Sigmoid curves have found broad applicability in biological sciences and biopharmaceutical research during the last decades. A well planned experiment design is essential to accurately estimate the parameters of the model. In contrast to a large literature and extensive results on optimal designs for linear models, research on the design for nonlinear, including sigmoid curve, models has not kept pace. Furthermore, most of the work in the optimal design literature for nonlinear models concerns the characterization of minimally supported designs. These minimal, optimal designs are frequently criticized for their inability to check goodness of fit, as there are no additional degrees of freedom for the testing. This design issue can be a serious problem, since checking the model adequacy is of particular importance when the model is selected without complete certainty. To assess for lack of fit, we must add at least one extra distinct design point to the experiment. The goal of this dissertation is to identify optimal or highly efficient designs capable of checking the fit for sigmoid curve models. In this dissertation, we consider some commonly used sigmoid curves, including logistic, probit and Gompertz models with two, three, or four parameters. We use D-optimality as our design criterion. We first consider adding one extra point to the design, and consider five alternative designs and discuss their suitability to test for lack of fit. Then we extend the results to include one more additional point to better understand the compromise among the need of detecting lack of fit, maintaining high efficiency and the practical convenience for the practitioners. We then focus on the two-parameter Gompertz model, which is widely used in fitting growth curves yet less studied in literature, and explore three-point designs for testing lack of fit under various error variance structures. One reason that nonlinear design problems are so challenging is that, with nonlinear models, information matrices and optimal designs depend on the unknown model parameters. We propose a strategy to bypass the obstacle of parameter dependence for the theoretical derivation. This dissertation also successfully characterizes many commonly studied sigmoid curves in a generalized way by imposing unified parameterization conditions, which can be generalized and applied in the studies of other sigmoid curves. We also discuss Gompertz model with different error structures in finding an extra point for testing lack of fit. / Statistics

Bayesian optimal designs have received increasing attention in recent years, especially in biomedical and clinical trials. Bayesian design procedures can utilize the available prior information of the unknown parameters so that a better design can be achieved. However, a difficulty in dealing with the Bayesian design is the lack of efficient computational methods. In this research, a hybrid computational method, which consists of the combination of a rough global optima search and a more precise local optima search, is proposed to efficiently search for the Bayesian D-optimal designs for multi-variable generalized linear models. Particularly, Poisson regression models and logistic regression models are investigated. Designs are examined for a range of prior distributions and the equivalence theorem is used to verify the design optimality. Design efficiency for various models are examined and compared with non-Bayesian designs. Bayesian D-optimal designs are found to be more efficient and robust than non-Bayesian D-optimal designs. Furthermore, the idea of the Bayesian sequential design is introduced and the Bayesian two-stage D-optimal design approach is developed for generalized linear models. With the incorporation of the first stage data information into the second stage, the two-stage design procedure can improve the design efficiency and produce more accurate and robust designs. The Bayesian two-stage D-optimal designs for Poisson and logistic regression models are evaluated based on simulation studies. The Bayesian two-stage optimal design approach is superior to the one-stage approach in terms of a design efficiency criterion. / Ph. D.

This paper is concerned with the problem of computing theapproximate D-optimal design for polynomial regression with weight function w(x)>0 on the design interval I=[m_0-a,m_0+a]. It is shown that if w'(x)/w(x) is a rational function on I and a is close to zero, then the problem of constructing D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix including a finite number of auxiliary unknown constants, which can be approximated by a Taylor expansion. We provide a recursive algorithm to compute Taylor expansion of these constants. Moreover, the D-optimal interior support points are the zeros of a polynomial which has coefficients that can be computed from a linear system.

