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 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.     

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.     

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.     

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.     

Designs for restricted exploration in mixture experiments

Extreme Vertices designs were developed by MCLEAN and ANDERSON (1966) for situations where components of a mixture are restricted by lower and upper bounds, SNEE and MARQUARDT (1974) and SNEE (1975) gave algorithms to construct optimum designs in these situations. SAXENA and NIGAM (1975) evolved a transformation which provides designs for restricted exploration using Symmetric Simplex designs. In this paper a procedure has been given which provides alternative designs with uniform exploration in constrained mixture experiments. The procedure is illustrated by an example.     

A graphical method for finding maximin efficiency designs

We consider the problem of designing an experiment when there are two competing optimality criteria. Designs that maximize the minimum efficiencies under the two criteria are proposed along with a graphical method for finding these maximin designs.     

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.     

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.     

On the Construction of Balanced and Partially Balanced Ternary Design

A method of constructing balanced and partially balanced ternary designs from balanced and partially balanced incomplete block designs, respectively, and two methods of constructing partially balanced ternary designs from association schemes are obtained. Two new and efficient balanced ternary designs having K < V and R ≦ 20 are obtained by the first method.     

Augmented p-rep designs

Early generation variety trials are very important in plant and tree breeding programs. Typically many entries are tested, often with very little resources available. Unreplicated trials using control plots are popular and it is common to repeat the trials at a number of locations. An alternative is to use p-rep designs, where a proportion of the test entries are replicated at each location; this can obviate the need for control plots. α-Designs are commonly used for replicated variety trials and we show how these can be adapted to produce efficient p-rep designs.     

Test‐compatible confidence intervals for adaptive two‐stage single‐arm designs with binary endpoint

Inference after two‐stage single‐arm designs with binary endpoint is challenging due to the nonunique ordering of the sampling space in multistage designs. We illustrate the problem of specifying test‐compatible confidence intervals for designs with nonconstant second‐stage sample size and present two approaches that guarantee confidence intervals consistent with the test decision. Firstly, we extend the well‐known Clopper–Pearson approach of inverting a family of two‐sided hypothesis tests from the group‐sequential case to designs with fully adaptive sample size. Test compatibility is achieved by using a sample space ordering that is derived from a test‐compatible estimator. The resulting confidence intervals tend to be conservative but assure the nominal coverage probability. In order to assess the possibility of further improving these confidence intervals, we pursue a direct optimization approach minimizing the mean width of the confidence intervals. While the latter approach produces more stable coverage probabilities, it is also slightly anti‐conservative and yields only negligible improvements in mean width. We conclude that the Clopper–Pearson‐type confidence intervals based on a test‐compatible estimator are the best choice if the nominal coverage probability is not to be undershot and compatibility of test decision and confidence interval is to be preserved.     

Cylindrically Rotatable Designs of Type 3: Further Considerations

The cylindrically rotatable designs of type 3 are generalised to consider the factors split up into any number of groups. The conditions which must be satisfied by such a design are derived from the moment generating function of the design. A method of construction of second-order cylindrically rotatable designs of type 3 with three or more groups of factors is provided.     

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.     

A note on the Design of Unreplicated Trials

One approach to the design of early generation trials is to have a number of replicated check varieties and a larger number of unreplicated or new varieties. For such trials we define the class of α–α‐designs as an extension to the augmented lattice square designs of Federer (2002). Efficient latinized α–α‐designs can be conveniently constructed using the design generation package CycDesigN.     

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 comparison of methods for adaptive treatment selection

Traditionally drug development is generally divided into three phases which have different aims and objectives. Recently so-called adaptive seamless designs that allow combination of the objectives of different development phases into a single trial have gained much interest. Adaptive trials combining treatment selection typical for Phase II and confirmation of efficacy as in Phase III are referred to as adaptive seamless Phase II/III designs and are considered in this paper. We compared four methods for adaptive treatment selection, namely the classical Dunnett test, an adaptive version of the Dunnett test based on the conditional error approach, the combination test approach, and an approach within the classical group-sequential framework. The latter two approaches have only recently been published. In a simulation study we found that no one method dominates the others in terms of power apart from the adaptive Dunnett test that dominates the classical Dunnett by construction. Furthermore, scenarios under which one approach outperforms others are described.     

Non-Iterative Least Squares Estimation of Missing Values in Hyper-Graeco-Latin Square Designs

In this paper an attempt has been made to obtain explicit expressions for the estimators of the several missing values in hyper-graeco-latin square designs. Further it has been shown that the estimates of the missing values in latin square designs and graeco-latin square designs are obtained as a particular case of the estimates of the missing values in hyper-graeco-latin square designs.     

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.