Os experimentos agronômicos que envolvem somente um fator longitudinal são bastante comuns. No entanto, existem casos em que as observações são tomadas considerando dois ou mais desses fatores, como nos casos em que são feitas medidas de uma variável resposta em profundidades diferentes ao longo do tempo, por exemplo. Admite-se que essas observações, tomadas de modo sistemático em cada unidade experimental, sejam correlacionadas e as variâncias nos diferentes níveis do fator longitudinal sejam heterogêneas. Com o uso de modelos mistos, essa correlação entre medidas repetidas e a heterogeneidade de variâncias podem ser modeladas convenientemente. Para que esses modelos sejam ajustados a um conjunto de dados envolvendo presença de dois fatores longitudinais, existe a necessidade de se adaptarem algumas estruturas de variâncias e covariâncias que são comuns em experimentos com somente um fator longitudinal. O objetivo do presente trabalho é utilizar a classe dos modelos lineares mistos para estudar a massa seca de raiz no solo de uma plantação de cana-de-açúcar. O experimento foi casualizado em blocos e as parcelas receberam quatro doses de nitrogênio. Foram feitas medidas repetidas ao longo de dois fatores longitudinais, sendo um qualitativo ordinal (profundidades) e um quantitativo (distâncias da linha de plantio). Por meio dos testes de razão de verossimilhanças, de Wald e utilizando os critérios de informação AIC e BIC, selecionou-se uma estrutura de covariâncias parcimoniosa e outra estrutura para explicar o comportamento médio das respostas. A verificação do ajuste foi feita por meio de gráficos de diagnósticos de resíduos. / Agronomic experiments involving only one longitudinal factor are quite common. However, there are cases that the observations are made by considering two or more of these factors such as where measurements are made in a response variable at different depths along the time, for example. It is admitted that these observations, taken in a systematic way in each experimental unit are correlated and variances are heterogeneous in different levels of longitudinal factor. Using mixed models, this correlation between repeated measures and heterogeneity of variances can be modeled conveniently. To fit these models to data set involving presence of two longitudinal factors, there is need to adapt some variance and covariance structures that are common in experiments with only one longitudinal factor. The objective of this work is to use the class of linear mixed models to study the dry root mass in the soil of a plantation of cane sugar. The experiment was the randomized complete blocks design and parcels received four doses of nitrogen. Repeated measurements were made along two longitudinal factors being one ordinal qualitative (depths) and one quantitative (distances from the row). With the aid of likelihood ratio, Wald tests and using the AIC and BIC information criteria, we selected a parsimonious covariance structure and another structure to explain the average behavior of the responses. Checking the fit was made using diagnostic graphics of residuals.

In HIV/AIDS studies, viral load (the number of copies of HIV-1 RNA) and CD4 cell counts are important biomarkers of the severity of viral infection, disease progression, and treatment evaluation. Recently, joint models, which have the capability on the bias reduction and estimates' efficiency improvement, have been developed to assess the longitudinal process, survival process, and the relationship between them simultaneously. However, the majority of the joint models are based on mean regression, which concentrates only on the mean effect of outcome variable conditional on certain covariates. In fact, in HIV/AIDS research, the mean effect may not always be of interest. Additionally, if obvious outliers or heavy tails exist, mean regression model may lead to non-robust results. Moreover, due to some data features, like left-censoring caused by the limit of detection (LOD), covariates with measurement errors and skewness, analysis of such complicated longitudinal and survival data still poses many challenges. Ignoring these data features may result in biased inference. Compared to the mean regression model, quantile regression (QR) model belongs to a robust model family, which can give a full scan of covariate effect at different quantiles of the response, and may be more robust to extreme values. Also, QR is more flexible, since the distribution of the outcome does not need to be strictly specified as certain parametric assumptions. These advantages make QR be receiving increasing attention in diverse areas. To the best of our knowledge, few study focuses on the QR-based joint models and applies to longitudinal-survival data with multiple features. Thus, in this dissertation research, we firstly developed three QR-based joint models via Bayesian inferential approach, including: (i) QR-based nonlinear mixed-effects joint models for longitudinal-survival data with multiple features; (ii) QR-based partially linear mixed-effects joint models for longitudinal data with multiple features; (iii) QR-based partially linear mixed-effects joint models for longitudinal-survival data with multiple features. The proposed joint models are applied to analyze the Multicenter AIDS Cohort Study (MACS) data. Simulation studies are also implemented to assess the performance of the proposed methods under different scenarios. Although this is a biostatistical methodology study, some interesting clinical findings are also discovered.

