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.     

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.     

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.     

Optimal designs for a multiresponse Emax model and efficient parameter estimation

The aim of dose finding studies is sometimes to estimate parameters in a fitted model. The precision of the parameter estimates should be as high as possible. This can be obtained by increasing the number of subjects in the study, N, choosing a good and efficient estimation approach, and by designing the dose finding study in an optimal way. Increasing the number of subjects is not always feasible because of increasing cost, time limitations, etc. In this paper, we assume fixed N and consider estimation approaches and study designs for multiresponse dose finding studies. We work with diabetes dose–response data and compare a system estimation approach that fits a multiresponse Emax model to the data to equation‐by‐equation estimation that fits uniresponse Emax models to the data. We then derive some optimal designs for estimating the parameters in the multi‐ and uniresponse Emax model and study the efficiency of these designs.     

Optimal fed batch experiment design for estimation of monod kinetics of Azospirillum brasilense: from theory to practice

In this paper the problem of reliable and accurate parameter estimation for unstructured models is considered. It is illustrated how a theoretically optimal design can be successfully translated into a practically feasible, robust, and informative experiment. The well-known parameter estimation problem of Monod kinetic parameters is used as a vehicle to illustrate our approach. As known for a long time, noisy batch measurements do not allow for unique and accurate estimation of the kinetic parameters of the Monod model. Techniques of optimal experiment design are, therefore, exploited to design informative experiments and to improve the parameter estimation accuracy. During the design process, practical feasibility has to be kept in mind. The designed experiments are easy to implement in practice and do not require additional monitoring equipment. Both design and experimental validation of informative fed batch experiments are illustrated with a case study, namely, the growth of the nitrogen-fixing bacteria Azospirillum brasilense.     

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.     

Sequential designs for ordinal phase I clinical trials

Sequential designs for phase I clinical trials which incorporate maximum likelihood estimates (MLE) as data accrue are inherently problematic because of limited data for estimation early on. We address this problem for small phase I clinical trials with ordinal responses. In particular, we explore the problem of the nonexistence of the MLE of the logistic parameters under a proportional odds model with one predictor. We incorporate the probability of an undetermined MLE as a restriction, as well as ethical considerations, into a proposed sequential optimal approach, which consists of a start‐up design, a follow‐on design and a sequential dose‐finding design. Comparisons with nonparametric sequential designs are also performed based on simulation studies with parameters drawn from a real data set.     

Design of experiments for the precise estimation of dose-response parameters: the Hill equation

Optimal experimental designs were evaluated for the precise estimation of parameters of the Hill model. The optimally effective designs were obtained by using the criterion of D-optimization. For the Hill model, optimal designs replicate 3 sampling points. These points were shown to be quite sensitive to the behavior of the experimental error. Since an investigator is often uncertain about error conditions in biological studies, a practical approach would use the sampling scheme calculated for an intermediate error condition. Thus, if the behavior of error variances is not known, precise parameters of the Hill model are obtained by choosing concentrations which yield fractional responses (responses divided by their asymptotic, maximum value) of 0.086, 0.581 and 1.0. When experimental constraints limit the maximum attainable concentration and response, all design points are lowered. Appropriate designs can be constructed based on the design which is optimal when constraints result in a maximum attainable fractional response of 0.5. The optimal designs were found to be robust when the parameter values assumed by the investigator did not equal their true values. The estimating efficiencies obtained by using two frequently applied designs were assessed. Uniformly spaced concentrations yielded imprecise parameters. Six-point, geometrically spaced designs gave generally good results. However, their estimating efficiency was generally exceeded by the recommended sampling schemes even in the presence of uncertainty about error conditions. The method exemplified in this paper can be used for other models.     

Using partition designs to enhance purification process understanding

Characterization of purification processes by identifying significant input parameters and establishing predictive models is vital to developing robust processes. Current experimental design approaches restrict analysis to one process step at a time, which can severely limit the ability to identify interactions between process steps. This can be overcome by the use of partition designs which can model multiple, sequential process steps simultaneously. This paper presents the application of partition designs to a monoclonal antibody purification process. Three sequential purification steps were modeled using both traditional experimental designs and partition designs and the results compared as a proof of concept study. The partition and traditional design approaches identified the same input parameters within each process step that significantly affected the product quality output examined. The partition design also identified significant interactions between input parameters across process steps that could not be uncovered by the traditional approach. Biotechnol. Bioeng. 2010;107: 814–824. © 2010 Wiley Periodicals, Inc.     

Minimax D-optimal designs for the logistic model

We propose an algorithm for constructing minimax D-optimal designs for the logistic model when only the ranges of the values for both parameters are assumed known. Properties of these designs are studied and compared with optimal Bayesian designs and Sitter's (1992, Biometrics, 48, 1145-1155) minimax D-optimal kk-designs. Examples of minimax D-optimal designs are presented for the logistic and power logistic models, including a dose-response design for rheumatoid arthritis patients.     

Trend tests for case-control studies of genetic markers: power,sample size and robustness

The Cochran-Armitage trend test is commonly used as a genotype-based test for candidate gene association. Corresponding to each underlying genetic model there is a particular set of scores assigned to the genotypes that maximizes its power. When the variance of the test statistic is known, the formulas for approximate power and associated sample size are readily obtained. In practice, however, the variance of the test statistic needs to be estimated. We present formulas for the required sample size to achieve a prespecified power that account for the need to estimate the variance of the test statistic. When the underlying genetic model is unknown one can incur a substantial loss of power when a test suitable for one mode of inheritance is used where another mode is the true one. Thus, tests having good power properties relative to the optimal tests for each model are useful. These tests are called efficiency robust and we study two of them: the maximin efficiency robust test is a linear combination of the standardized optimal tests that has high efficiency and the MAX test, the maximum of the standardized optimal tests. Simulation results of the robustness of these two tests indicate that the more computationally involved MAX test is preferable.     

