On theE-optimality of certain asymmetrical designs under mixed effects model

TheE-optimality of the following designs within the class of all proper and connected designs with givenb, k andv under mixed effects model are established.

1. A group divisible design with λ₂ = λ₁ + 1.
2. A group divisible design with λ₁ = λ₂ + 1 and group size 2.
3. A linked block design.
4. The dual of design (i)
5. The dual of design (ii).

All these designs are known to satisfy the same optimality property under fixed effects model whenk<v, while the design (i) is known to beE-optimal even whenk>v. From the results proved here, theE-optimality of designs (ii, (iii), (iv) and (v) under fixed effects model in the situation whenk >v also follows.     

A note on the determination of A-optimal type block designs under fixed and mixed effects models

In this paper, we consider the problem of estimating a selected set of contrasts between v treatments using a block design consisting of b blocks of size k. Traditionally, the construction of A-optimal block designs for such situations has been carried out assuming a fixed effects model. In this paper, we show that A-optimal designs constructed under a fixed effects model are robust in the sense that these designs have maximal minimal efficiency when considered among all available designs and under all possible mixed effects models. AMS 1991 subject classifications: Primary 62K05; Secondary 62K10     

Comparison of test vs. control treatments using distance optimality criterion

We consider the problem of comparison of one test treatment (τ₀) with a set of v control treatments (τ₁, τ₂, …, τ_v) using distance optimality [DS-optimality] criterion introduced by Sinha (1970) in some treatment-connected design settings. It turns out that the nature of DS-optimal designs is quite similar to that for the usual A−, D− and E− optimality criteria. However, the optimality problem is quite complicated in most situations. First we deal with the CRD model and derive DS-optimal allocations for a given set of treatments. The results are almost identical to the A-optimal allocations for such problems. Then we consider a block design set-up and examine the nature of DS-optimal designs. In the process, we introduce the method of weighted coverage probability and maximize the resulting expression to obtain an optimal design. Received: December 1999     

Screening properties of certain two-level designs

Two-level screening designs are appropriate for situations where a large number of factors (q) is examined but relatively few (k) of these are expected to be important. It is not knownwhich of theq factors will be the important ones, that is, it is not known whichk dimensions of the experimental space will be of further interest. After the results of the design have received a first analysis, the design will be projected into thek dimensions of interest. These projections are investigated for Plackett and Burman type-screening designs withq23 factors, andk=3, 4, and 5.     

Optimal extended complete block designs for dependent observations

Optimal designs under general dependence structures are usually difficult to specify theoretically or find algorithmically. However, they can sometimes be found for a specific dependence structure and a particular parameter value. In this paper, a class of generalized binary block designs with t treatments and b blocks of size k>t is considered. Each block consists of h consecutive complete blocks and, at the end, an incomplete block of size k−ht (if k > ht). For a suitable number of blocks, a universally optimal design is found for a first-order stationary autoregressive process with positive correlations. Optimal generalized binary designs and balanced block designs are also considered. Some constructions for a universally optimal design are described. A negative dependence parameter, and some other dependence structures, are also considered.     

Optimal block designs for diallel crosses

Dey and Midha (Biometrika 83(2):484–489, 1996) constructed optimal block designs for complete diallel cross experiment using triangular partially balanced incomplete block designs with two associate classes. They listed optimal block designs for the lines in the range from 5 ≤ v ≤ 10. In this paper, we are also proposing additional optimal block designs for complete diallel cross experiment using two associate class partially balanced block designs for some additional values of v. Our method yields designs for proper and non-proper settings for complete diallel cross experiments. The proper and non proper designs are optimal in the sense of Kempthorne (Genetics 41:451–459, 1956) and non-proper designs are universally optimal in the sense of Kiefer (A survey of statistical design and linear models, North Holland, Amsterdam, 1975). The list of practically important designs is also given.     

