Being able to explore a relationship between two life events is of great interest to scientists from different disciplines. Some issues of particular concern are, for example, the connection between smoking cessation and pregnancy (Thompson and Pantoja-Galicia 2003), the interrelation between entry into marriage for individuals in a consensual union and first pregnancy (Blossfeld and Mills 2003), and the association between job loss and divorce (Charles and Stephens 2004, Huang 2003 and Yeung and Hofferth 1998). Establishing causation in observational studies is seldom possible. Nevertheless, if one of two events tends to precede the other closely in time, a causal interpretation of an association between these events can be more plausible. The role of longitudinal surveys is crucial, then, since they allow sequences of events for individuals to be observed. Thompson and Pantoja-Galicia (2003) discuss in this context several notions of temporal association and ordering, and propose an approach to investigate a possible relationship between two lifetime events. In longitudinal surveys individuals might be asked questions of particular interest about two specific lifetime events. Therefore the joint distribution might be advantageous for answering questions of particular importance. In follow-up studies, however, it is possible that interval censored data may arise due to several reasons. For example, actual dates of events might not have been recorded, or are missing, for a subset of (or all) the sampled population, and can be established only to within specified intervals. Along with the notions of temporal association and ordering, Thompson and Pantoja-Galicia (2003) also discuss the concept of one type of event "triggering" another. In addition they outline the construction of tests for these temporal relationships. The aim of this thesis is to implement some of these notions using interval censored data from longitudinal complex surveys. Therefore, we present some proposed tools that may be used for this purpose. This dissertation is divided in five chapters, the first chapter presents a notion of a temporal relationship along with a formal nonparametric test. The mechanisms of right censoring, interval censoring and left truncation are also overviewed. Issues on complex surveys designs are discussed at the end of this chapter. For the remaining chapters of the thesis, we note that the corresponding formal nonparametric test requires estimation of a joint density, therefore in the second chapter a nonparametric approach for bivariate density estimation with interval censored survey data is provided. The third chapter is devoted to model shorter term triggering using complex survey bivariate data. The semiparametric models in Chapter 3 consider both noncensoring and interval censoring situations. The fourth chapter presents some applications using data from the National Population Health Survey and the Survey of Labour and Income Dynamics from Statistics Canada. An overall discussion is included in the fifth chapter and topics for future research are also addressed in this last chapter.

Being able to explore a relationship between two life events is of great interest to scientists from different disciplines. Some issues of particular concern are, for example, the connection between smoking cessation and pregnancy (Thompson and Pantoja-Galicia 2003), the interrelation between entry into marriage for individuals in a consensual union and first pregnancy (Blossfeld and Mills 2003), and the association between job loss and divorce (Charles and Stephens 2004, Huang 2003 and Yeung and Hofferth 1998). Establishing causation in observational studies is seldom possible. Nevertheless, if one of two events tends to precede the other closely in time, a causal interpretation of an association between these events can be more plausible. The role of longitudinal surveys is crucial, then, since they allow sequences of events for individuals to be observed. Thompson and Pantoja-Galicia (2003) discuss in this context several notions of temporal association and ordering, and propose an approach to investigate a possible relationship between two lifetime events. In longitudinal surveys individuals might be asked questions of particular interest about two specific lifetime events. Therefore the joint distribution might be advantageous for answering questions of particular importance. In follow-up studies, however, it is possible that interval censored data may arise due to several reasons. For example, actual dates of events might not have been recorded, or are missing, for a subset of (or all) the sampled population, and can be established only to within specified intervals. Along with the notions of temporal association and ordering, Thompson and Pantoja-Galicia (2003) also discuss the concept of one type of event "triggering" another. In addition they outline the construction of tests for these temporal relationships. The aim of this thesis is to implement some of these notions using interval censored data from longitudinal complex surveys. Therefore, we present some proposed tools that may be used for this purpose. This dissertation is divided in five chapters, the first chapter presents a notion of a temporal relationship along with a formal nonparametric test. The mechanisms of right censoring, interval censoring and left truncation are also overviewed. Issues on complex surveys designs are discussed at the end of this chapter. For the remaining chapters of the thesis, we note that the corresponding formal nonparametric test requires estimation of a joint density, therefore in the second chapter a nonparametric approach for bivariate density estimation with interval censored survey data is provided. The third chapter is devoted to model shorter term triggering using complex survey bivariate data. The semiparametric models in Chapter 3 consider both noncensoring and interval censoring situations. The fourth chapter presents some applications using data from the National Population Health Survey and the Survey of Labour and Income Dynamics from Statistics Canada. An overall discussion is included in the fifth chapter and topics for future research are also addressed in this last chapter.