Hierarchical commensurate and power prior models for adaptive incorporation of historical information in clinical trials

Bayesian clinical trial designs offer the possibility of a substantially reduced sample size, increased statistical power, and reductions in cost and ethical hazard. However when prior and current information conflict, Bayesian methods can lead to higher than expected type I error, as well as the possibility of a costlier and lengthier trial. This motivates an investigation of the feasibility of hierarchical Bayesian methods for incorporating historical data that are adaptively robust to prior information that reveals itself to be inconsistent with the accumulating experimental data. In this article, we present several models that allow for the commensurability of the information in the historical and current data to determine how much historical information is used. A primary tool is elaborating the traditional power prior approach based upon a measure of commensurability for Gaussian data. We compare the frequentist performance of several methods using simulations, and close with an example of a colon cancer trial that illustrates a linear models extension of our adaptive borrowing approach. Our proposed methods produce more precise estimates of the model parameters, in particular, conferring statistical significance to the observed reduction in tumor size for the experimental regimen as compared to the control regimen.     

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.     

Efficiency of population structures for mapping of Mendelian and imprinted quantitative trait loci in outbred pigs using variance component methods

In a simulation study different designs for a pure line pig population were compared for efficiency of mapping QTL using the variance component method. Phenotypes affected by a Mendelian QTL, a paternally expressed QTL, a maternally expressed QTL or by a QTL without an effect were simulated. In all alternative designs 960 progeny were phenotyped. Given the limited number of animals there is an optimum between the number of families and the family size. Estimation of Mendelian and parentally expressed QTL is more efficient in a design with large family sizes. Too small a number of sires should be avoided to minimize chances of sires to be non-segregating. When a large number of families is used, the number of haplotypes increases which reduces the accuracy of estimating the QTL effect and thereby reduces the power to show a significant QTL and to correctly position the QTL. Dense maps allow for smaller family size due to exploitation of LD-information. Given the different possible modes of inheritance of the QTL using 8 to16 boars, two litters per dam was optimal with respect to determining significance and correct location of the QTL for a data set consisting of 960 progeny. The variance component method combining linkage disequilibrium and linkage analysis seems to be an appropriate choice to analyze data sets which vary in marker density and which contain complex family structures.     

Two-phase sampling for simultaneous prevalence estimation and case detection

Two-phase designs for estimation of prevalence, where the first-phase classification is fallible and the second is accurate but relatively expensive, are not necessarily justified on efficiency grounds. However, they might be advantageous for dual-purpose studies, for example where prevalence estimation is followed by a clinical trial or case-control study, if they can identify cases of disease for the second study in a cost-effective way. Alternatively, they may be justified on ethical grounds if they can identify more, previously undetected but treatable cases of disease, than a simple random sample design. An approach to sampling is proposed, which formally combines the goals of efficient prevalence estimation and case detection by setting different notional study costs for investigating cases and noncases. Two variants of the method are compared with an "ethical" two-phase scheme proposed by Shrout and Newman (1989, Biometrics 45, 549-555), and with the most efficient scheme for prevalence estimation alone, in terms of the standard error of the prevalence estimate, the expected number of cases, and the fraction of cases among second-phase subjects, given a fixed budget. One variant yields the highest fraction and expected number of cases but also the largest standard errors. The other yields a higher fraction than Shrout and Newman's scheme and a similar number of cases but appears to do so more efficiently.     

The model of Monod, Wyman, and Changeux generates two sets of parameters when applied to oxygen binding in hemoglobin

The model of Monod, Wyman and Changeux is applied to binding phenomena where the Mass Law and its expansion are employed. In this communication the model of Monod, Wyman and Changeux (MWC) is applied to analyze the oxygen binding reaction in hemoglobin. The symmetrical structure of the MWC model with its three parameters is such that two sets of these parameters, rather than one, fit experimental data for the binding of oxygen to hemoglobin.     

An Approximation to the Bayes Optimal Multi-Stage Design in the Colton Model for Clinical Trials

A simple and operationally convenient approximation is proposed for the Bayes optimal multistage design within the Colton decision-theoretic model for the comparison of two medical treatments. The two- and three-stage designs are developed in full; the latter is found to be superior to several existing designs including some involving sequential adaptive sampling.     

A Bayesian adaptive design for dual-agent phase I–II oncology trials integrating efficacy data across stages

Combination of several anticancer treatments has typically been presumed to have enhanced drug activity. Motivated by a real clinical trial, this paper considers phase I–II dose finding designs for dual-agent combinations, where one main objective is to characterize both the toxicity and efficacy profiles. We propose a two-stage Bayesian adaptive design that accommodates a change of patient population in-between. In stage I, we estimate a maximum tolerated dose combination using the escalation with overdose control (EWOC) principle. This is followed by a stage II, conducted in a new yet relevant patient population, to find the most efficacious dose combination. We implement a robust Bayesian hierarchical random-effects model to allow sharing of information on the efficacy across stages, assuming that the related parameters are either exchangeable or nonexchangeable. Under the assumption of exchangeability, a random-effects distribution is specified for the main effects parameters to capture uncertainty about the between-stage differences. The inclusion of nonexchangeability assumption further enables that the stage-specific efficacy parameters have their own priors. The proposed methodology is assessed with an extensive simulation study. Our results suggest a general improvement of the operating characteristics for the efficacy assessment, under a conservative assumption about the exchangeability of the parameters a priori.