Robust analysis of variance for a randomized block design

In this paper we study the asymptotic theory of M-estimates and their associated tests for a one-factor experiment in a randomized block design. In this case one natural asymptotic theory corresponds to leaving the number of treatments fixed and letting the number of blocks tend to infinity. The classic asymptotic theory of M-estimates does not apply here, because the number of parameters and the number of observations are of the same order. In this paper we prove the consistency and asymptotic normality of the estimators of the treatment effects. It turns out that the asymptotic covariance matrix of the treatment effects estimators differs from the one derived from the classic theory of M-estimates for the linear model with a fixed number of parameters. We also study a test for treatment effects derived from M-estimates and we compare by Monte Carlo simulation the efficiency of this test with respect to the F-test, the Friedman test and the test based on aligned ranks.     

Interblock information in multidimensional block designs

In many experimental situations, d-way heterogeneity among experimental units may be controlled through use of multiple blocking criteria. In some cases it is reasonable to regard some or all of the block effects as random. Then the model is mixed and observations within blocks are correlated. Very general estimators of treatment effects and their dispersion matrix with recovery of interblock information are provided. They apply to designs with d > 1 blocking criteria that may be crossed, nested, or a combination thereof. These general results may be specialized to provide analyses of new classes of MBD's or used directly for numerical analyses of designs in the general class, perhaps through use as the basis for very general computer programs. Estimation of variance components is discussed, and an example is provided to illustrate adaptation of the general results.     

Super-simple group divisible designs with block size 4 and index 9

A design is said to be super-simple if the intersection of any two blocks has at most two elements. In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple GDDs are useful in constructing super-simple BIBDs. The existence of super-simple (4,λ)‐GDDs has been determined for λ=2-6. In this paper, we investigate the existence of a super-simple (4,9)-GDD of group type g^u and show that such a design exists if and only if u≥4, g(u−2)≥18 and u(u−1)g²≡0 (mod 4).     

The Construction of Optimal Imcomplete Block Designs When Observations Within a Block are Correlated

An algorithmic method is described for the construction of optimal incomplete block designs when a known correlation structure is assumed for observations from plots in the same block. The method is applicable to a wide class of designs and correlation structures. Some examples are given to illustrate the procedure.     

Spacing of information in polynomial regression:a simple solution

Consider a k polynomial regression on a single real variable. If n uncorrelated observations are to be taken in a design with support on more than k+1 points, there is an approximate experiment, ν, with support on k+1 points and n observations such that both designs have the same information matrix for the model. A proof of this result is provided. A method to obtain the approximate design ν is given and illustrated by an example. The source of disagreement between Kiefer (1959) and De La Garza (1954) in the solution of this problem is clarified.     

On Combining Information in Generally Balanced Nested Block Designs

Explicit formulae for analyzing an experiment carried out in a nested block design having the general balance property of some desirable pattern are presented. They follow from a randomization-derived mixed model, decomposed into stratum submodels. Attention is confined here to the combined analysis allowing the information from different strata to be joined together. This article is an extension of some results presented in Caliński and Kageyama (2000 Caliński , T. , Kageyama , S. ( 2000 ). Block Designs: A Randomization Approach, vol. I: Analysis . Lecture Notes in Statistics , vol. 150. New York : Springer .[Crossref] , [Google Scholar]), where the theory underlying the considered analysis is given.     

Construction of neighbor designs of block size 4 for all values of v ≥ 5 by using the method of addition

In this article, a general method of construction of neighbor block designs is given. The designs are constructed using variation of a simple method which we refer to as the method of addition (renamed as the method of cyclic shifts). We give complete solution of neighbor balanced designs for k = 4 for any value of v. We also give many series of generalized neighbor designs (GNDs). In the last section, we have constructed GNDs in a sequential manner (as Did John 1981) for v ≤ 50 and r is multiple of k.     

On the asymptotic efficiency of aligned-rank tests in randomized block designs

This paper is concerned with the ranking-after-alignment procedure, the alignment being made on the mean, in randomized block designs. The asymptotic efficiencies, as the number of blocks goes to infinity, of a class of aligned-rank tests, relative to the maximin most powerful test based on aligned observations, are established and studied. Some asymptotic efficiencies under doubleexponentiality are also obtained using Monte Carlo methods.     

Symmetric designs of types V and VI

In Butler (1984a) a semi-translation block was defined and a classification given of all symmetric 2-(υ,k,λ) designs with λ>1, which contain more than one such block. In this paper we consider symmetric designs of type V and VI. We show that symmetric designs of type V are also of type VI, and in addition we show that all such designs can be obtained from a P_n,q by a construction which we give. Finally examples of proper symmetric designs of type V which are not of type VI are given.     

Construction on non-adaptive hypergeometric group testing desings for identifying at most two defectives

In the literature a systematic method of obtaining a group testing design is not available at present. Weideman and Raghavarao (1987a, b) gave methods for the construction of non - adaptive hypergeometric group testing designs for identifying at most two defectives by using a dual method. In the present investigation we have developed a method of construction of group testing designs from (i) Hypercubic Designs for t ≡ 3 (mod 6) and (ii) Balanced Incomplete Block Designs for t ≡ 1 (mod 6) and t ≡ 3 (mod 6). These constructions are accomplished by the use of dual designs. The designs so constructed satisfy specified properties and attained an optimal bound as discussed by Weidman and Raghavarao (1987a,b). Here it is also shown that the condition for pairwise disjoint sets of BIBD for t ≡ 1 (mod 6) given by Weideman and Raghavarao (1987b) is not true for all such designs.     

