Optimal Circular Block Designs for Neighbouring Competition Effects

Competition or interference occurs when the responses to treatments in experimental units are affected by the treatments in neighbouring units. This may contribute to variability in experimental results and lead to substantial losses in efficiency. The study of a competing situation needs designs in which the competing units appear in a predetermined pattern. This paper deals with optimality aspects of circular block designs for studying the competition among treatments applied to neighbouring experimental units. The model considered is a four-way classified model consisting of direct effect of the treatment applied to a particular plot, the effect of those treatments applied to the immediate left and right neighbouring units and the block effect. Conditions have been obtained for the block design to be universally optimal for estimating direct and neighbour effects. Some classes of balanced and strongly balanced complete block designs have been identified to be universally optimal for the estimation of direct, left and right neighbour effects and a list of universally optimal designs for v<20 and r<100 has been prepared.     

A method for obtaining randomized block designs in preclinical studies with multiple quantitative blocking variables

A method is proposed for block randomization of treatments to experimental units that can accommodate both multiple quantitative blocking variables and unbalanced designs. Hierarchical clustering in conjunction with leaf‐order optimization is used to block experimental units in multivariate space. The method is illustrated in the context of a diabetic mouse assay. A simulation study is presented to explore the utility of the proposed randomization method relative to that of a completely randomized approach, both in the presence and absence of covariate adjustment. An example R function is provided to illustrate the implementation of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

A modified Friedman test for randomized complete block designs

The Friedman test is often used for a randomized complete block design when the normality assumption is not satisfied or the data are ordinal. The Friedman test can be viewed as an extension of the sign test for multiple measurements within each subject or block. We propose a modified Friedman test based on the Wilcoxon sign rank approach. Coincidentally, Tukey proposed a test statistic similar to our proposed test, but blocks are ranked by the minimum difference within each block. In the proposed test, we use the variance of block to rank the blocks, with the least variance being ranked the smallest. In both Tukey test and the modified Friedman test, linear ranks are used for blocks and treatments. The Tukey test belongs to the family of weighted-ranking test from Quade (1979 Quade, D. (1979). Using weighted rankings in the analysis of complete blocks with additive block effects. Journal of the American Statistical Association 74(367):680–683. [Google Scholar]).The modified Friedman test, the Friedman test and the Tukey test are compared under various conditions and the results indicate that the proposed test is generally more powerful than the Friedman test and the Tukey test when the number of groups is small.     

Fourth exact moment results for mrbp tests with two or three treatments

MRBP tests were proposed by Mielke and Iyer (1982) to analyze multivariate data for the randomized block design, based on permutation procedures. They obtained the first three exact moments of the MRBP test statistic to approximate its permutation distribution. Tracy and Khan (1991) derived its fourth exact moment, to obtain a better approximating distribution, when there are four or more treatments. In this paper we obtain the fourth exact moment when the number of treatments is less than four.     

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.     

ON CONSTRUCTION OF GENERALIZED GROUP DIVISIBLE DESIGNS WITH TWO GROUPS

ABSTRACT

Many simple and general methods of construction, using α-resolvable balanced incomplete block designs, for generalized group divisible designs with two groups (GGDD(2)) are given. The E-optimality of all constructed designs is also discussed.     

The mad estimator: when is it non-linear? applications to two-way design models

The mean absolute deviation (MAD) estimator has recently received a great deal of attention as applied to full-rank linear regression models. This paper provides a necessary and sufficient condition for the MAD estimator to be a non-linear estimator, in which case conditions for the variance of the MAD estimator to be larger or smaller than those for OLS are, in general, unknown. The non-linearity of the MAD estimator is examined for several two-way designs; in particular (1) randomized block design (2) two-way nested design (3) two-way classification with interaction and (4) partially balanced incomplete block design     

The Use of Balanced Incomplete Block Designs in Designing Randomized Response Surveys

This paper investigates the block total response method proposed by Raghavarao and Federer for providing accurate estimates of the base rates of sensitive characteristics during surveys. It determines the best balanced incomplete block design to use to estimate the base rates for three, four, five and six sensitive attributes respectively, given a maximum total number of 13 questions. The estimates obtained from this method have smaller variance than estimates obtained using the similar, but more popular, unmatched count technique.     

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.     

A study of several useful tests for ordered alternatives in the randomized block design

This study investigates the small sample powers of several tests designed against ordered location alternatives in randomized block experiments. The results are intended to aid the researcher in the selection process. Toward this end the small sample powers of three classes of rank tests — tests based on ‘within-blocks’ rankings (W-tests), ‘among-b locks’ rankings (A-tests), and ‘ranking after alignment’ within blocks (RAA-tests)— are compared and contrasted with the asymptotic properties given by Pirie (1974) as well as with the empirical powers of competing parametric procedures.     

