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Orthogonal arrays (OAs) are widely used in industrial experiments for factor screening. Suppose that only a few of the factors in the experiments turn out to be important. An OA can be used not only for screening factors, but also for detecting interactions among a subset of active factors. In this article a set of optimality criteria is proposed to assess the performance of designs for factor screening, projection, and interaction detection, and a three-step approach is proposed to search for optimal designs. Combinatorial and algorithmic construction methods are proposed for generating new designs. Permutations of levels are used for improving the eligibility and estimation efficiency of the projected designs. The techniques are then applied to search for best three-level designs with 18 and 27 runs. Many new, efficient, and practically useful nonregular designs are found and their properties are discussed.

In some practical situations the choice of defining generators matters even for the same defining contrast subgroup. Two such examples are blocking schemes for full and fractional factorial designs and split-plot fractional factorial designs. We propose an algorithm to find defining generators with extreme lengths for any s n − k designs, s being a prime power. Some illustrations of the method are given.

Blocking is commonly used in design of experiments to reduce systematic variation and increase precision of effect estimation. For 2n-k designs in 2p blocks, others have studied the confounding pattern between blocking effects and treatment effects and proposed some criteria to choose optimal blocking schemes. We propose alternative criteria that allow estimation of more lower-order effects. Extensions to three-level fractional factorial design are considered, and tables of useful designs are given.

In a quantal response study, there may be insufficient knowledge of the response relationship for the stimulus (or dose) levels to be chosen properly. Information from such a study can be scanty or even unreliable. A two-stage design is proposed for such studies, which can determine whether and how a follow-up (i.e., second-stage) study should be conducted to select additional stimulus levels to compensate for the scarcity of information in the initial study. These levels are determined by using optimal design theory and are based on the fitted model from the data in the initial study. Its advantages are demonstrated using a fishery study.

A general approach is proposed for constructing response surface designs of economical size for qualitative and quantitative factors. It starts with an efficient design (e.g. central composite design) for the quantitative factors and then partitions the design points into groups corresponding to different level combinations of the qualitative factors. Good designs are selected to ensure high estimation efficiency for models that reflect some stated objectives. The method is illustrated with the construction of designs for one qualitative factor. Some designs for two qualitative factors are constructed and extensions considered. Algorithmic construction of designs is briefly discussed.

SUMMARY An N x N Hadamard matrix can be used to construct a saturated 2 N-1 design with N runs. Furthermore, if an interaction column for two of the columns of the matrix is not fully aliased with any column of the matrix, this interaction column can be used as a supplementary column for studying an additional factor. For some small Hadamard matrices studied by Plackett & Burman (1946), the number of such interaction columns is very large, thus allowing the construction of supersaturated designs whose number of factors far exceeds the number of runs. A general method of construction along these lines is proposed. The efficiency of the constructed designs is studied by using three criteria.

SUMMARY For stratified samples with arbitrary numbers of primary sampling units per stratum, one can divide the units in the stratum into two groups and then apply the balanced half-samples method to the two groups thus formed. The efficiency loss of the grouped balanced half-samples variance estimator can be substantial. Use of a mixed orthogonal array, whose numbers of symbols correspond to the numbers of units per stratum, gives a set of balanced subsamples. Unlike that of Gupta & Nigam (1987), the proposed method of analysis gives a consistent variance estimator and bias reduction for nonlinear statistics. To further reduce the number of replicates, use of nearly orthogonal arrays is considered. The balanced half-samples method (McCarthy, 1966, 1969) is commonly used for estimating the variance of nonlinear statistics from stratified samples with two primary sampling units per stratum. These samples arise naturally if deep stratification is desired as in some large-scale surveys. The method is extended (Gurney & Jewett, 1975) to the case of p primary sampling units per stratum for prime p. This extension for p > 2 is not adopted in common practice since it requires a much larger number of replicates. Also, in many survey designs the numbers of primary sampling units are unequal. A common practice is to divide the units in the stratum into two groups and then to apply the balanced half-samples method to the two groups thus formed. This has gained popularity because the software for obtaining balanced half-samples is readily available and the number of replicates required is more economical. As shown in ? 2, the resulting variance estimators can be quite inefficient and their bias based on a conditional analysis can be considerable. The conditional analysis also suggests a method of grouping to reduce the bias. The degree of grouping depends on the tradeoff between computational economy and statistical efficiency. If the efficiency loss from grouping is not justifiable, one may draw replicates by using mixed orthogonal arrays with variable numbers of symbols to accommodate the different numbers of units or groups of units in the strata. As long as an orthogonal array is used for drawing replicates, we call the method balanced repeated replications. In ? 3 we show that the analysis method of Gupta & Nigam (1987) based on mixed orthogonal arrays leads to an inconsistent variance estimator for nonlinear statistics, and propose a method that is valid for variance estimation and bias reduction

