DT-optimum designs for model discrimination and parameter estimation

The paper introduces DT-optimum designs that provide a specified balance between model discrimination and parameter estimation. An equivalence theorem is presented for the case of two models and extended to an arbitrary number of models and of combinations of parameters. A numerical example shows the properties of the procedure. The relationship with other design procedures for parameter estimation and model discrimination is discussed.     

Optimal block designs for three treatments when observations are correlated

E$E$-optimal designs for comparing three treatments in blocks of size three are identified, where intrablock observations are correlated according to a first order autoregressive error process with parameter ρ∈(0,1)$\rho \in (0,1)$. For number of blocks b   of the form b=3n+1$b=3n+1$, there are two distinct optimal designs depending on the value of ρ$\rho$, with the best design being unequally replicated for large ρ$\rho$. For other values of b$b$, binary, equireplicate designs with specified within-block assignment patterns are best. In many cases, the stronger majorization optimality is established.     

Cost-cautious designs for confirmatory bioassay

Confirmatory bioassay experiments take place in late stages of the drug discovery process when a small number of compounds have to be compared with respect to their properties. As the cost of the observations may differ considerably, the design problem is well specified by the cost of compound used rather than by the number of observations. We show that cost-efficient designs can be constructed using useful properties of the minimum support designs. These designs are particularly suited for studies where the parameters of the model to be estimated are known with high accuracy prior to the experiment, although they prove to be robust against typical inaccuracies of these values. When the parameters of the model can only be specified with ranges of values or by a probability distribution, we use a Bayesian criterion of optimality to construct the required designs. Typically, the number of their support points depends on the prior knowledge for the model parameters. In all cases we recommend identifying a set of designs with good statistical properties but different potential costs to choose from.     

New 3-level response surface designs constructed from incomplete block designs

Box and Behnken [1958. Some new three level second-order designs for surface fitting. Statistical Technical Research Group Technical Report No. 26. Princeton University, Princeton, NJ; 1960. Some new three level designs for the study of quantitative variables. Technometrics 2, 455–475.] introduced a class of 3-level second-order designs for fitting the second-order response surface model. These 17 Box–Behnken designs (BB designs) are available for 3–12 and 16 factors. Although BB designs were developed nearly 50 years ago, they and the central-composite designs of Box and Wilson [1951. On the experimental attainment of optimum conditions. J. Royal Statist. Soc., Ser. B 13, 1–45.] are still the most often recommended response surface designs. Of the 17 aforementioned BB designs, 10 were constructed from balanced incomplete block designs (BIBDs) and seven were constructed from partially BIBDs (PBIBDs). In this paper we show that these seven BB designs constructed from PBIBDs can be improved in terms of rotatability as well as average prediction variance, D$D$- and G$G$-efficiency. In addition, we also report new orthogonally blocked solutions for 5, 8, 9, 11 and 13 factors. Note that an 11-factor BB design is available but cannot be orthogonally blocked. All new designs can be found at http://www.math.montana.edu/∼$\sim$jobo/bbd/.     

Predictability of designs which adjust for imbalances in prognostic factors

Minimisation is a method often used in clinical trials to balance the treatment groups with respect to some prognostic factors. In the case of two treatments, the predictability of this method is calculated for different numbers of factors, different numbers of levels of each factor and for different proportions of the population at each level. It is shown that if we know nothing about the previous patients except the last treatment allocation, the next treatment can be correctly guessed more than 60% of the time if no biased coin is used. If the two previous assignments are known to have been the same, the next treatment can be guessed correctly around 80% of the time. Therefore, it is suggested that a biased coin should always be used with minimisation. Different choices of biased coin are investigated in terms of the reduction in predictability and the increase in imbalance that they produce. An alternative design to minimisation which makes use of optimum design theory is also investigated, by means of simulation, and does not appear to have any clear advantages over minimisation with a biased coin.     

A constrained optimum experimental design problem for model discrimination with a continuously varying factor

We present the theoretical background and the numerical procedure for calculating optimum experimental designs for non-linear model discrimination in the presence of constraints. The design support points consist of two kinds of factors: a continuous function of time and discrete levels of other quantitative factors. That is, some of the experimental conditions are allowed to continually vary during the experimental run. We implement the theory in a chemical kinetic model discrimination problem.     

An optimality criterion for supersaturated designs with quantitative factors

A supersaturated design (SSD) is a factorial design in which the degrees of freedom for all its main effects exceed the total number of distinct factorial level-combinations (runs) of the design. Designs with quantitative factors, in which level permutation within one or more factors could result in different geometrical structures, are very different from designs with nominal ones which have been treated as traditional designs. In this paper, a new criterion is proposed for SSDs with quantitative factors. Comparison and analysis for this new criterion are made. It is shown that the proposed criterion has a high efficiency in discriminating geometrically nonisomorphic designs and an advantage in computation.     