The main objective in sampling is to select a sample from a population in order to estimate some unknown population parameter, usually a total or a mean of some interesting variable. When the units in the population do not have the same probability of being included in a sample, it is called unequal probability sampling. The inclusion probabilities are usually chosen to be proportional to some auxiliary variable that is known for all units in the population. When unequal probability sampling is applicable, it generally gives much better estimates than sampling with equal probabilities. This thesis consists of six papers that treat unequal probability sampling from a finite population of units. A random sample is selected according to some specified random mechanism called the sampling design. For unequal probability sampling there exist many different sampling designs. The choice of sampling design is important since it determines the properties of the estimator that is used. The main focus of this thesis is on evaluating and comparing different designs. Often it is preferable to select samples of a fixed size and hence the focus is on such designs. It is also important that a design has a simple and efficient implementation in order to be used in practice by statisticians. Some effort has been made to improve the implementation of some designs. In Paper II, two new implementations are presented for the Sampford design. In general a sampling design should also have a high level of randomization. A measure of the level of randomization is entropy. In Paper IV, eight designs are compared with respect to their entropy. A design called adjusted conditional Poisson has maximum entropy, but it is shown that several other designs are very close in terms of entropy. A specific situation called real time sampling is treated in Paper III, where a new design called correlated Poisson sampling is evaluated. In real time sampling the units pass the sampler one by one. Since each unit only passes once, the sampler must directly decide for each unit whether or not it should be sampled. The correlated Poisson design is shown to have much better properties than traditional methods such as Poisson sampling and systematic sampling.

Le problème d'intérêt est d'estimer la fonction de concentration et l'aire sous la courbe (AUC) à travers l'estimation des paramètres d'un modèle de régression linéaire avec un processus d'erreur autocorrélé. On construit un estimateur linéaire sans biais simple de la courbe de concentration et de l'AUC. On montre que cet estimateur construit à partir d'un plan d'échantillonnage régulier approprié est asymptotiquement optimal dans le sens où il a exactement la même performance asymptotique que le meilleur estimateur linéaire sans biais (BLUE). De plus, on montre que le plan d'échantillonnage optimal est robuste par rapport à la misspecification de la fonction d'autocovariance suivant le critère du minimax. Lorsque des observations répétées sont disponibles, cet estimateur est consistant et a une distribution asymptotique normale. Les résultats obtenus sont généralisés au processus d'erreur de Hölder d'indice compris entre 0 et 2. Enfin, pour des tailles d'échantillonnage petites, un algorithme de recuit simulé est appliqué à un modèle pharmacocinétique avec des erreurs corrélées. / The problem of interest is to estimate the concentration curve and the area under the curve (AUC) by estimating the parameters of a linear regression model with autocorrelated error process. We construct a simple linear unbiased estimator of the concentration curve and the AUC. We show that this estimator constructed from a sampling design generated by an appropriate density is asymptotically optimal in the sense that it has exactly the same asymptotic performance as the best linear unbiased estimator (BLUE). Moreover, we prove that the optimal design is robust with respect to a misspecification of the autocovariance function according to a minimax criterion. When repeated observations are available, this estimator is consistent and has an asymptotic normal distribution. All those results are extended to the error process of Hölder with index including between 0 and 2. Finally, for small sample sizes, a simulated annealing algorithm is applied to a pharmacokinetic model with correlated errors.

