In this dissertation a methodology is proposed for simultaneously evaluating the population bioequivalence (PBE) of a generic drug to a pre-licensed drug, or the bioequivalence of two formulations of a drug using multiple correlated pharmacokinetic metrics. The univariate criterion that is accepted by the food and drug administration (FDA) for testing population bioequivalence is generalized. Very few approaches for testing multivariate extensions of PBE have appeared in the literature. One method uses the trace of the covariance matrix as a measure of total variability, and another uses a pooled variance instead of the reference variance. The former ignores the correlation between the measurements while the later is not equivalent to the criterion proposed by the FDA in the univariate case, unless the variances of the test and reference are identical, which reduces the PBE to the average bioequivalence. The confidence interval approach is used to test the multivariate population bioequivalence by using a parametric bootstrap method to evaluate the 100% (1-alpha) confidence interval. The performance of the multivariate criterion is evaluated by a simulation study. The size and power of testing for bioequivalence using this multivariate criterion are evaluated in a simulation study by altering the mean differences, the variances, correlations between pharmacokinetic variables and sample size. A comparison between the two published approaches and the proposed criterion is demonstrated. Using nonlinear models and nonlinear mixed effects models, the multivariate population bioequivalence is examined. Finally, the proposed methods are illustrated by simultaneously testing the population bioequivalence for AUC and Cmax in two datasets.

The initial portion of this research focuses on appropriate parameter estimators within a general context of multivariate quality control. The goal of Phase I analysis of multivariate quality control data is to identify multivariate outliers and step changes so that the estimated control limits are sufficiently accurate for Phase II monitoring. High breakdown estimation methods based on the minimum volume ellipsoid (MVE) or the minimum covariance determinant (MCD) are well suited to detecting multivariate outliers in data. Because of the inherent difficulties in computation many algorithms have been proposed to obtain them. We consider the subsampling algorithm to obtain the MVE estimators and the FAST-MCD algorithm to obtain the MCD estimators. Previous studies have not clearly determined which of these two estimation methods is best for control chart applications. The comprehensive simulation study here gives guidance for when to use which estimator. Control limits are provided. High breakdown estimation methods such as MCD and MVE can be applied to a wide variety of multivariate quality control data. The final, lengthier portion of this research considers profile monitoring. Profile monitoring is a relatively new technique in quality control used when the product or process quality is best represented by a profile (or a curve) at each time period. The essential idea is often to model the profile via some parametric method and then monitor the estimated parameters over time to determine if there have been changes in the profiles. Because the estimated parameters may be correlated, it is convenient to monitor them using a multivariate control method such as the T-squared statistic. Previous modeling methods have not incorporated the correlation structure within the profiles. We propose the use of mixed models (both linear and nonlinear) to monitor linear and nonlinear profiles in order to account for the correlation structure within a profile. We consider various data scenarios and show using simulation when the mixed model approach is preferable to an approach that ignores the correlation structure. Our focus is on Phase I control chart applications. / Ph. D.

Product reliability is an important characteristic for all manufacturers, engineers and consumers. Industrial statisticians have been planning experiments for years to improve product quality and reliability. However, rarely do experts in the field of reliability have expertise in design of experiments (DOE) and the implications that experimental protocol have on data analysis. Additionally, statisticians who focus on DOE rarely work with reliability data. As a result, analysis methods for lifetime data for experimental designs that are more complex than a completely randomized design are extremely limited. This dissertation provides two new analysis methods for reliability data from life tests. We focus on data from a sub-sampling experimental design. The new analysis methods are illustrated on a popular reliability data set, which contains sub-sampling. Monte Carlo simulation studies evaluate the capabilities of the new modeling methods. Additionally, Monte Carlo simulation studies highlight the principles of experimental design in a reliability context. The dissertation provides multiple methods for statistical inference for the new analysis methods. Finally, implications for the reliability field are discussed, especially in future applications of the new analysis methods. / Ph. D.