A primary focus of wildlife research is to understand how habitat conditions and human activities impact the health of wild animals. External factors, both natural and anthropogenic that impact the ability of an animal to acquire food and build energy reserves have important implications for reproductive success, avoidance of predators, and the ability to withstand disease, and periods of food scarcity. In the analyses presented here, I quantify the impacts of habitat quality and anthropogenic disturbance on indicators of health for individuals in a threatened grizzly bear population in Alberta, Canada. The first analysis relates spatial patterns of hair cortisol concentrations, a promising indicator of long-term stress in mammals, measured from 304 grizzly bears to a variety of continuous environmental variables representative of habitat quality (e.g., crown closure, landcover, and vegetation productivity), topographic conditions (e.g., elevation and terrain ruggedness), and anthropogenic disturbances (e.g., roads, forest harvest blocks, and oil and gas well-sites). Hair cortisol concentration point data were integrated with continuous variables by creating a stress surface for male and female bears using kernel density estimation validated through bootstrapping. The relationships between hair cortisol concentrations for males and females and environmental variables were quantified using random forests, and landscape scale stress levels for both genders was predicted based on observed relationships. Low female stress levels were found to correspond with regions with high levels of anthropogenic disturbance and activity. High female stress levels were associated primarily with high-elevation parks and protected areas. Conversely, low male stress levels were found to correspond with parks and protected areas and spatially limited moderate to high stress levels were found in regions with greater anthropogenic disturbance. Of particular concern for conservation is the observed relationship between low female stress and sink habitats which have high mortality rates and high energetic costs. Extending the first analysis, the second portion of this research examined the impacts of scale-specific habitat selection and relationships between biology, habitat quality, and anthropogenic disturbance on body condition in 85 grizzly bears represented using a body condition index. Habitat quality and anthropogenic variables were represented at multiple scales using isopleths of a utilization distribution calculated using kernel density estimation for each bear. Several hypotheses regarding the influence of biology, habitat quality, and anthropogenic disturbance on body condition quantified using linear mixed-effects models were evaluated at each habitat selection scale using the small sample Aikake Information Criterion. Biological factors were influential at all scales as males had higher body condition than females, and body condition increased with age for both genders. At the scale of most concentrated habitat selection, the biology and habitat quality hypothesis had the greatest support and had a positive effect on body condition. A component of biology, the influence of long-term stress, which had a negative impact on body condition, was most pronounced within the biology and habitat quality hypothesis at this scale. As the scale of habitat selection was represented more broadly, support for the biology and anthropogenic disturbance hypothesis increased. Anthropogenic variables of particular importance were distance decay to roads, density of secondary linear features, and density of forest harvest areas which had a negative relationship with body condition. Management efforts aimed to promote landscape conditions beneficial to grizzly bear health should focus on promoting habitat quality in core habitat and limiting anthropogenic disturbance within larger grizzly bear home ranges. / Graduate / 0768 / 0463 / 0478 / mathieub@uvic.ca

Cette thèse est principalement consacrée au développement de méthodes de sélection de modèles par maximum de vraisemblance pénalisée dans le cadre de données complexes. Un premier travail porte sur la sélection des modèles linéaires généralisés dans le cadre de données stratifiées, caractérisées par la mesure d’observations ainsi que de covariables au sein de différents groupes (ou strates). Le but de l’analyse est alors de déterminer quelles covariables influencent de façon globale (quelque soit la strate) les observations mais aussi d’évaluer l’hétérogénéité de cet effet à travers les strates.Nous nous intéressons par la suite à la sélection des modèles non linéaires à effets mixtes utilisés dans l’analyse de données longitudinales comme celles rencontrées en pharmacocinétique de population. Dans un premier travail, nous décrivons un algorithme de type SAEM au sein duquel la pénalité est prise en compte lors de l’étape M en résolvant un problème de régression pénalisé à chaque itération. Dans un second travail, en s’inspirant des algorithmes de type gradient proximaux, nous simplifions l’étape M de l’algorithme SAEM pénalisé précédemment décrit en ne réalisant qu’une itération gradient proximale à chaque itération. Cet algorithme, baptisé Stochastic Approximation Proximal Gradient algorithm (SAPG), correspond à un algorithme gradient proximal dans lequel le gradient de la vraisemblance est approché par une technique d’approximation stochastique.Pour finir, nous présentons deux travaux de modélisation statistique, réalisés au cours de cette thèse. / This thesis is mainly devoted to the development of penalized maximum likelihood methods for the study of complex data.A first work deals with the selection of generalized linear models in the framework of stratified data, characterized by the measurement of observations as well as covariates within different groups (or strata). The purpose of the analysis is then to determine which covariates influence in a global way (whatever the stratum) the observations but also to evaluate the heterogeneity of this effect across the strata.Secondly, we are interested in the selection of nonlinear mixed effects models used in the analysis of longitudinal data. In a first work, we describe a SAEM-type algorithm in which the penalty is taken into account during step M by solving a penalized regression problem at each iteration. In a second work, inspired by proximal gradient algorithms, we simplify the M step of the penalized SAEM algorithm previously described by performing only one proximal gradient iteration at each iteration. This algorithm, called Stochastic Approximation Proximal Gradient Algorithm (SAPG), corresponds to a proximal gradient algorithm in which the gradient of the likelihood is approximated by a stochastic approximation technique.Finally, we present two statistical modeling works realized during this thesis.