A modified least-failures sampling procedure for bernoulli subset selection

We restrict attention to a class of Bernoulli subset selection procedures which take observations one-at-a-time and can be compared directly to the Gupta-Sobel single-stage procedure. For the criterion of minimizing the expected total number of observations required to terminate experimentation, we show that optimal sampling rules within this class are not of practical interest. We thus turn to procedures which, although not optimal, exhibit desirable behavior with regard to this criterion. A procedure which employs a modification of the so-called least-failures sampling rule is proposed, and is shown to possess many desirable properties among a restricted class of Bernoulli subset selection procedures. Within this class, it is optimal for minimizing the number of observations taken from populations excluded from consideration following a subset selection experiment, and asymptotically optimal for minimizing the expected total number of observations required. In addition, it can result in substantial savings in the expected total num¬ber of observations required as compared to a single-stage procedure, thus it may be de¬sirable to a practitioner if sampling is costly or the sample size is limited.     

Measures of non orthogonality in block designs

Orthogonality is an important concept in block design. Necessary and sufficient condition for a connected block design to be orthogonal is well known. However, when a design is not orthogonal, it is not clear how much it deviates from orthogonality. In this paper, an attempt has been made to first define the measures of or indices to non orthogonality in block design and then to characterize designs possessing minimum non orthogonality indices. It is shown that a Balanced Incomplete Block Design (BIBD) and a Balanced Block Design (BBD), if exist, possess this property.     

Constructions of efficiency-balanced designs

We present a number of methods of constructing efficiency-balanced binary block designs which are design patterns for simplification of statistical analysis. Furthermore, a method of construction of an efficiency-balanced block design with v+1 treatments from one with v treatments is generally characterized.     

Minimization of model inadequacy when observations subject to shift in means and variances

In this article optimality of experimental design for fitting a lower-order polynomial to a higher order response function for the situation in which observations may be subject to shift in means as well as in variances is considered. It is found that Karson, Manson and Hader‘s (1969) optimum designs provide pro-tection, in some sense, against model inadequacies even when observations are subject to shift in means and variances.     

Upper Bounds on the True Coverage of Bootstrap Percentile Type Confidence Intervals

A simple derivation of expected mean squares is given for the randomized (complete) block design, showing that “experimental error,” the error term for testing treatments, is comprised of three sources of variability: block by treatment interaction, within block plot-to-plot variability, and within experimental plot sampling variation. The approach could readily be extended to incorporate measurement error as a fourth component of experimental error.     

On a robustness property of PBIBD

In this paper we study a robustness property of partially balanced incomplete block designs based on association schemes with m classes (PBIBD(m)) against the unavailability of data in the sense that, when any t (a positive integer) observations are unavailable the design remains connected w.r.t. treatment. We characterize the robustness property of PBIBD(m) completely for m=2 and partially for m=3.     

E-optimal design in irregular BIBD settings

When the necessary conditions for a BIBD are satisfied, but no BIBD exists, there is no simple answer for the optimal design problem. This paper identifies the E-optimal information matrices for any such irregular BIBD setting when the number of treatments is no larger than 100. A- and D-optimal designs are typically not E-optimal. An E-optimal design for 15 treatments in 21 blocks of size 5 is found.     

E- and R-optimality of block designs for treatment-control comparisons

Abstract

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs, which is characterized by simple linear constraints. Based on this characterization, we obtain a class of E-optimal exact designs for unequal block sizes. In the studied model, we provide a statistical interpretation for wide classes of E-optimal designs. Moreover, we show that all approximate A-optimal designs and a large class of A-optimal exact designs for treatment-control comparisons are also R-optimal. This reinforces the observation that A-optimal designs perform well even for rectangular confidence regions.     

Algorithmic and analytical construction of efficient designs in small blocks for comparing consecutive pairs of treatments

Optimal block designs in small blocks are explored under the A-, E- and D-criteria when the treatments have a natural ordering and interest lies in comparing consecutive pairs of treatments. We first formulate the problem via approximate theory which leads to a convenient multiplicative algorithm for obtaining A-optimal design measures. This, in turn, yields highly efficient exact designs, under the A-criterion, even when the number of blocks is rather small. Moreover, our approach is seen to allow nesting of such efficient exact designs which is an advantage when the resources for the experiment are available in possibly several stages. Illustrative examples are given and tables of A-optimal design measures are provided. Approximate theory is also seen to yield analytical results on E- and D-optimal design measures.     

On the asymptotic distribution of a class of aligned rank order test statistics in randomized block designs

In the present investigation, the unconditional asymptotic distribution of a class of aligned rank order test statistics for randomized block designs is derived under the null hypothesis and for nearby alternatives, as the number of blocks tends to infinity. The proofs of these results are based on the asymptotic equivalence in quadratic mean between aligned observations and their ranks and thus are quite similar to the Hájek and SKidák (1967) approach.