La modélisation de la croissance des plantes a vu le jour à la fin du XXème siècle, à l’intersection de trois disciplines : l’agronomie, la botanique et l’informatique. Après un premier élan qui a donné naissance à un grand nombre de modèles, un deuxième courant a vu le jour au cours de la dernière décennie pour donner à ces modèles un formalisme mathématique et statistique rigoureux. Les travaux développés dans cette thèse s’inscrivent dans cette démarche et proposent deux axes de développement, l’un autour de l’évaluation et de la comparaison de modèles, et l’autre autour de l’étude de la variabilité inter-plantes.Dans un premier temps, nous nous sommes intéressés à la capacité prédictive des modèles de croissance de plantes, en appliquant une méthodologie permettant de construire et d’évaluer des modèles qui seront utilisés comme outils prédictifs. Une première étape d’analyse de sensibilité permet d’identifier les paramètres les plus influents afin d’élaborer une version plus robuste de chaque modèle, puis les capacités prédictives des modèles sont comparées à l’aide de critères appropriés. Cette étude a été appliquée au cas de la betterave sucrière mais peut se généraliser à d’autres plantes.La deuxième partie de la thèse concerne la prise en compte de la variabilité inter-individuelle dans les populations de plantes. Il existe en effet une forte variabilité entre plantes, d’origine génétique ou environnementale, dont il est nécessaire de tenir compte. Nous proposons dans cette thèse une approche basée sur l’utilisation de modèles (non linéaires) à effets mixtes pour caractériser la variabilité inter- individuelle. L’estimation paramétrique par maximum de vraisemblance nécessite l’utilisation de versions stochastiques de l’algorithme d’Espérance Maximisation basées sur des simulations de type Monte Carlo par Chaîne de Markov. Après une première application au cas de l’organogenèse chez la betterave sucrière, nous proposons une extension du modèle structure-fonction Greenlab à l’échelle de la population, appliqué aux cas de la betterave sucrière et du colza. / The modelling of plant growth and development was born at the end of the XXth century at the intersection of three disciplines: agronomy, botany and computer science. After a first period corresponding to the emergence of a lot of different models, a new trend has been initiated in the last decade to give these models a rigorous mathematical and statistical formalism. This thesis focuses on two main areas of development: (i) models evaluation and comparison, and (ii) inter-individual variability in plant populations.In the first part of the thesis, we study the predictive capacity of plant growth models, and we apply a two-step methodology to build and evaluate different models in a predictive perspective. In a first step, a sensitivity analysis is conducted to identify the most influential parameters and elaborate a more robust version of each model, and in a second step the predictive capacities of the models are compared using appropriate criteria. This study is carried out on sugar beet crops but can be easily generalized to other species.The second part of this thesis concerns the inter-individual variability in plant populations, which can be very high due to genetics or environmental varying conditions. This variability is rarely accounted for despite the major impact it can have at the agrosystem level. We proposed to take it into account using (nonlinear) mixed models, for which parameter estimation using maximum likelihood method relies on the use of stochastic variants of the Expectation-Maximization algorithm, based on Markov Chain Monte Carlo simulation techniques. We first apply this approach to the case of organogenesis in sugar beet populations, and secondly, we develop an extension of the functional-structural plant growth model Greenlab, from the individual to the population scale.