Os experimentos agronômicos que envolvem somente um fator longitudinal são bastante comuns. No entanto, existem casos em que as observações são tomadas considerando dois ou mais desses fatores, como nos casos em que são feitas medidas de uma variável resposta em profundidades diferentes ao longo do tempo, por exemplo. Admite-se que essas observações, tomadas de modo sistemático em cada unidade experimental, sejam correlacionadas e as variâncias nos diferentes níveis do fator longitudinal sejam heterogêneas. Com o uso de modelos mistos, essa correlação entre medidas repetidas e a heterogeneidade de variâncias podem ser modeladas convenientemente. Para que esses modelos sejam ajustados a um conjunto de dados envolvendo presença de dois fatores longitudinais, existe a necessidade de se adaptarem algumas estruturas de variâncias e covariâncias que são comuns em experimentos com somente um fator longitudinal. O objetivo do presente trabalho é utilizar a classe dos modelos lineares mistos para estudar a massa seca de raiz no solo de uma plantação de cana-de-açúcar. O experimento foi casualizado em blocos e as parcelas receberam quatro doses de nitrogênio. Foram feitas medidas repetidas ao longo de dois fatores longitudinais, sendo um qualitativo ordinal (profundidades) e um quantitativo (distâncias da linha de plantio). Por meio dos testes de razão de verossimilhanças, de Wald e utilizando os critérios de informação AIC e BIC, selecionou-se uma estrutura de covariâncias parcimoniosa e outra estrutura para explicar o comportamento médio das respostas. A verificação do ajuste foi feita por meio de gráficos de diagnósticos de resíduos. / Agronomic experiments involving only one longitudinal factor are quite common. However, there are cases that the observations are made by considering two or more of these factors such as where measurements are made in a response variable at different depths along the time, for example. It is admitted that these observations, taken in a systematic way in each experimental unit are correlated and variances are heterogeneous in different levels of longitudinal factor. Using mixed models, this correlation between repeated measures and heterogeneity of variances can be modeled conveniently. To fit these models to data set involving presence of two longitudinal factors, there is need to adapt some variance and covariance structures that are common in experiments with only one longitudinal factor. The objective of this work is to use the class of linear mixed models to study the dry root mass in the soil of a plantation of cane sugar. The experiment was the randomized complete blocks design and parcels received four doses of nitrogen. Repeated measurements were made along two longitudinal factors being one ordinal qualitative (depths) and one quantitative (distances from the row). With the aid of likelihood ratio, Wald tests and using the AIC and BIC information criteria, we selected a parsimonious covariance structure and another structure to explain the average behavior of the responses. Checking the fit was made using diagnostic graphics of residuals.

1 AbstraktEN The diploma thesis deals with linear mixed effects models. In the first chap- ter, we discuss parameter estimation and hypothesis testing in the linear mixed effects models. The second chapter is dedicated to graphical diagnostics. We look at the suitable diagnostic plots for residuals and random effects estimates. It is closely described, how the violations of assumptions affect the diagnostic plots. In the third chapter we have consequences of the violations of assumptions on the parameter estimates and results of hypothesis testing for fixed effects. 1

