A more powerful test for incomplete block designs

We studied the behavior of a statistic that tests for treatment effects in incomplete block designs, Simulation was used to estimate the first four moments of a test statistic that combines intra- and inter-block estimates of treatment contrasts. For a wide range of alternative hypotheses, the moments are remarkably close to those of an F distribution with one degree of freedom in the numerator, and the denominator degrees of freedom equal to that of the mean square error in the usual intra-block analysis, The noncentrality parameter depends upon the block and unit variances and the eigenvalues of the incidence matrix of the design, The power of the test statistic can be estimated from standard power function charts.     

Universally optimal designs under interference models with and without block effects

The concept of neighbor designs was introduced and defined by Rees (1967 Rees, D.H. (1967). Some designs of use in serology. Biometrics 23:779–791.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) along with giving some methods of their construction. Henceforth, many methods of construction of neighbor designs as well as of their generalizations are available in the literature. However, there are only few results on their optimality. Therefore, the purpose of this article is to give an overview of study on this problem. Recent results on optimality of specified neighbor balanced designs under various interference models with block effects are presented and then these results are compared with respective models where block effects are not significant.     

Robust designs for linear mixed effects models

Summary.  In health sciences, medicine and social sciences linear mixed effects models are often used to analyse time-structured data. The search for optimal designs for these models is often hampered by two problems. The first problem is that these designs are only locally optimal. The second problem is that an optimal design for one model may not be optimal for other models. In this paper the maximin principle is adopted to handle both problems, simultaneously. The maximin criterion is formulated by means of a relative efficiency measure, which gives an indication of how much efficiency is lost when the uncertainty about the models over a prior domain of parameters is taken into account. The procedure is illustrated by means of three growth studies. Results are presented for a vocabulary growth study from education, a bone gain study from medical research and an epidemiological decline in height study. It is shown that, for the mixed effects polynomial models that are applied to these studies, the maximin designs remain highly efficient for different sets of models and combinations of parameter values.     

Robust integer-valued designs for linear random intercept models

This article considers the robust design problem for linear random intercept models with both departures from fixed effects and correlated errors on a finite design space. Two strategies are proposed. One is a worst-case method minimizing the maximum value of the MSE of estimates for the fixed effects over the departure. The other is an average-case method minimizing the average value of the MSE with respect to some priors for the class of departure functions and correlation structures of random errors. Two examples are given to show robust designs for two polynomial models.     

Block designs for random treatment effects

A Bayesian design criterion for selection experiments in plant breeding is derived using a utility function that minimizes the risk of an incorrect selection. A prior distribution on the heritability parameter is used to complete the definition of the design optimality criterion. An example is given with evaluations of the criterion for different prior distributions on the heritability. Though coming from a genetic motivation this criterion should prove useful for any other types of experiments with random treatment effects.     

On a class of variance balanced incomplete block designs

In this paper variance balanced incomplete block designs have been constructed for situations when suitable BIB designs do not exist for a given number of treatments, because of the contraints bk=vr, λ(v-1) = r(k-l). These variance balanced designs are in unequal block sizes and unequal replications.     

Block designs robust against the presence of positional effects

The problem considered is to find optimum designs for treatment effects in a block design (BD) setup, when positional effects are also present besides treatment and block effects, but they are ignored while formulating the model. In the class of symmetric balanced incomplete block designs, the Youden square design is shown to be optimal in the sense of minimizing the bias term in the mean squared error (MSE) of the best linear unbiased estimators of the full set of orthonormal treatment contrasts, irrespective of the value of the positional effects.     

On totally balanced block designs for competition effects

Competition between neighbouring units in field experiments is a serious source of bias. The study of a competing situation needs construction of an environment in which it can happen and the competing units have to appear in a predetermined pattern. This paper describes methods of constructing incomplete block designs balanced for neighbouring competition effects. The designs obtained are totally balanced in the sense that all the effects, direct and neighbours, are estimated with the same variance. The efficiency of these designs has been computed as compared to a complete block design balanced for neighbours and a catalogue has also been prepared.     