O trabalho se estruturou em torno de dois estudos. O primeiro avaliou o ajuste de um modelo não linear de efeito misto para descrever o afilamento do tronco de árvores clonais de eucalipto. O modelo utilizado para descrever as variações da altura em função do raio foi o logístico de quatro parâmetros que, por integração permitiu a estimação do volume das árvores. A incorporação de funções de variância no processo de ajuste resultou em redução significativa no valor do Critério de informação de Akaike, mas os resíduos não apresentaram melhorias notáveis. Com a finalidade de compatibilizar precisão e parcimônia, o modelo que considera as variações do afilamento como uma função da altura total e do raio à altura do peito mostrou-se como o mais indicado para a estimativa do volume de árvores por funções de afilamento. O segundo estudo analisou uma nova proposta para inventários florestais em plantios clonais de eucalipto que integra modelagem geoestatística, medições de circunferência das árvores em campo e a tecnologia LiDAR aeroembarcada. As estatísticas propostas mostraram que o modelo geoestatístico com função para média foi estatisticamente superior ao modelo com média constante, com erros reduzidos em até 40%. A altura das árvores que compuseram o grid de predição para aplicação do modelo geoestatístico foi obtida pelo processamento da nuvem de pontos dos dados LiDAR. Obtidos os pares de diâmetro e altura, aplicou-se o modelo de afilamento selecionado no primeiro artigo em que se observaram diferenças médias na predição do volume próximas a 0,7%, e 0,18% para contagem de árvores, ambas com tendências de subestimativas. Diante dos resultados obtidos, o método é considerado como promissor e trabalhos futuros visam gerar um banco de parcelas permanentes que propiciem estudos de crescimento e produção florestal. / This study investigates the use of mixed-effect model and the use of LiDAR based model to estimate volume from eucalyptus forest plantation. At the first part, this study evaluates nonlinear mixed-effects to model stem taper of monoclonal Eucalyptus trees. The relation between radius and height variation was described by the four-parameter logistic model that integration returns stem volume. Embedding variance functions to the estimation process decreased significantly the Akaike\'s Information Criterion but did not improve the residual analysis. The best model to estimate stem volume from taper equations explained the stem taper as a function of the commercial height and the radius at breast height. The second part investigated the volume estimation fusing geostatistic derived from field information and airborne laser scanning data. The model based on geostatistic assumptions was statistically superior to the traditional one, with errors 40% lower. Thus, the geostatistical model was applied over tree heights extracted from the laser cloud. To each combination of diameter and height, the taper equation form the first part of this study was used. The volume and the number of trees were underestimated in 0.7% and 0.18%, respectively. The results look promising, and more permanent plots are necessary to allow studies about growth and yield of forest.

O trabalho se estruturou em torno de dois estudos. O primeiro avaliou o ajuste de um modelo não linear de efeito misto para descrever o afilamento do tronco de árvores clonais de eucalipto. O modelo utilizado para descrever as variações da altura em função do raio foi o logístico de quatro parâmetros que, por integração permitiu a estimação do volume das árvores. A incorporação de funções de variância no processo de ajuste resultou em redução significativa no valor do Critério de informação de Akaike, mas os resíduos não apresentaram melhorias notáveis. Com a finalidade de compatibilizar precisão e parcimônia, o modelo que considera as variações do afilamento como uma função da altura total e do raio à altura do peito mostrou-se como o mais indicado para a estimativa do volume de árvores por funções de afilamento. O segundo estudo analisou uma nova proposta para inventários florestais em plantios clonais de eucalipto que integra modelagem geoestatística, medições de circunferência das árvores em campo e a tecnologia LiDAR aeroembarcada. As estatísticas propostas mostraram que o modelo geoestatístico com função para média foi estatisticamente superior ao modelo com média constante, com erros reduzidos em até 40%. A altura das árvores que compuseram o grid de predição para aplicação do modelo geoestatístico foi obtida pelo processamento da nuvem de pontos dos dados LiDAR. Obtidos os pares de diâmetro e altura, aplicou-se o modelo de afilamento selecionado no primeiro artigo em que se observaram diferenças médias na predição do volume próximas a 0,7%, e 0,18% para contagem de árvores, ambas com tendências de subestimativas. Diante dos resultados obtidos, o método é considerado como promissor e trabalhos futuros visam gerar um banco de parcelas permanentes que propiciem estudos de crescimento e produção florestal. / This study investigates the use of mixed-effect model and the use of LiDAR based model to estimate volume from eucalyptus forest plantation. At the first part, this study evaluates nonlinear mixed-effects to model stem taper of monoclonal Eucalyptus trees. The relation between radius and height variation was described by the four-parameter logistic model that integration returns stem volume. Embedding variance functions to the estimation process decreased significantly the Akaike\'s Information Criterion but did not improve the residual analysis. The best model to estimate stem volume from taper equations explained the stem taper as a function of the commercial height and the radius at breast height. The second part investigated the volume estimation fusing geostatistic derived from field information and airborne laser scanning data. The model based on geostatistic assumptions was statistically superior to the traditional one, with errors 40% lower. Thus, the geostatistical model was applied over tree heights extracted from the laser cloud. To each combination of diameter and height, the taper equation form the first part of this study was used. The volume and the number of trees were underestimated in 0.7% and 0.18%, respectively. The results look promising, and more permanent plots are necessary to allow studies about growth and yield of forest.