Les performances en mathématiques sont associées à de nombreux enjeux dans notre société, des enjeux de nature individuelle et des enjeux de nature sociétale. Malgré le fait que les élèves québécois réussissent très bien sur la scène internationale en mathématiques, dans le cadre des évaluations à grande échelle en éducation, les analyses selon le genre, elles, font état d’écarts préoccupants. En effet, à plusieurs des cycles de ces grandes enquêtes, les filles ont obtenu des résultats statistiquement inférieurs à ceux des garçons. L’anxiété mathématique, pourrait expliquer, à tout le moins en partie, les écarts de performances observés entre les filles et les garçons (Stoet et al., 2016). En effet, les recherches menées sur le sujet depuis les années 70 révèlent que, de manière générale, chez les adolescents et les adultes, les filles ont tendance à se dire davantage anxieuses à l’égard des mathématiques que les garçons (p.ex. Else-Quest et al., 2010; Hyde et al., 1990; Stoet et al. 2016). Également, dans ces mêmes groupes d’âge, une corrélation linéaire négative est observée entre le niveau d’anxiété mathématique des individus et leurs performances en mathématiques (p.ex. Barroso et al.,2021; Hembree, 1990; Ma, 1999; Zhang et al., 2019). En 2003 et en 2012, le PISA s’est intéressé à l’anxiété mathématique. En effet, comme les mathématiques ont constitué le domaine majeur d’évaluation lors de ces deux enquêtes, l’anxiété mathématique a été documentée, à ces occasions, au même titre que bon nombre d’autres facteurs non-cognitifs liés aux performances dans le domaine. Le portrait canadien, issu du PISA de 2003 et de 2012, supporte et renforce l’idée selon laquelle l’anxiété mathématique pourrait contribuer aux écarts de performance observés entre les garçons et les filles, en mathématiques, au Québec. Devant l’absence de portrait québécois en regard du phénomène, et souhaitant, à terme, fournir des leviers en vue de réduire les écarts de performances observés entre les filles et les garçons québécois, nous énonçons l’objectif général de recherche comme suit : quantifier les écarts d’anxiété mathématique entre les filles et les garçons, étudier le lien anxiété mathématique/performances en mathématiques et ensuite, identifier des facteurs ayant le potentiel de réduire les écarts d’anxiété mathématique observés, chez les élèves québécois francophones de 15 ans, à partir d’une analyse secondaire des données du PISA de 2003 et de 2012. Afin de répondre à cet objectif général de recherche, nous définissons trois objectifs spécifiques de recherche. Chacun d’eux est traité dans un des trois articles de cette thèse par articles. Le premier objectif spécifique de notre recherche vise à identifier les considérations méthodologiques inhérentes aux données issues du PISA et à proposer des techniques d’analyse qui permettent de les traiter, adéquatement. Cet objectif spécifique est traité dans le premier article, un article de nature méthodologique. Les deuxième objectif spécifique de notre recherche vise à quantifier les écarts d’anxiété mathématique entre les garçons et les filles francophones de 15 ans du Québec, à partir d’une analyse secondaire des données du PISA de 2003 et de 2012, et à étudier le lien négatif anxiété mathématique/performances en mathématiques, chez ces mêmes élèves. Cet objectif spécifique est traité dans le second article de la thèse, un article de nature empirique, qui prend appui sur les fondements méthodologiques proposés dans l’article 1. Le troisième objectif spécifique de notre recherche vise à identifier des facteurs qui permettent d’expliquer les écarts d’anxiété mathématique observés entre les garçons et les filles francophones du Québec et qui ont le potentiel de guider, à terme, la mise en œuvre d’interventions visant à réduire les écarts observés. Répondre à cet objectif spécifique fait l’objet du troisième article de la thèse. Cet article prend également appui sur les fondements méthodologiques exposés dans l’article 1. Des résultats issus de l’article 2, il ressort que : 1) en moyenne, les filles francophones de 15 ans du Québec font état d’un niveau d’anxiété mathématique statistiquement plus élevé que les garçons et 2) le lien entre l’anxiété mathématique et les performances en mathématiques est un lien négatif et équivalent entre les garçons et les filles francophones du Québec. Des résultats issus de l’article 3, il ressort que les écarts d’anxiété mathématique observés entre les filles et les garçons québécois francophones ayant participé au PISA de 203 et de 2012 pourraient s’expliquer par : 1) un concept de soi en mathématiques plus faible chez les filles et 2) un lien direct genre/anxiété mathématique. Des implications scientifiques et pratiques de ces résultats