A supersaturated design is a design whose run size is not large enough for estimating all the main effects. The goodness of multi-level supersaturated designs can be judged by the generalized minimum aberration criterion proposed by Xu and Wu [Ann. Statist. 29 (2001) 1066--1077]. A new lower bound is derived and general construction methods are proposed for multi-level supersaturated designs. Inspired by the Addelman--Kempthorne construction of orthogonal arrays, several classes of optimal multi-level supersaturated designs are given in explicit form: Columns are labeled with linear or quadratic polynomials and rows are points over a finite field. Additive characters are used to study the properties of resulting designs. Some small optimal supersaturated designs of 3, 4 and 5 levels are listed with their properties., Published at http://dx.doi.org/10.1214/009053605000000688 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A new design criterion, called the B criterion, is given for the situation where each experimental unit has some (categorical) covariate information. It directly addresses the question of balanced assignment over covariates, is simple in form, and does not involve matrix inversion. A simple and efficient algorithm for generating nearly baianced assignments is given. The basic element of the algorithm consists of switching and exchanging treatment assignments. The algorithm is illustrated on two examples.

A class of nearly balanced incomplete block designs is defined. This extends the concept of regular graph designs of John and Mitchell (1977) to the unequally replicated case. Some necessary conditions on the existence of such designs are derived. Methods of construction are given for some special cases. For five or six varieties, the 'best' nearly balanced incomplete block designs and their efficiencies are tabulated. (Author)

SUMMARY If an experiment is designed sequentially, repeated-sampling inference may not necessarily be made using distributional results that are valid for fixed designs. A few simple illustrative examples and alternative approaches are given. Sometimes the sequential nature of design can be ignored asymptotically. Links are forged with inference for stochastic processes and missing data problems.

In comparing two treatments, suppose the suitable subjects arrive sequentially and must be treated at once. Known or unknown to the experimenter there may be nuisance factors systematically affecting the subjects. Accidental bias is a measure of the influence of these factors in the analysis of data. We show in this paper that the random allocation design minimizes the accidental bias among all designs that allocate n, out of 2n, subjects to each treatment and do not prefer either treatment in the assignment. When the final imbalance is allowed to be nonzero, optimal and efficient designs are given. In particular the random allocation design is shown to be very efficient in this broader setup.

Let S be a random nonnegative definite matrix and let G be an orthogonal matrix such that S=GDG′, where D is the diagonal matrix of the latent roots of S. In this note, we prove the a.s. existence of G−111, where G=[ G 11 G 12 G 21 G 22 ] , under some weak conditions on the distribution of S.

SUMMARY It is shown that confidence regions for nonlinear parameters constructed by the repeated-sampling principle, are asymptotically valid for sequential designs in general linear models. The related questions of consistency of parameter estimators and convergence of sequential design to an optimal design are answered positively. An empirical finding of Ford & Silvey (1980) is given a theoretical justification.

SUMMARY A general class of estimators of the variance of the ratio estimator is considered, which includes two standard estimators vo and v2 and approximates another estimator VH suggested by Royall & Eberhardt (1975). Asymptotic expansions for the variances and biases of the proposed estimators are obtained. Based on this we obtain optimal variance estimators in the class and compare the relative merits of three estimators vo, v1 and v2 without any model assumption. Under a simple regression model a more definite comparison of vo, v1 and v2 is made in terms of variance and bias. Some key word8: Asymptotic expansion; Bias; Optimal variance estimator; Ratio estimator; Superpopulation; Variance.

When the saving of sample size is an important consideration, sequential design of experiments is often used. By efficiently utilizing the information in the past experiments, it determines how the next experiment should be conducted. Statistical theory for sequential designs has been developed for normal and binomial variations. For the problem of determining the solution of an unknown nonlinear equation, we have developed a class of sequential design procedures that can handle very general variations described by the generalized linear models. In special cases it includes a new adaptive version of the Robbins-Monro stochastic approximation and a maximum likelihood recursion scheme for quantal responses. Its relation to the stochastic approximation and the role the link function plays are studied. Theoretical issues such as consistency, robustness, asymptotic normality and second-order properties are discussed.

The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles. This well-known deficiency can be rectified by using a more general jackknife with $d$, the number of observations deleted, depending on a smoothness measure of the point estimator. Our general theory explains why jackknife works or fails. It also shows that (i) for "sufficiently smooth" estimators, the jackknife variance estimators with bounded $d$ are consistent and asymptotically unbiased and (ii) for "nonsmooth" estimators, $d$ has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness. Improved results are obtained for several classes of estimators. In particular, for the sample $p$-quantiles, the jackknife variance estimators with $d$ satisfying $n^{1/2}/d \rightarrow 0$ and $n - d \rightarrow \infty$ are consistent and asymptotically unbiased.

For a linear regression model, the necessary and sufficient condition for the asymptotic consistency of the least squares estimator is known. An analogous condition for the nonlinear model is considered in this paper. The condition is proved to be necessary for the existence of any weakly consistent estimator, including the least squares estimator. It is also sufficient for the strong consistency of the nonlinear least squares estimator if the parameter space is finite. For an arbitrary compact parameter space, its sufficiency for strong consistency is proved under additional conditions in a sense weaker than previously assumed. The proof involves a novel use of the strong law of large numbers in $C(S)$. Asymptotic normality is also established.