Choice of optimal initial designs in sequential experiments

Combined-optimal designs (Li and Lin, 2003) are obviously the best choices for the initial designs if we partition the experiment into two parts with equal size to obtain some information about the process, especially for the case not considering the blocking factor. In this paper, the definition of combined-optimal design is extended to the case when blocking factor is significant, and this new class of designs is called blocked combined-optimal designs. Some general results are obtained which relate 2^k−p_III initial designs with their complementary designs when , where n=2^k−p. By applying these results, we are able to characterize 2^k−p_III combined-optimal designs or blocked combined-optimal designs in terms of their complementary designs. It is also proved that both 2^k−p_III combined-optimal and blocked combined-optimal designs are not minimum aberration designs when and n−1−k > 2. And some combined-optimal and blocked combined-optimal designs with 16 and 32 runs are constructed for illustration. 2000 Mathematics Subject Classifications: 62K15, 62K05     

Multi-level k-circulant Supersaturated Designs

In this paper we present a new method for constructing multi-level supersaturated designs with n rows, m columns and the equal occurrence property. We investigate the existence of multi-level supersaturated designs using a single generator vector and its k-cyclic permutations as rows. We find the conditions needed, in order this vector to generate a balanced supersaturated design. These conditions are simplified for the case of three level supersaturated designs. By using the proposed method three level supersaturated designs are constructed and explored. Moreover, many new, optimal and near optimal, multi-level supersaturated designs are provided as well.     

Robustness of Williams Square change-over designs

In literature, Williams Square change-over designs balanced for first residual effects of treatments have been proved to be universally optimal. Here, an attempt has been made to show that these designs are robust against missing of last α [≤v−1; v being the number of periods in the design for v treatments] observations from an experimental unit. Received May 2000     

SomeE andMV-optimal row-column designs having equal numbers of rows and columns

In this paper we consider experimental situations in whichv treatments are to be tested using a row-column design consisting ofb columns andb rows and wherev does not divideb ². Some sufficient conditions are obtained for a design to beE orMV-optimal in such an experimental setting and methods for constructing row-column designs satisfying the sufficient conditions obtained are also given. This research was supported by NSF Grant No. DMS-8401943.     

Evaluation of inequivalent projections of Hadamard matrices of order 24

Screening designs are useful for situations where a large number of factors (q) is examined but only few (k) of these are expected to be important. It is of practical interest for a given k to know all the inequivalent projections of the design into the k dimensions. In this paper we give all the (combinatorially) inequivalent projections of inequivalent Hadamard matrices of order 24 into k=3,4 and 5 dimensions, as well as their frequencies. Then, we sort these projections according to their generalized resolution, generalized aberration and centered L₂-discrepancy measure of uniformity. Then, we study the hidden projection properties of these designs as they are introduced by Wang and Wu (1995). The hidden projection property suggests that complex aliasing allows some interactions to be estimated without making additional runs.     

A note on reliability properties of k-record statistics

LetY _k,n denote the nth (upper) k-record value of an infinite sequence of independent, identically distributed random variables with common continuous distribution function F. We show that if the nth k-record valueY _k,n has an increasing failure rate (IFR), thenY _l,n (l<k) andY _k+1,n+1(n≤k+1) also have IFR distributions. On the other hand, ifY _k,n has a decreasing failure rate (DFR), thenY _l,n (1>k) has also a DFR distribution. We also present some results concerning log convexity and log concavity ofY _k,n .     

A note on connectedness of block designs

Summary We give a simple proof for the well known result that a block design (for an additive, fixed effects block model withv treatments inb blocks) is connected iff its ℰ-matrix has rankv-1.     

On the application of the Minkowski-Farkas theorem to sampling designs

In the paper, we consider the following problem: Let {π_k} be a sequence satisfying 0π_kΣ1 (k=1,…, N) and π=n.Tben, is there an unordered sampling design such that, for each k=1,…N, the inclusion probability of unit k is equal to π? It is shown that it can be solved by the straightforward application of the Minkowski-Farkas theorem.     

Optimality and construction of some rectangular designs

We obtain a sufficient condition forE-optimality of equireplicate designs. As an application, we proveE-optimality of certain types of three-class PBIBDs based on rectangular association scheme — in short — rectangular designs. These designs turn out to be highly efficient with respect to theA-criterion as well. We also observe that these designs, though themeselves not regular graph designs (RGD's) are yet strictlyE-better than every competing RGD, wheneverv≥26 andv=2 (mod 4). This provides an infinite series of counter examples to the conjecture of John and Mitchell (1977). We also present two methods of construction of the rectangular designs. Apart from providing infinitely many examples of the designs provedE-optimal in this paper and in Cheng and Constantine (1986), this construction also provides — as a special case — the first known infinite series of most balanced group divisible designs, which were proved optimal with respect to all type 1 criteria by Cheng (1978).     