Some aspects of simplicity in the analysis of block designs

In this article, we develop and examine a group-theoretic approach to the notion of simplicity in the analysis of block designs. This calls for invariance considerations (in a specialized sense). The defining conditions (for simplicity) in this approach, as they stand, appear to be quite rigid. We have, therefore, imposed some weaker conditions and examined further aspects of the problem. The results seem to be quite interesting.     

Inequality for equireplicated n-ary block designs with unequal block sizes

It is shown that certain inequalities known for binary, equireplicated, equiblock-sized block designs remain valid for equireplicated n-ary block designs with unequal block sizes. The approach used here is based on the spectral expansion of the C-matrix of the block design. The main theorems include some useful and combinatorially interesting results.     

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.     

Optimal block designs with non additive mixed effects interference

This paper deals with optimality aspects of block designs balanced for interference effects from neighboring units on both sides under a general non additive model along with random block effects. Here, a class of complete, circular block designs strongly balanced for interference effects has been shown to be universally optimal for the estimation of direct effects and interference effects (left and right) of treatments under a non additive mixed effects model.     

Optimal regression designs in the presence of random block effects

A- and D-optimal regression designs under random block-effects models are considered. We first identify certain situations where D- and A-optimal designs do not depend on the intra-block correlation and can be obtained easily from the optimal designs under uncorrelated models. For example, for quadratic regression on [−1,1], this covers D-optimal designs when the block size is a multiple of 3 and A-optimal designs when the block size is a multiple of 4. In general, the optimal designs depend on the intra-block correlation. For quadratic regression, we provide expressions for D-optimal designs for any block size. A-optimal designs with blocks of size 2 for quadratic regression are also obtained. In all the cases considered, robust designs which do not depend on the intrablock correlation can be constructed.     

A characterization of a universally optimal design within a class of block designs

It is shown that within the class of connected binary designs with arbitrary block sizes and arbitrary replications only a symmetic balanced incomplete block design produces a completely symmetric information matrix for the treatment effects whenever the number of blocks is equal to the number of treatments and the number of experimental units is an integer multiple of the number of treatments. Such a design is known to be universally optimal.     

Optimal cross-over designs for nonlinear mixed models using a first-order expansion

We consider the construction of optimal cross-over designs for nonlinear mixed effect models based on the first-order expansion. We show that for AB/BA designs a balanced subject allocation is optimal when the parameters depend on treatments only. For multiple period, multiple sequence designs, uniform designs are optimal among dual balanced designs under the same conditions. As a by-product, the same results hold for multivariate linear mixed models with variances depending on treatments.     

Multiresponse surface models with random block effects

A multiresponse experiment is one in which a number of responses are measured for each setting of a group of input (control) variables. Quite often, the experimental units are subdivided into groups — or blocks — in order to control an extraneous source of variation. This necessitates adding block effects to the hypothesized multiresponse surface model, which typically contains fixed polynomial effects. Khuri and Valeroso (1998, J. Statist. Plann. Inference, Vol. 73, pp. 7–20) discussed the analysis of such a model when the block effects are considered fixed. There are many situations, however, where these effects should more appropriately be treated as random. In the present article, we address the analysis of a multiresponse model in the latter situations. In particular, we discuss the estimation of the model's polynomial effects in two cases:

• 1.The block effects are additive in the model.
• 2.The blocks have interactive effects with the polynomial portion of the model.

Multiresponse optimization in the presence of blocks will also be discussed. A numerical example is presented for illustrative purposes.     

New combinatorial designs and their applications to group testing

A class of designs with property C(t) are introduced for the first time, and their applications in group testing of samples are studied.     

Methods of constructing group divisible designs with k = 3 and 4

The paper provides methods of construction of group divisible designs with block sizes three and four through balanced incomplete block and partially balanced incomplete block designs of block sizes 3 and 4. Furthermore, four non-isomorphic solutions are given.     

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.     

Three methods of constructing pbib designs

Three construction methods of two- or three-associate partially balanced incomplete block (PBIB) designs are presented.