Construction of Resolvable Designs with Nested Treatment Structure

The construction of resolvable incomplete block and row-column designs is investigated when the treatments have a nested structure. Some theoretical results are derived for lattice designs. Efficient designs for unequal-sized treatment groups are obtained by defining a multiple objective function and carrying out a computer search using an interchange algorithm.     

Extension of the rank sum test for clustered data: two-group comparisons with group membership defined at the subunit level

The Wilcoxon rank sum test is widely used for two-group comparisons for nonnormal data. An assumption of this test is independence of sampling units both between and within groups. In ophthalmology, data are often collected on two eyes of an individual, which are highly correlated. In ophthalmological clinical trials, randomization is usually performed at the subject level, but the unit of analysis is the eye. If the eye is used as the unit of analysis, then a modification to the usual Wilcoxon rank sum variance formula must be made to account for the within-cluster dependence. For some clustered data designs, where the unit of analysis is the subunit, group membership may be defined at the subunit level. For example, in some randomized ophthalmologic clinical trials, different treatments may be applied to fellow eyes of some patients, while the same treatment may be applied to fellow eyes of other patients. In general, binary eye-specific covariates may be present (scored as exposed or unexposed) and one wishes to compare nonnormally distributed outcomes between exposed and unexposed eyes using the Wilcoxon rank sum test while accounting for the clustering. In this article, we present a corrected variance formula for the Wilcoxon rank sum statistic in the setting of eye (subunit)-specific covariates. We apply it to compare ocular itching scores in ocular allergy patients between eyes treated with active versus placebo eye drops, where some patients receive the same eye drop in both eyes, while other patients receive different eye drops in fellow eyes. We also present comparisons between the clustered Wilcoxon test and each of the signed rank tests and mixed model approaches and show dramatic differences in power in favor of the clustered Wilcoxon test for some designs.     

A Simple Procedure for Constructing Experiment Designs with Incomplete Blocks of Sizes 2 and 3

In order to maximize control of heterogeneity within complete blocks, an experimenter could use incomplete blocks of size k = 2 or 3. In certain situations, incomplete blocks of this nature would eliminate the need for such spatial types of analyses as nearest neighbor. The intrablock efficiency factors for such designs are relatively low. However, with recovery of interblock information, FEDERER and SPEED (1987) have presented measures of design efficiency factors which demonstrate that efficiency factors approach unity for certain ratios of the intrablock and interblock variance components. Hence with recovery of interblock information, even incomplete block designs with k = 2 or 3 have relatively high efficiency factors. The reduction in the intrablock error variance over the complete block error variance in many situations will provide designs with high efficiency. A simple procedure for constructing incomplete blocks of sizes 2 and 3 is presented. It is shown how to obtain additional zero-one association confounding arrangements when v = 4 t, t an integer, and for v = pk, k ≤ p. It is indicated how to do the statistical analysis for these designs.     

On Construction of Some Augmented Block Designs

Augmented block designs are frequently used in practice. In this paper procedures of construction of some types of augmented block designs are discussed using the concept of efficiency-balance and partially efficiency-balance.     

Experiment with Split-Plot Generated by PBIB Designs

The paper refers to the construction of two-factor experiments carried out in incomplete split-plot design in which levels of both factors occur as treatments in PBIB designs. It presents also the analysis of variance of these experiments based on partial analyses in the so called strata. These strata are connected with the variability between subplots inside wholeplots, between the wholeplots inside blocks and between blocks inside the total experiment.     

Optimal nested row-column designs with specified components

We consider nested row-column designs where each of the rowand column component designs is specified. For the case thateach of the component designs has second-order balance, we definesuch a nested row-column design to be special if it is generallybalanced, with the smallest possible number of canonical treatmentcontrasts having the lower canonical efficiency factor in bothcomponents. We show that if any special row-column design existsthen it is A-optimal over all nested row-column designs withthe given components.     

Response Adaptive Designs with a Variance‐penalized Criterion

We consider a response adaptive design of clinical trials with a variance‐penalized criterion. It is shown that this criterion evaluates the performance of a response adaptive design based on both the number of patients assigned to the better treatment and the power of the statistical test. A new proportion of treatment allocation is proposed and the doubly biased coin procedure is used to target the proposed proportion. Under reasonable assumptions, the proposed design is demonstrated to generate an asymptotic variance of allocation proportions, which is smaller than that of the drop‐the‐loser design. Simulation comparisons of the proposed design with some existing designs are presented.     

Third-order Minimax Designs for the Difference Between Two Estimated Responses

Minimization of the variance of the difference between estimated responses at two points maximized over pairs of points in the region of interest is taken as the design criterion. Optimal third-order designs are derived for spherical co-centric regions of experimentation and interest with some restrictions imposed on the pairs of points under consideration.     