A new interpretation of optimality forE-optimal designs in linear regression models

The optimal design problem for the estimation of several linear combinationsc′ _l ϑ (l=1, …,m) is considered in the usual linear regression modely=f′(x)ϑ (f(x) ∈ ℝ ^k ,ϑ ∈ ℝ ^k ). An optimal design minimizes a (weighted)p-norm of the variances of the least squares estimates for the different linear combinationsc′ _l ϑ. A generalized Elfving theorem is used to derive the relation of the new optimality criterion to theE-optimal design problem. It is shown that theE-optimal design for the parameterϑ minimizes such a (weighted)p-norm whenever the vectorc=(c′ ₁, …, c′_k)′ is an inball vector of a symmetric convex and compact “Elfving set” in.     

U-Optimum row-columns designs for the comparison of two treatments

Summary AllU-optimumu x v designs for the comparison of two treatments are to be characterized here — the row and column figuresu andv being arbitrary.     

A Lower Bound for the Centered <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>-Discrepancy on Asymmetric Factorials and its Application

The role of uniformity measured by the centered L ₂-discrepancy (Hickernell 1998a) has been studied in fractional factorial designs. The issue of a lower bound for the centered L ₂-discrepancy is crucial in the construction of uniform designs. Fang and Mukerjee (2000) and Fang et al. (2002, 2003b) derived lower bounds for fractions of two- and three-level factorials. In this paper we report some new lower bounds for the centered L ₂-discrepancy for a set of asymmetric fraction factorials. Using these lower bounds helps to measure uniformity of a given design. In addition, as an application of these lower bounds, we propose a method to construct uniform designs or nearly uniform designs with asymmetric factorials.     

Streuungszerlegung für Paar-Vergleiche

Zusammenfassung Pl?ne für Paar-Vergleich Versuche werden gegeben und die Auswertung solcher Versuche wird beschrieben. Die Arbeit st?tzt sich auf das vonBock (1958) abge?nderte Modell vonThurstone undMosteller (1951a). Die Pl?ne sind so, da? sie zu unabh?ngig verteilten Fehlern führen. Zu ihrer Auswertung kann bedenkenlos die Streuungszerlegung angewendet werden. Die Konstruktion von ausgewogenen Pl?nen (allek(k−1)/2 m?glichen Paare werden verglichen) wird beschrieben und teilweise ausgewogene Pl?ne (nicht alle m?glichen Paare werden verglichen) für bis zu 10 Objekte sind tabelliert. Es wird gezeigt, da? die Auswertung solcher Versuche mit der Auswertung von Versuchen in unvollst?ndigen Bl?cken identisch ist. Zur leichteren Auswertung der teilweise ausgewogenen Versuche sind die Inversen der Koeffizientenmatrix der Normalgleichungen tabelliert. Ein Anwendungsbeispiel aus der Wollforschung wird durchgerechnet.

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Summary Designs for paired comparisons are given and their analysis is discussed. The paper is based onBock’s modification (1958) ofThurstone andMosteller’s (1951a) model. The designs lead to independently distributed errors and analysis of variance can be applied. The construction of balanced designs (in which allk(k−1)/2 possible comparisons are made) is described and partially balanced designs (in which not all possible comparisons are made) for up to 10 objects are tabulated. It is shown that the analysis of paired comparison experiments is identical to the analysis of incomplete block experiments. To simplify the analysis of the partially balanced designs the inverses of the matrices comprising the coefficients of the normal equations are tabulated. A worked example of application to a problem in wool research is given.

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Die Arbeit wurde im South African Wool Textile Research Institute, Rhodes University, Grahamstown, Südafrika, geschrieben.     

A note on generalized wald’s method

Let {v _n(θ)} be a sequence of statistics such that whenθ =θ ₀,v _n(θ ₀) N _p(0,Σ), whereΣ is of rankp andθ εR ^d. Suppose that underθ =θ ₀, {Σ _n} is a sequence of consistent estimators ofΣ. Wald (1943) shows thatv _n ^T (θ ₀)Σ _n ⁻¹ v _n(θ ₀) x ²(p). It often happens thatv _n(θ ₀) N _p(0,Σ) holds butΣ is singular. Moore (1977) states that under certain assumptionsv _n ^T (θ ₀)Σ _n ⁻ v _n(θ ₀) x ²(k), wherek = rank (Σ) andΣ _n ⁻ is a generalized inverse ofΣ _n. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (Σ _n) =k forn sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier to verify, assumptions. Research partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University.