estimation and growth prediction

This is a follow up of a recent paper on the study of the optimality aspects of linear growth models with correlated errors. In this paper, we examine optimality aspects of quadratic growth models with correlated errors and provide optimal designs for parameter estimation and growth prediction. In the former case we examine A-, D- and E-optimal designs and in the latter case we examine A-optimal designs, under two different correlation structures. In the process, we also examine the robustness of optimal designs with respect to the values of the correlation coefficient.

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.

Reliability estimation based on ranked set sampling

In this paper we consider the problem of estimating the reliability of an exponential component based on a Ranked Set Sample (RSS) of size n. Given the first r observations of that sample, 1≤r≤n, we construct an unbiased estimator for this reliability and we show that these n unbiased estimators are the only ones in a certain class of estimators. The variances of some of these estimators are compared. By viewing the observations of the RSS of size n as the lifetimes of n independent k-out-of-n systems, 1≤k≤n, we are able to utilize known properties of these systems in conjunction with the powerful tools of majorization and Schur functions to derive our results.