découlent des recommandations en vue de réduire les écarts d’anxiété mathématique observés entre les filles et les garçons francophones du Québec, de même que des avenues de recherche à explorer dans le futur. Ces recommandations et avenues de recherches sont présentées au terme de la thèse. / Performance in mathematics is linked to a number of issues in our society, both individual and societal. Even though Quebec students are doing very well internationally in mathematics, in large-scale educational assessments, gender-based analyses reveal disturbing gaps. Over the course of several cycles of these large-scale surveys, girls achieved statistically lower results than boys. Mathematics anxiety could at least partially explain the performance gaps observed between girls and boys (Stoet et al., 2016). Indeed, research conducted since the 1970s has shown that, in general, in adolescents and adults, girls tend to report greater anxiety about mathematics than boys (e.g.Else-Quest et al., 2010; Hyde et al., 1990; Stoet et al. 2016). Furthermore, in these same age groups, there is a negative linear correlation between individuals' level of anxiety in mathematics and their performance in mathematics (e.g.Hembree, 1990; Ma, 1999; Zhang et al., 2019). In 2003 and 2012, PISA turned its attention to mathematics anxiety. Indeed, since mathematics was the main domain of assessment in both surveys, mathematical anxiety was documented on these occasions, along with many other non-cognitive factors related to performance in the domain. The Canadian portrait, from both the 2003 and 2012 PISA surveys, supports and reinforces the idea that mathematics anxiety could explain, at least in part, the observed performance gaps between boys and girls in mathematics in Quebec. Given the absence of a Quebec portrait of the phenomenon, and wishing, in the long run, to provide levers for reducing the observed performance gaps between Quebec girls and boys, we state the general research objectives as follows: 1) quantify the gaps between girls and boys with respect to mathematics anxiety, 2) study the link between mathematics anxiety and mathematical performance, and finally 3) identify factors that may reduce the gaps observed in mathematical anxiety among 15-year-old French-speaking students in Québec, based on a secondary analysis of the 2003 and 2012 PISA data. 7 In order to meet this overall research objective, three specific research objectives are identified. Each of these is discussed in one of the three articles of the present thesis per article. The first specific objective of our research is to identify the methodological considerations inherent to the PISA data and to propose analytical techniques that will allow us to adequately address them. This specific objective is discussed in the first article, a methodology article. The second specific objective of our research is to quantify the differences in mathematical anxiety between 15-year-old French-speaking boys and girls in Quebec, based on a secondary analysis of the 2003 and 2012 PISA data, and to investigate the negative relation between mathematical anxiety and mathematical performance in these same students. This specific objective is discussed in the second article of the thesis, an empirical article, which is based on the methodological bases proposed in Article 1. The third specific objective of our research is to identify factors that contribute to explaining the differences observed in mathematical anxiety between French-speaking boys and girls in Quebec and have the potential to guide, over the long run, the implementation of interventions aimed at reducing the observed gaps. The third paper in the thesis focuses on this specific objective. This article is also based on the methodological bases set out in Article 1. The results of Article 2 show that: 1) on average, French-speaking 15-year-old girls in Quebec report a statistically higher level of mathematical anxiety than boys and 2) the relationship between mathematical anxiety and mathematical performance is negative and equivalent between French-speaking boys and girls in Quebec. From the results of Article 3, it appears that the differences in mathematical anxiety observed between Quebec French-speaking girls and boys who participated in PISA 2003 and 2012 can be explained by: 1) a lower perception of mathematical competence among girls and 2) a direct gender/mathematical anxiety link. The scientific and practical implications of these results lead to recommendations to reduce the differences observed in mathematical anxiety among French-speaking girls and boys in Quebec, as well as lines of future research. These recommendations and lines of research are presented at the end of the thesis.