Four Factor Balanced Optimal Designs of Resolution V in Terms of a Modified Trace Criterion

Balanced optimal two level fractional factorial designs of resolution V have been reported in series of papers by SRIVASTAVA and CHOPRA (1971a, 1971b, 1974) and CHOPRA and SRIVASTAVA (1973a, 1973b). The optimality criterion considered by them is the trace optimality which implies minimisation of the average variance of the estimated effects which include the general mean μ, main effects, A_i, and first order interactions, A_ij. In the present work some new optimal designs have been obtained on the basis of a modified criterion of trace optimality. The criterion requires minimisation of the average variance of each of the main effects, two factor interactions, etc. separately instead of minimising average variance of all the estimated effects.     

Statistical designs for two-color spotted microarray experiments

Two-color cDNA or oligonucleotide-based spotted microarrays have been commonly used in measuring the expression levels of thousands of genes simultaneously. To realize the immense potential of this powerful new technology, budgeted within limited resources or other constraints, practical designs with high efficiencies are in demand. In this study, we address the design issue concerning the arrangement of the mRNA samples labeled with fluorescent dyes and hybridized on the slides. A normalization model is proposed to characterize major sources of systematic variation in a two-color microarray experiment. This normalization model establishes a connection between designs for two-color microarray experiments with a particular class of classical row-column designs. A heuristic algorithm for constructing A-optimal or highly efficient designs is provided. Statistical optimality results are found for some of the designs generated from the algorithm. It is believed that the constructed designs are the best or very close to the best possible for estimating the relative gene expression levels among the mRNA samples of interest.     

The design and analysis of breeders' trials of sugar beet (Beta vulgaris) using two-dimensional blocking structures

A method for generating two-dimensional blocking designs to fit any shape and size of experimental area is presented. The method takes an alpha design appropriate to the experimental area and imposes additional blocking perpendicular to the alpha blocks. Improvements to the design are then made by repositioning entries within the alpha blocks. Although this method of construction is less sophisticated than for other two-dimensional designs, the designs are particularly suited to large scale breeders trials where no alternative two-dimensional design may exist. The designs have been used for 3 yr in a sugar beet breeding programme, and have given improvements in efficiency over one-dimensional alpha designs.     

Adaptive web sampling

A flexible class of adaptive sampling designs is introduced for sampling in network and spatial settings. In the designs, selections are made sequentially with a mixture distribution based on an active set that changes as the sampling progresses, using network or spatial relationships as well as sample values. The new designs have certain advantages compared with previously existing adaptive and link-tracing designs, including control over sample sizes and of the proportion of effort allocated to adaptive selections. Efficient inference involves averaging over sample paths consistent with the minimal sufficient statistic. A Markov chain resampling method makes the inference computationally feasible. The designs are evaluated in network and spatial settings using two empirical populations: a hidden human population at high risk for HIV/AIDS and an unevenly distributed bird population.     

Group Divisible Second Order Rotatable Designs

The Group Divisible Rotatable (GDR) designs are the designs in which the factors get divided into groups such that for the factors within group, the designs are rotatable. In the present paper we have obtained a series of Group Divisible Second Order Rotatable designs, by decomposing the v-dimensional space corresponding to v-factors under consideration into three mutually orthogonal spaces. We have given the least squares estimates of the parameters, the analysis and construction of such designs.     

Uniform magnetic fields and double-wrapped coil systems: improved techniques for the design of bioelectromagnetic experiments.

A common mistake in biomagnetic experimentation is the assumption that Helmholtz coils provide uniform magnetic fields; this is true only for a limited volume at their center. Substantial improvements on this design have been made during the past 140 years with systems of three, four, and five coils. Numerical comparisons of the field uniformity generated by these designs are made here, along with a table of construction details and recommendations for their use in experiments in which large volumes of uniform intensity magnetic exposures are needed. Double-wrapping, or systems of bifilar windings, can also help control for the non-magnetic effects of the electric coils used in many experiments. In this design, each coil is wrapped in parallel with two separate, adjacent strands of copper wire, rather than the single strand used normally. If currents are flowing in antiparallel directions, the magnetic fields generated by each strand will cancel and yield virtually no external magnetic field, whereas parallel currents will yield an external field. Both cases will produce similar non-magnetic effects of ohmic heating, and simple measures can reduce the small vibration and electric field differences. Control experiments can then be designed such that the only major difference between treated and untreated groups is the presence or absence of the magnetic field. Double-wrapped coils also facilitate the use of truly double-blind protocol, as the same apparatus can be used either for experimental or control groups.     