Introduction: The current development strategy of population pharmacokinetic models is a complex and iterative process that is manually performed by modellers. Such a strategy is time-demanding, subjective, and dependent on the modellers’ experience. This thesis presents a novel model building tool that automates the development process of pharmacokinetic (PK) structural models. Methods: Modelsearch is a tool in Pharmpy library, an open-source package for pharmacometrics modelling, that searches for the best structural model using an exhaustive stepwise search algorithm. Given a dataset, a starting model and a pre-specified model search space of structural model features, the tool creates and fits a series of candidate models that are then ranked based on a selection criterion, leading to the selection of the best model. The Modelsearch tool was used to develop structural models for 10 clinical PK datasets (5 orally and 5 i.v. administered drugs). A starting model for each dataset was generated using the assemblerr package in R, which included a first-order (FO) absorption without any absorption delay for oral drugs, one-compartment disposition, FO elimination, a proportional residual error model, and inter-individual variability on the starting model parameters with a correlation between clearance (CL) and central volume of distribution (VC). The model search space included aspects of absorption and absorption delay (for oral drugs), distribution and elimination. In order to understand the effects of different IIV structures on structural model selection, five model search approaches were investigated that differ in the IIV structure of candidate models: 1. naïve pooling, 2. IIV on starting model parameters only, 3. additional IIV on mean delay time parameter, 4. additional diagonal IIVs on newly added parameters, and 5. full block IIVs. Additionally, the implementation of structural model selection in the workflow of the fully automatic model development was investigated. Three strategies were evaluated: SIR, SRI, and RSI depending on the development order of structural model (S), IIV model (I) and residual error model (R). Moreover, the NONMEM errors encountered when using the tool were investigated and categorized in order to be handled in the automatic model building workflow. Results: Differences in the final selected structural models for each drug were observed between the five different model search approaches. The same distribution components were selected through Approaches 1 and 2 for 6/10 drugs. Approach 2 has also identified an absorption delay component in 4/5 oral drugs, whilst the naïve pooling approach only identified an absorption delay model in 2 drugs. Compared to Approaches 1 and 2, Approaches 3, 4 and 5 tended to select more complex models and more often resulted in minimization errors during the search. For the SIR, SRI and RSI investigations, the same structural model was selected in 9/10 drugs with a significant higher run time in RSI strategy compared to the other strategies. The NONMEM errors were categorized into four categories based on the handling suggestions which is valuable to further improve the tool in its automatic error handling. Conclusions: The Modelsearch tool was able to automatically select a structural model with different strategies of setting the IIV model structure. This novel tool enables the evaluation of numerous combinations of model components, which would not be possible using a traditional manual model building strategy. Furthermore, the tool is flexible and can support multiple research investigations for how to best implement structural model selection in a fully automatic model development workflow.

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In longitudinal studies, data collected within a subject or cluster are somewhat correlated by their very nature and special cares are needed to account for such correlation in the analysis of data. Under the framework of longitudinal studies, three topics are being discussed in this thesis. In chapter 2, the joint modelling of multivariate longitudinal process consisting of different types of outcomes are discussed. In the large cohort study of UK north Stafforshire osteoarthritis project, longitudinal trivariate outcomes of continuous, binary and ordinary data are observed at baseline, year 3 and year 6. Instead of analysing each process separately, joint modelling is proposed for the trivariate outcomes to account for the inherent association by introducing random effects and the covariance matrix G. The influence of covariance matrix G on statistical inference of fixed-effects parameters has been investigated within the Bayesian framework. The study shows that by joint modelling the multivariate longitudinal process, it can reduce the bias and provide with more reliable results than it does by modelling each process separately. Together with the longitudinal measurements taken intermittently, a counting process of events in time is often being observed as well during a longitudinal study. It is of interest to investigate the relationship between time to event and longitudinal process, on the other hand, measurements taken for the longitudinal process may be potentially truncated by the terminated events, such as death. Thus, it may be crucial to jointly model the survival and longitudinal data. It is popular to propose linear mixed-effects models for the longitudinal process of continuous outcomes and Cox regression model for survival data to characterize the relationship between time to event and longitudinal process, and some standard assumptions have been made. In chapter 3, we try to investigate the influence on statistical inference for survival data when the assumption of mutual independence on random error of linear mixed-effects models of longitudinal process has been violated. And the study is conducted by utilising conditional score estimation approach, which provides with robust estimators and shares computational advantage. Generalised sufficient statistic of random effects is proposed to account for the correlation remaining among the random error, which is characterized by the data-driven method of modified Cholesky decomposition. The simulation study shows that, by doing so, it can provide with nearly unbiased estimation and efficient statistical inference as well. In chapter 4, it is trying to account for both the current and past information of longitudinal process into the survival models of joint modelling. In the last 15 to 20 years, it has been popular or even standard to assume that longitudinal process affects the counting process of events in time only through the current value, which, however, is not necessary to be true all the time, as recognised by the investigators in more recent studies. An integral over the trajectory of longitudinal process, along with a weighted curve, is proposed to account for both the current and past information to improve inference and reduce the under estimation of effects of longitudinal process on the risk hazards. A plausible approach of statistical inference for the proposed models has been proposed in the chapter, along with real data analysis and simulation study.

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