Les grandes entreprises ferroviaires intègrent au niveau du matériel roulant une grande variété de systèmes complexes qui se doivent d’être fiables et ce, dès le démarrage du service commercial. Ce travail de thèse propose une méthodologie expérimentale pour l’amélioration de la robustesse d’un système prédominant, à savoir l’accès voyageurs. L'objectif est d'améliorer sa fiabilité intrinsèque dans un laps de temps raisonnable dans le cadre de projet industriel contraint par le temps. La méthodologie expérimentale proposée s’appuie sur la méthode des essais aggravés et accélérés de fiabilité, et se veut être optimisée grâce à l’utilisation de plans d’expériences D-optimaux. Après une analyse bibliographique, suivie d’une étude sur l’utilisation des plans d’expériences D-optimaux, ce travail expose les méthodes et moyens expérimentaux mis en place pour utiliser les plans d’expériences dans un contexte industriel. La dernière partie de cette thèse contient les résultats quantitatifs et qualitatifs issus des expérimentations réaliséessur le banc d'essais du système accès voyageurs développé par Bombardier. / The train manufacturer companies handle a large number of complex systems in their trains. These systems must be reliable to ensure reliability of the final product as soon the entry into commercial services. This thesis provides an experimental solution to improve robustness of a predominate system, namely passengers access system. The goal is to improve inherent reliability in a reasonable amount of time to be integrated in a project phase. This experimental method is based on testing aggravated and accelerated method temporally optimized with using of D-optimal design of experiment. After a literature review, followed by a focus on the definition and use of experimental design D-optimal, this work will present experimental methods and means used to set up the experimental design in an industrial context. The last part of this thesis contains quantitative and qualitative results performed on the passenger’s access test bench developed by Bombardier.

<p>New techniques and approaches for organic synthesis, purification and biological testing are enabling pharmaceutical industries to produce and test increasing numbers of compounds every year. Surprisingly, this has not led to more new drugs reaching the market, prompting two questions – why is there not a better correlation between their efforts and output, and can it be improved? One possible way to make the drug discovery process more efficient is to ensure, at an early stage, that the tested compounds are diverse, representative and of high quality. In addition the biological evaluation systems have to be relevant and reliable. The diversity of the tested compounds could be ensured and the reliability of the biological assays improved by using Design Of Experiments (DOE) more frequently and effectively. However, DOE currently offers insufficient options for these purposes, so there is a need for new, tailor-made DOE strategies. The aim of the work underlying this thesis was to develop and evaluate DOE approaches for diverse compound selection and efficient assay optimisation. This resulted in the publication of two new DOE strategies; D-optimal Onion Design (DOOD) and Rectangular Experimental Designs for Multi-Unit Platforms (RED-MUP), both of which are extensions to established experimental designs.</p><p>D-Optimal Onion Design (DOOD) is an extension to D-optimal design. The set of possible objects that could be selected is divided into layers and D-optimal selection is applied to each layer. DOOD enables model-based, but not model-dependent, selections in discrete spaces to be made, since the selections are not only based on the D-optimality criterion, but are also biased by the experimenter’s prior knowledge and specific needs. Hence, DOOD selections provide controlled diversity.</p><p>Assay development and optimisation can be a major bottleneck restricting the progress of a project. Although DOE is a recognised tool for optimising experimental systems, there has been widespread unwillingness to use it for assay optimisation, mostly because of the difficulties involved in performing experiments according to designs in 96-, 384- and 1536- well formats. The RED-MUP framework combines classical experimental designs orthogonally onto rectangular experimental platforms, which facilitates the execution of DOE on these platforms and hence provides an efficient tool for assay optimisation.</p><p>In combination, these two strategies can help uncovering the crossroads between biology and chemistry in drug discovery as well as lead to higher information content in the data received from biological evaluations, providing essential information for well-grounded decisions as to the future of the project. These two strategies can also help researchers identify the best routes to take at the crossroads linking biological and chemical elements of drug discovery programs.</p>

In this paper we investigate (d + 1)-point D-optimal designs for d-th degree polynomial regression with weight function w(x) > 0 on the interval [a, b]. Suppose that w'(x)/w(x) is a rational function and the information of whether the optimal support contains the boundary points a and b is available. Then the problem of constructing (d + 1)-point D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix with k auxiliary unknown constants. We characterize the weight functions corresponding to the cases when k= 0 and k= 1. Then, we can solve (d + 1)-point D-optimal designs directly from differential equation (k = 0) or via eigenvalue problems (k = 1). The numerical results show us an interesting relationship between optimal designs and ordered eigenvalues.

Consider the minimally-supported D-optimal designs for dth degree polynomial regression with bounded and positive weight function on a compact interval. We show that the optimal design converges weakly to the arcsin distribution as d goes to infinity. Comparisons of the optimal design with the arcsin distribution and D-optimal arcsin support design by D-efficiencies are also given. We also show that if the design interval is [−1, 1], then the minimally-supported D-optimal design converges to the D-optimal arcsin support design with the specific weight function 1/¡Ô(£\-x^2), £\>1, as £\¡÷1+.

