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.     

A Simple Approach to Analyse Non-Orthogonal Data

The present paper deals with an alternative and simple procedure to analyse the non-orthogonal data. The procedure is general in nature but has some advantages for the non-orthogonal data due to some missing observations. The procedure is applied to (i) two way classification with unequal number of observations per cell; (ii) randomized block designs with some missing observations and (iii) balanced incomplete block designs and also illustrated with the help of numerical examples.     

Construction of resolvable spatial row-column designs

Resolvable row-column designs are widely used in field trials to control variation and improve the precision of treatment comparisons. Further gains can often be made by using a spatial model or a combination of spatial and incomplete blocking components. Martin, Eccleston, and Gleeson presented some general principles for the construction of robust spatial block designs which were addressed by spatial designs based on the linear variance (LV) model. In this article we define the two-dimensional form of the LV model and investigate extensions of the Martin et al. principles for the construction of resolvable spatial row-column designs. The computer construction of efficient spatial designs is discussed and some comparisons made with designs constructed assuming an autoregressive variance structure.     

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.     

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.     

Fractional simplex designs for interaction screening in complex mixtures

In mixture experiments, one may be interested in estimating not only main effects but also some interactions. Main effects and significant interactions in a mixture may be estimated through appropriate mixture experiments, such as simplex-centroid designs. However, for mixtures with a large number of factors, the run size for these designs becomes impractically large. A subset of a full simplex-centroid design may be used, but the problem remains regarding which factor-level settings should be selected. In this paper, we propose a solution that considers design points with either one or p individual nonzero factor-level settings. These fractional simplex designs provide a means of screening for interactions and of investigating the behavior of many-component mixtures as a whole while greatly reducing the run size compared with full simplex-centroid designs. The means of construction of the design arrays is described, and designs for < or = 31 factors are presented. Some of the proposed methodology is illustrated using generated data.     

Conditional Point Estimation in Adaptive Group Sequential Test Designs

It is well known that point estimates in group sequential designs are biased. This also applies to adaptive designs that enable, e.g., data driven reassessments of group sample sizes. For triangular designs, Whitehead (1986) (Biometrika 73 , 573–581) proposed a bias adjusted estimate. But this estimate is not feasible in adaptive designs although it is in group sequential designs. Furthermore, there is a waste of information because it does not use the information at which stage the trial was stopped. We present a modification which does use this information and which is applicable to adaptive designs. The modified estimate achieves an improvement in group sequential designs and shows similar results in adaptive designs.     

Adaptive Two Stage Designs and the Conditional Error Function

Proschan and Hunsberger (1995) suggest the use of a conditional error function to construct a two stage test that meets the α level and allows a very flexible reassessment of the sample size after the interim analysis. In this note we show that several adaptive designs can be formulated in terms of such an error function. The conditional power function defined similarly provides a simple method for sample size reassessment in adaptive two stage designs.     

On Analysis of Some Augmented Block Designs

Augmented designs are frequently used in practice. In this paper, a simplified procedure of analysis of some types of augmented designs is discussed using the concept of efficiency-balance and partially efficiency-balance.     

Systematic designs for experiments with mixtures

In this paper systematic designs for experiments with mixtures are developed. The plan of analysis of the experiment is discussed for a quadratic model of SCHEFF É (1958) for q-component mixture with orthogonal polynomials of third degree describing the time trends.     

A general mixture model approach for mapping quantitative trait loci from diverse cross designs involving multiple inbred lines

Most current statistical methods developed for mapping quantitative trait loci (QTL) based on inbred line designs apply to crosses from two inbred lines. Analysis of QTL in these crosses is restricted by the parental genetic differences between lines. Crosses from multiple inbred lines or multiple families are common in plant and animal breeding programmes, and can be used to increase the efficiency of a QTL mapping study. A general statistical method using mixture model procedures and the EM algorithm is developed for mapping QTL from various cross designs of multiple inbred lines. The general procedure features three cross design matrices, W, that define the contribution of parental lines to a particular cross and a genetic design matrix, D, that specifies the genetic model used in multiple line crosses. By appropriately specifying W matrices, the statistical method can be applied to various cross designs, such as diallel, factorial, cyclic, parallel or arbitrary-pattern cross designs with two or multiple parental lines. Also, with appropriate specification for the D matrix, the method can be used to analyse different kinds of cross populations, such as F2 backcross, four-way cross and mixed crosses (e.g. combining backcross and F2). Simulation studies were conducted to explore the properties of the method, and confirmed its applicability to diverse experimental designs.     

Adaptive two-stage designs for single-arm phase IIA cancer clinical trials

The main purpose of a phase IIA trial of a new anticancer therapy is to determine whether the therapy has sufficient promise against a specific type of tumor to warrant its further development. The therapy will be rejected for further investigation if the true response rate is less than some uninteresting level and the test of hypothesis is powered at a specific target response rate. Two-stage designs are commonly used for this situation. However, in many situations investigators often express concern about uncertainty in targeting the alternative hypothesis to study power at the planning stage. In this article, motivated by a real example, we propose a strategy for adaptive two-stage designs that will use the information at the first stage of the study to either reject the therapy or continue testing with either an optimistic or a skeptic target response rate, while the type I error rate is controlled. We also introduce new optimal criteria to reduce the expected total sample size.     

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.     

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.     

Estimation of Several Missing Values in F-Square Designs

The present study deals with the estimation of several missing values in F-square designs. The estimating equations for the non-iterative least squares estimation of Missing Values and explicit expressions for the estimators of the particular patterns of Missing Values are presented. This procedure is illustrated with the help of a numerical example.     

A New Class of Matrices with Application in Experimental Designs

A very general class of balanced matrices is defined in this paper. These matrices are generalisations of the balanced orthogonal designs of Rao (1970) and generalised balanced matrices of Dey and Midha (1976). Some methods of construction of these balanced matrices are discussed. An application of these matrices in experimental designs is also given.     

Robustness and Efficiency of D‐optimal Experimental Designs in a Growth Problem

To assess tree growth, for example in diameter, a forester typically measures the trees at regular time points. We call such designs equidistant. In this paper we look at the robustness and efficiency of several experimental designs, using the D‐optimality criterion, in a case study of diameter growth in cork oaks. We compare D‐optimal designs (unrestricted and replication‐free) with equidistant designs. We further compare designs in different experimental regions. Results indicate that the experimental region should be adequate to the problem, and that D‐optimal designs are substantially more efficient than equidistant designs, even under parameter mis‐specification.     

Non-Iterative Least Squares Estimation of Missing Values in Cross-Over Designs Without Residual Effect

In the present study an attempt has been made to esttimate several missing values in cross-over designs. When the observations are missing in a particular pattern, explicit expressions are given for the estimators of the missing values. This procedure is illustrated with the help of a numerical example.     

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.     

Simultaneous confidence intervals for pairwise multiple comparisons in a two-way unbalanced design

Turkey's (1953, The Problem of Multiple Comparisons, unpublished report, Princeton University) procedure is widely used for pairwise multiple comparisons in one-way ANOVA. It provides exact simultaneous pairwise confidence intervals (SPCI) for balanced designs and conservative SPCI for unbalanced designs. In this paper, we will extend Turkey's procedure to two-way unbalanced designs. Both the exact and the conservative methods will be introduced. The application of the new procedure is illustrated with sample data from two experiments.