Accelerated life test planning with independent lognormal competing risks

In accelerated life testing (ALT), products are exposed to stress levels higher than those at normal use in order to obtain information in a timely manner. Past work on planning ALT is predominantly on a single cause of failure. This article presents methods for planning ALT in the presence of k competing risks. Expressions for computing the Fisher information matrix are presented when risks are independently distributed lognormal. Optimal test plans are obtained under criteria that are based on determinants and maximum likelihood estimation. The proposed method is demonstrated on ALT of motor insulation.     

Defining equations for two-level factorial designs

Defining equations are introduced in the context of two-level factorial designs and they are shown to provide a concise specification of both regular and nonregular designs. The equations are used to find orthogonal arrays of high strength and some optimal designs. The latter optimal designs are formed in a new way by augmenting notional orthogonal arrays which are allowed to have some runs with a negative number of replicates before augmentation. Defining equations are also shown to be useful when the factorial design is blocked.     

Kernel density estimators for random fields satisfying an interlaced mixing condition

For a sequence of strictly stationary random fields that are uniformly ρ′-mixing$\rho \prime \mathrm{-}\mathrm{mixing}$ and satisfy a Lindeberg condition, a central limit theorem is obtained for sequences of “rectangular” sums from the given random fields. The “Lindeberg CLT” is then used to prove a CLT for some kernel estimators of probability density for some strictly stationary random fields satisfying ρ′-mixing$\rho \prime \mathrm{-}\mathrm{mixing}$, and whose probability density and joint densities are absolutely continuous.     

D-optimal designs for logistic models with three and four parameters

Logistic functions are used in different applications, including biological growth studies and assay data analysis. Locally D-optimal designs for logistic models with three and four parameters are investigated. It is shown that these designs are minimally supported. Efficiencies are computed for equally spaced and uniform designs.     

Statistical analysis for rounded data

When random variables do not take discrete values, observed data are often the rounded values of continuous random variables. Errors caused by rounding of data are often neglected by classical statistical theories. While some pioneers have identified and made suggestions to rectify the problem, few suitable approaches were proposed. In this paper, we propose an approximate MLE (AMLE) procedure to estimate the parameters and discuss the consistency and asymptotic normality of the estimates. For our illustration, we shall consider the estimates of the parameters in AR(p)$\mathit{AR}(p)$ and MA(q)$\mathit{MA}(q)$ models for rounded data.     

Negative dependence and the Simes inequality

It is shown that the Simes inequality is reversed for a broad class of negatively dependent distributions.     

Selecting and checking a binomial response generalized linear model using phi-divergences

This work introduces specific tools based on phi-divergences to select and check generalized linear models with binary data. A backward selection criterion that helps to reduce the number of explanatory variables is considered. Diagnostic methods based on divergence measures such as a new measure to detect leverage points and two indicators to detect influential points are introduced. As an illustration, the diagnostics are applied to human psychology data.     

Prediction intervals for general balanced linear random models

The main interest of prediction intervals lies in the results of a future sample from a previously sampled population. In this article, we develop procedures for the prediction intervals which contain all of a fixed number of future observations for general balanced linear random models. Two methods based on the concept of a generalized pivotal quantity (GPQ) and one based on ANOVA estimators are presented. A simulation study using the balanced one-way random model is conducted to evaluate the proposed methods. It is shown that one of the two GPQ-based and the ANOVA-based methods are computationally more efficient and they also successfully maintain the simulated coverage probabilities close to the nominal confidence level. Hence, they are recommended for practical use. In addition, one example is given to illustrate the applicability of the recommended methods.     

Pilot–pivotal trials for average bioequivalence

Before carrying out a full scale bioequivalence trial, it is desirable to conduct a pilot trial to decide if a generic drug product shows promise of bioequivalence. The purpose of a pilot trial is to screen test formulations, and hence small sample sizes can be used. Based on the outcome of the pilot trial, one can decide whether or not a full scale pivotal trial should be carried out to assess bioequivalence. This article deals with the design of a pivotal trial, based on the evidence from the pilot trial. A two-stage adaptive procedure is developed in order to determine the sample size and the decision rule for the pivotal trial, for testing average bioequivalence using the two one-sided test (TOST). Numerical implementation of the procedure is discussed in detail, and the required tables are provided. Numerical results indicate that the required sample sizes could be smaller than that recommended by the FDA for a single trial, especially when the pilot study provides strong evidence in favor of bioequivalence.