Design effect of randomized systematic sampling

In statistical practice, systematic sampling (SYS) is used in many modifications due to its simple handling. In addition, SYS may provide efficiency gains if it is well adjusted to the structure of the population under study. However, if SYS is based on an inappropriate picture of the population a high decrease of efficiency, i.e. a high increase in variance may result by changing from simple random sampling to SYS. In the context of two-stage designs SYS so far seems often in use for subsampling within the primary units. As an alternative to this practice, we propose to randomize the order of the primary units, then to select systematically a number of primary units and, thereafter, to draw secondary units by simple random sampling without replacement within the primary units selected. This procedure is more efficient than simple random sampling with replacement from the whole population of all secondary units, i.e. the variance of an adequate estimator for a total is never increased by changing from simple random sampling to randomized SYS whatever be the values associated by a characteristic with the secondary units, while there are values for which the variance decreases for the change mentioned. This result should hold generally, even if our proof, so far, is not complete for general sample sizes.     

On use of symmetrical balanced incomplete block design in construction of sampling design realising pre-assigned sets of inclusion probabilities of first two orders.

Gupta, Nigam and Kumar (1982) proposed a sampling scheme using certain combinatorial properties of balanced incomplete block design (BIBD) which realises first order inclusion probabilities proportional to measure of size(IPPS). Here their results have been extended by presenting samplng schemes realising pre-assigned sets of inclusion probabilities of first two orders.     

Design-based analysis of embedded experiments with applications in the Dutch Labour Force Survey

Summary.  Previous research has proposed a design-based analysis procedure for experiments that are embedded in complex sampling designs in which the ultimate sampling units of an on-going sample survey are randomized over different treatments according to completely randomized designs or randomized block designs. Design-based Wald and t -statistics are applied to test whether sample means that are observed under various survey implementations are significantly different. This approach is generalized to experimental designs in which clusters of sampling units are randomized over the different treatments. Furthermore, test statistics are derived to test differences between ratios of two sample estimates that are observed under alternative survey implementations. The methods are illustrated with a simulation study and real life applications of experiments that are embedded in the Dutch Labour Force Survey. The functionality of a software package that was developed to conduct these analyses is described.     

Fourth moment results for mrbp and related power performance

A permutation test for analysing randomized block data was proposed by Mielke and Iyer (1982). They obtained the first three exact moments of this test statistic and approximated its permutation distribution by the Pearson type III distribution. Tracy and Khan (1991) derived the fourth exact moment of this test statistic to obtain a better approximating distribution. Here we obtain the simplified form of the fourth moment result for some special cases of this test statistic. Empirical powers for four treatments are compared, using this additional information, with those based on the three moment results, after simulating data from some underlying populations.     

On constructing general minimum lower order confounding two-level block designs

In practice, to reduce systematic variation and increase precision of effect estimation, a practical design strategy is then to partition the experimental units into homogeneous groups, known as blocks. It is an important issue to study the optimal way on blocking the experimental units. Blocked general minimum lower order confounding (B¹-GMC) is a new criterion for selecting optimal block designs. The paper considers the construction of optimal two-level block designs with respect to the B¹-GMC criterion. By utilizing doubling theory and MaxC2 design, some optimal block designs with respect to the B¹-GMC criterion are obtained.     

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.     

Γ-Optimal decision procedures for selecting the best population in randomized complete block design

In randomized complete block design, we face the problem of selecting the best population. If some partial information about the unknown parameters is available, then we wish to delermine the optimal decisin rule to select the best population.

In this paper, in the class of natural selection rules, we employ the Γ-optimal criterion to determine optimal decision rules that will minimize the maximum expected risk over the class of some partial information. Furthermore, the traditional hypothesis testing is briefly discussed from the view point of ranking and selecting.     

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.     

On the recovery of intra- and interblock information in n-ary designs

This note presents an extension of Q-method of analysis for binary designs given by Rao (1956) to n-ary balanced and partially balanced block designs. Here a linked n-ary block (LNB) design is defined as the dual of balanced n-ary (BN) design. Having a note on Yates’ (1939, 1940) method of P-analysis, we further extend the expressions for binary linked block (LB) designs given by Rao (1956) to linked n-ary block (LNB) designs which admit easy estimation of parameters for these type of all n-ary designs.