Agnes M. Herzberg

Fourth order rotatability

Fourth order rotatable designs are discussed. A general k, design moment inequality is given. The variance function for two-factor designs is derived, and plotted for a specific design. A minimum point set requirement for two-factor designs is established, thus enabling one to form an infinity of such designs. Some difficulties in obtaining deLigns for k>2 are described. Some questions are posed for future work.

The robustness of chain block designs and coat-of-mail designs

Chain block designs are relatively vulnerable to loss of information when missing values or outliers may occur. An alternative class of designs, coat-of-mail designs, are proposed and the relative robustness of the two types of design are compared.

Equi‐information robust designs: Which designs are possible?

The effects of missing observations in a designed experiment are reviewed. Conditions are determined for a design to retain equal information, i.e. the same generalized variance of unknown parameters, when either any single or any pair of observations is lost. Some examples of designs with this property are given. Although there are many designs which retain equal information for the loss of exactly t observations, where t = 1,2,3,…, it is shown that it is not possible to obtain any design which retains equal information when any one and any two and also any three observations are missing. n n n nNous rappelons les effets que peuvent causer les observations manquantes dans le cadre dune experience planifiee. Sous certaines conditions, nous demontrons que linformation contenue dans le plan dexperience est constante, en ce sens que la variance generalisee des parametres inconnus demeure la měme quelle que soit lobservation ou la paire dobservations manquantes. Nous donnons quelques exemples de plans dexperience dotes de cette propriete. De facon generale, il existe aussi des plans a information constante dans les situations ou exactement t = 1,2,3,…, observations sont declaree manquantes. Toutefois, nous demontrons quil nexiste aucun plan dexperience qui renferme la měme information pour toute observation, paire dobservations et triplet dobservations manquantes.

Equally spaced design points in polynomial regression: A comparison of systematic sampling methods with the optimal design of experiments

In some situations an experimenter may desire to have equally spaced design points. Three methods of obtaining such points on the interval [—1,1]—namely systematic random sampling, centrally located systematic sampling, and a purposive systematic sampling method which includes the endpoints - 1 and 1 as two of the design points-are evaluated under the D-optimal and G-optimal criteria. These methods are also compared to the optimal designs in polynomial regression and to the limiting designs of Kiefer and Studden (1976). n n n nDans certains cas, un experimentateur peut juger necessaire que le plan de son experience consiste en des points espaces egalement. Cet article evalue, selon les criteres de D-optimalite et de G-optimalite, trois methodes permettant dobtenir un plan de points espaces egalement sur lintervalle [—1,1]. Les methodes etudiees sont les suivantes: lechantillonnage aleatoire systematique, lechantillonnage systematique central ainsi quune methode dechantillonnage systematique qui inclut au nombre des points du plan les valeurs extrěmes — 1 et 1. On etablit aussi une cornparaison des trois methodes de plan proposees avec celles des plans optimaux en regression polynomiale et des plans avec limites de Kiefer & Studden (1976).

A comparison of equally spaced designs with different correlation structures in one and more dimensions

Equally spaced designs are compared using the generalized variance as a measure of efficiency. Results for polynomial models are derived on the increased efficiency arising from increasing the number of design points when the regions are fixed and when the regions are expanded. The effects of dependence among the observations on these results are studied by considering a particular family of stationary correlated error structures.

The design of experiments for correlated error structures: Layout and robustness†

Observations with correlated error structures are sometimes unavoidable. Appropriate designs and analyses are reviewed for such situations. Serious problems can occur if conventional designs and analyses are used when correlated errors and layout are ignored or when the error structure is not known. Robust designs are discussed which guard against these problems. n n n nIl nest pas toujours possible deviter les situations qui engendrent des observations correlees, et lapplication inconsideree dune analyse conventionnelle a ce genre de situation peut provoquer de serieuses difficultes. Cet article presente des schemas experimentaux qui tiennent compte de la dependance entre les erreurs et traite de leur analyse. Certains schemas sont suffisamment robustes pour sappliquer aux cas ou la structure des erreurs est inconnue.

An investigation of first‐order and second‐order designs for extrapolation outside a hypersphere

Methods for selecting a first-order or second-order rotatable response surface design when both variance and bias error exist are applied to a situation in which it is desired to extrapolate the fitted model in all directions outside of a sphere within which all the experiments are to be made. The extrapolation region is a spherical shell. n n n nDans cet article on utilise des methodes pour choisir un modele a rotations de premier ou deuxieme degre pour la surface representant la reaction en fonction des stimuli quand il y a des erreurs dans le biais et la variance pour extrapoler, dans toutes les directions, le modele ajuste a lexterieur dune sphere dans laquelle toutes les observations ont ete obtenues. La region dextrapolation est une coquille spherique.

A Ridge-Regression Sidelight

Abstract A useful matrix result is applied in ridge regression to cast light on the form of the ridge-regression residual sum of squares as a power series in the ridge parameter. An example illustrates that the series may diverge or else converge very slowly, depending on the value of the ridge parameter.

The monte-carlo comparison of two criteria for the selection of models

A Monte-Carlo comparison is made of two criteriaCp and the jackknife, for the selection of polynomial models. Several measures of comparison of these criteria are introduced. Examples with polynomial and nonlinear models are given.