Robust and efficient design of experiments for the Monod model

In this paper the problem of designing experiments for the Monod model, which is frequently used in microbiology, is studied. The model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, environmental research, pharmacokinetics, and plant physiology. The designs presented so far in the literature are local optimal designs, which depend sensitively on a preliminary guess of the unknown parameters, and are for this reason in many cases not robust with respect to their misspecification. Uniform designs and maximin optimal designs are considered as a strategy to obtain robust and efficient designs for parameter estimation. In particular, standardized maximin D- and E-optimal designs are determined and compared with uniform designs, which are usually applied in these microbiological models. It is demonstrated that maximin optimal designs are substantially more efficient than uniform designs. Parameter variances can be decreased by a factor of two by simply sampling at optimal times during the experiment. Moreover, the maximin optimal designs usually provide the possibility for the experimenter to check the model assumptions, because they have more support points than parameters in the Monod model.     

Evaluation of experimental designs and spatial analyses in wheat breeding trials

Thirty-three wheat breeding trials were conducted from 1994 to 1996 in the Northern Grains Region (QLD and Northern NSW) of Australia to evaluate the influence of experimental designs and spatial analyses on the estimation of genotype effects for yield and their impact on selection decisions. The relative efficiency of the alternative designs and analyses was best measured by the average standard error of difference between line means. Both more effective designs and spatial analyses significantly improved the efficiency relative to the randomised complete block model, with the preferred model (which combined the design information and spatial trends) giving an average relative efficiency of 138% over all 33 trials. When the Czekanowski similarity coefficient was used, none of the studied models were in full agreement with the randomised complete block model in the selection of the top lines. The agreement was influenced by selection proportions. Hence, the use of these methodologies can impact on the selection decisions in plant breeding. Received: 17 December 1998 / Accepted: 29 July 1999     

Maximin D-optimal designs for longitudinal mixed effects models

In this article, the optimal selection and allocation of time points in repeated measures experiments is considered. D-optimal cohort designs are computed numerically for the first- and second-degree polynomial models with random intercept, random slope, and first-order autoregressive serial correlations. Because the optimal designs are locally optimal, it is proposed to use a maximin criterion. It is shown that, for a large class of symmetric designs, the smallest relative efficiency over the model parameter space is substantial.     

On the analysis of phylogenetically paired designs

As phylogenetically controlled experimental designs become increasingly common in ecology, the need arises for a standardized statistical treatment of these datasets. Phylogenetically paired designs circumvent the need for resolved phylogenies and have been used to compare species groups, particularly in the areas of invasion biology and adaptation. Despite the widespread use of this approach, the statistical analysis of paired designs has not been critically evaluated. We propose a mixed model approach that includes random effects for pair and species. These random effects introduce a “two-layer” compound symmetry variance structure that captures both the correlations between observations on related species within a pair as well as the correlations between the repeated measurements within species. We conducted a simulation study to assess the effect of model misspecification on Type I and II error rates. We also provide an illustrative example with data containing taxonomically similar species and several outcome variables of interest. We found that a mixed model with species and pair as random effects performed better in these phylogenetically explicit simulations than two commonly used reference models (no or single random effect) by optimizing Type I error rates and power. The proposed mixed model produces acceptable Type I and II error rates despite the absence of a phylogenetic tree. This design can be generalized to a variety of datasets to analyze repeated measurements in clusters of related subjects/species.     

A Variance Components Model for Analyzing Repeated Measures Designs with Non-Stationary Error Structure

When data are collected in the form of multiple measurements on several subjects, they are often analyzed as repeated measures data with some stationary error structure assumed for the errors. For data with non-stationary error structure, the multivariate model is often used. The multivariate model imposes restrictions that are often not met in practice by data of such type. At the same time, they ignore valuable information in the data that are related to time dependencies and time relations. In this paper, we propose a model that is a reparametrization of the multivariate model and is suitable to analyze general repeated measures designs with non-stationary error structure. The model is shown to be a variance components model whose components are estimated using the method of maximum likelihood. Several other properties of the model are derived and discussed including tests of significance. Finally, an example on neurological data is included to demonstrate its application in biological sciences.     

Optimum Biased Spring Balance Weighing Designs Under Equal Correlations of Errors

The paper deals with the problem of estimating the individual weights of objects under a biased spring balance weighing design with equal correlations of errors in the model. A lower bound for the variance of each of the estimated weights resulting from this biased spring balance weighing design is obtained and a necessary and sufficient condition for this lower bound to be attained is given. The incidence matrix of a BIB design has been used to construct optimum biased spring balance weighing designs.