Mike Jacroux

The Construction of Trend-Free Run Orders of Two-Level Factorial Designs

Abstract In experimental situations where a factorial design with all factors occurring at two levels is to be run in a time sequence, the usual advice given to the experimenter is that the order of runs should be randomized before the experiment is performed; however, randomization may lead to an undesirable run order. For example, in a factory experiment, there may be a certain learning process that occurs over time as a result of making level changes in the factors being studied, or there may be equipment wear-out. In either case the observations obtained will be affected by uncontrollable variables that are highly correlated with the time or position in which they occur, and randomization may lead to a run order in which the estimates of factor effects are adversely effected by the presence of the trend. Joiner and Campbell (1976) gave some more specific examples as to when trend effects can occur in sequential experiments. Therefore, it is important to consider systematic designs in which the estimat...

On the optimality of chemical balance weighing designs

Abstract In this paper we consider the problem of optimally weighing n objects with N weighings on a chemical balance. Several previously known results are generalized. In particular, the designs shown by Ehlich (1964a) and Payne (1974) to be D-optimal in various classes of weighing designs where N≡2 (mod4) are shown to be optimal with respect to any optimality criterion of Type I as defined in Cheng (1980). Several results on the E-optimality of weighing designs are also given.

On the determination and construction of MV-optimal block designs for comparing test treatments with a standard treatment

Abstract In this paper we consider the problem of determining optimal block designs in experimental situations where υ test treatments of interest are to be compared with some standard treatment in b blocks of size k . A design is said to be MV-optimal in such an experimental setting if it minimizes the maximal variance with which the elementary treatment contrasts involving the standard treatment are estimated. A general method for finding an MV-optimal design is suggested and some sufficient conditions are given under which a design obtained using the proposed method will be MV-optimal. Some examples are also given to show how the results obtained can be applied.

Some sufficient conditions for the type 1 optimality of block designs

Abstract In this paper, we investigate the problem of determining block designs which are optimal under type 1 optimality criteria within various classes of designs having υ treatments arranged in b blocks of size k. The solutions to two optimization problems are given which are related to a general result obtained by Cheng (1978) and which are useful in this investigation. As one application of the solutions obtained, the definition of a regular graph design given in Mitchell and John (1977) is extended to that of a semi-regular graph design and some sufficient conditions are derived for the existence of a semi-regular graph design which is optimal under a given type 1 criterion. A result is also given which shows how the sufficient conditions derived can be used to establish the optimality under a specific type 1 criterion of some particular types of semi- regular graph designs having both equal and unequal numbers of replicates. Finally,some sufficient conditions are obtained for the dual of an A- or D-optimal design to be A- or D-optimal within an appropriate class of dual designs.

The A-Optimality of Block Designs for Comparing Test Treatments with a Control

Abstract In this article we consider experimental settings in which it is desired to optimally compare v test treatments to a standard or control treatment and the experimental units are to be arranged in b blocks of size k. This problem has received a good deal of attention in recent years. Majumdar and Notz (1983) developed some sufficient conditions that can be used to establish the A-optimality of balanced treatment block designs (BTBDs) in these situations, and several authors have since used these sufficient conditions to show the A-optimality of some specific BTBDs as well as characterize some infinite families of A-optimal BTBDs. Here we develop some further sufficient conditions for A-optimality that generalize those given by Majumdar and Notz (1983) and can often be used to establish the A-optimality of BTBDs not covered by the results of Majumdar and Notz (1983), as well as the A-optimality of certain types of designs called group-divisible treatment designs. Several examples are given to i...

On the usage of refined linear models for determining N -way classification designs which are optimal for comparing test treatments with a standard treatment

SummaryIn this paper we consider experimental settings in whichv test treatments are to be compared to some control or standard treatment and where heterogeneity needs to be eliminated inn-directions. Using techniques similar to those used by Kunnert (1983,Ann. Statist.,11, 247–257) concerning the determination of optimal designs under a refined linear model, some methods are given for constructingn-way classification designs which areA- andMV-optimal for estimating elementary treatment differences involving the standard treatment fromm-way classification designs,m<n, which areA- andMV-optimal for estimating the same treatment differences. Examples are given for the casen=2 to show how the results obtained can be applied.

A note on the determination and construction of minimal orthogonal main-effect plans

In this article, I derive sufficient conditions for an orthogonal main-effect plan having k factors at S i levels, i = 1, …, k, to have a minimal number of observations. These sufficient conditions are then used to show that many of the orthogonal main-effect plans given prcviously in the literature have minimal numbers of observations.

Some optimal designs for comparing a set of test treatments with a set of controls

In this paper, we give a detailed study of the problem of optimally comparing a set of t test treatments to a set of s controls under a 0-way elimination of heterogeneity model. The relationships between designs that are A and MV-optimal for comparing the test treatments to the controls and those that are A and MV-optimal for comparing all treatments are also studied.

On the determination and construction of optimal row-column designs having unequal row and column sizes

In this paper we consider experimental situations requiring usage of a row-column design where v treatments are to be applied to experimental units arranged in b1 rows and b2 columns where row i has size k1i, i=1,..., b1 and column j has size k2j, j=1,..., b2. Conditions analogous to those given in Kunert (1983, Ann. Statis., 11, 247–257) and Cheng (1978, Ann. Statist., 6, 1262–1272) are given which can often be used to establish the optimality of a given row-column design from the optimality of an associated block design. In addition, sufficient conditions are derived which guarantee the existence of an optimal row-column design which can be constructed by appropriately arranging treatments within blocks of an optimal block design.

Some sufficient conditions for the E- and MV-optimality of block designs having blocks of unequal size

SummaryIn this paper we consider experimental situations in which ν treatments are to be tested inb blocks wherebi blocks containki experimental units,i=1,...,p, k1<k2<...<kp. The idea of a group divisible (GD) design is extended to that of a group divisible design with unequal block sizes (GDUB design) and then a number of results concerning the E- and MV-optimality of GD designs are generalized to the case of GDUB designs.