Consider the D-optimal designs for the dth-degree polynomial regression model with a bounded and positive weight function on a compact interval. As the degree of the model goes to infinity, we show that the D-optimal design converges weakly to the arcsin distribution. If the weight function is equal to 1, we derive the formulae of the values of the D-criterion for five classes of designs including (i) uniform density design; (ii) arcsin density design; (iii) J_{1/2,1/2} density design; (iv) arcsin support design and (v) uniform support design. The comparison of D-efficiencies among these designs are investigated; besides, the asymptotic expansions and limits of their D-efficiencies are also given. It shows that the D-efficiency of the arcsin support design is the highest among the first four designs.

New techniques and approaches for organic synthesis, purification and biological testing are enabling pharmaceutical industries to produce and test increasing numbers of compounds every year. Surprisingly, this has not led to more new drugs reaching the market, prompting two questions – why is there not a better correlation between their efforts and output, and can it be improved? One possible way to make the drug discovery process more efficient is to ensure, at an early stage, that the tested compounds are diverse, representative and of high quality. In addition the biological evaluation systems have to be relevant and reliable. The diversity of the tested compounds could be ensured and the reliability of the biological assays improved by using Design Of Experiments (DOE) more frequently and effectively. However, DOE currently offers insufficient options for these purposes, so there is a need for new, tailor-made DOE strategies. The aim of the work underlying this thesis was to develop and evaluate DOE approaches for diverse compound selection and efficient assay optimisation. This resulted in the publication of two new DOE strategies; D-optimal Onion Design (DOOD) and Rectangular Experimental Designs for Multi-Unit Platforms (RED-MUP), both of which are extensions to established experimental designs. D-Optimal Onion Design (DOOD) is an extension to D-optimal design. The set of possible objects that could be selected is divided into layers and D-optimal selection is applied to each layer. DOOD enables model-based, but not model-dependent, selections in discrete spaces to be made, since the selections are not only based on the D-optimality criterion, but are also biased by the experimenter’s prior knowledge and specific needs. Hence, DOOD selections provide controlled diversity. Assay development and optimisation can be a major bottleneck restricting the progress of a project. Although DOE is a recognised tool for optimising experimental systems, there has been widespread unwillingness to use it for assay optimisation, mostly because of the difficulties involved in performing experiments according to designs in 96-, 384- and 1536- well formats. The RED-MUP framework combines classical experimental designs orthogonally onto rectangular experimental platforms, which facilitates the execution of DOE on these platforms and hence provides an efficient tool for assay optimisation. In combination, these two strategies can help uncovering the crossroads between biology and chemistry in drug discovery as well as lead to higher information content in the data received from biological evaluations, providing essential information for well-grounded decisions as to the future of the project. These two strategies can also help researchers identify the best routes to take at the crossroads linking biological and chemical elements of drug discovery programs.

假設二元反應曲線為簡單線性羅吉斯迴歸模型(Simple Linear Logistic Regression Model)，在樣本數為偶數的前題下，所謂的最佳D型設計(D-Optimal Design)是直接將半數的樣本點配置在第17.6個百分位數，而另一半則配置在第82.4個百分位數。很遺憾的是，這兩個位置在參數未知的情況下是無法決定的，因此逐次實驗設計法(Sequential Experimental Designs)在應用上就有其必要性。在大樣本的情況下，本文所探討的逐次實驗設計法在理論上具有良好的漸近最佳D型性質(Asymptotic D-Optimality)。尤其重要的是，這些特性並不會因為起始階段的配置不盡理想而消失，影響的只是收斂的快慢而已。但是在實際應用上，這些大樣本的理想性質卻不是我們關注的焦點。實驗步驟收斂速度的快慢，在小樣本的考慮下有決定性的重要性。基於這樣的考量，本文將提出三種起始階段設計的方法並透過模擬比較它們之間的優劣性。 / The D-optimal design is well known to be a two-point design for the simple linear logistic regression function model. Specif-ically , one half of the design points are allocated at the 17.6- th percentile, and the other half at the 82.4-th percentile. Since the locations of the two design points depend on the unknown parameters, the actual 2-locations can not be obtained. In order to dilemma, a sequential design is somehow necessary in practice. Sequential designs disscused in this context have some good properties that would not disappear even the initial stgae is not good enough under large sample size. The speed of converges of the sequential designs is influenced by the initial stage imposed under small sample size. Based on this, three initial stages will be provided in this study and will be compared through simulation conducted by C++ language.