Mendelian randomization is an instrumentation method that uses genetic instruments to estimate, via two-stage least squares regression for example, a causal relationship between an exposure and an outcome when the relationship is confounded by one or more unmeasured confounders. Mendelian randomization can handle confounding bias provided that the instruments are valid, i.e., that they meet three key assumptions. While two of the three assumptions can usually be satisfied, the third assumption is often invalidated by a genetic phenomenon called pleiotropy. In the presence of invalid instruments, the estimate of the causal effect of exposure on the outcome may be severely biased. To assess the potential presence of an invalid instrument in single-instrument studies, Glymour et al. (2012) proposed a method, hereinafter referred to as the simple difference approach, which uses the sign of the difference between the ordinary least squares estimator of the outcome on the exposure and the two-stage least squares estimator calculated using the instrument. Based on this approach, we introduce three methods applicable to Mendelian randomization with multiple instruments. The first method is the global difference approach and corresponds to a simple generalization of the simple difference approach to the case of multiple instruments that aims to detect whether one or more instruments are invalid. Next, we introduce the individual differences and the grouped differences approaches, two methods that generalize the simple difference approach to identify potentially invalid instruments and provide new estimates of the causal effect of the exposure on the outcome. The methods are evaluated using a theoretical investigation of the impact that invalid instruments have on the convergence of the ordinary least squares and two-stage least squares estimators as well as with a simulation study that compares the accuracy of the respective estimators and the ability of the corresponding methods to detect invalid instruments.

A few bootstrap approaches exist to create confidence intervals. Some difficulties appear for the specific case of a proportion when the usual estimator, the proportion of success in a sample, is 0. In the classical case where the observations are independently and identically distributed (i.i.d.) from a Bernoulli distribution, the bootstrap samples only contain zeros with probability 1 and the resulting bootstrap confidence intervals are degenerate at the value 0. We are facing the same problem in the survey sampling case when we apply the bootstrap method to a sample with all observations equal to 0. A possible solution is suggested by the estimator found in the confidence intervals of [Wilson, 1927] and [Agresti et Coull, 1998] where they use ˜p the proportion of success in a augmented sample consisting of adding two successes and two failures to the original sample. The proposed solution is to use the bootstrap method on ˆp but where the bootstrap is based on the augmented sample with two additional successes and failures, whether the sample comes from i.i.d. Bernoulli variables or from a simple random sample. Results show that a version of the percentile method is the most efficient bootstrap method to construct confidence intervals for a proportion both in the classical setting or in the case of a simple random sample. Our results also show that this percentile interval can compete with the best traditional methods.

This thesis introduces smoothed pseudo-population bootstrap methods for the purposes of variance estimation and the construction of confidence intervals for finite population quantiles. In an i.i.d. context, Hall et al. (1989) have shown that resampling from a smoothed estimate of the distribution function instead of the usual empirical distribution function can improve the convergence rate of the bootstrap variance estimator of a sample quantile. We extend the smoothed bootstrap to the survey sampling framework by implementing it in pseudo-population bootstrap methods. Given a kernel function and a bandwidth, it consists of smoothing the pseudo-population from which bootstrap samples are drawn using the original sampling design. Two designs are discussed, namely simple random sampling and Poisson sampling. The implementation of the proposed algorithms requires the specification of the bandwidth. To do so, we develop a plug-in selection method along with grid search selection methods based on bootstrap estimates of two performance metrics. We present the results of a simulation study which provide empirical evidence that the smoothed approach is more efficient than the standard approach for estimating the variance of a quantile estimator together with mixed results regarding confidence intervals.

In this thesis, we compare by simulation various bootstrap methods for evaluating the accuracy of a survey estimate for three sampling designs: simple random sampling, Poisson sampling and conditional Poisson sampling. Most bootstrap methods have been derived to reproduce the usual variance estimate for a linear parameter such as the average or the total of a population. We evaluate the method on their capacity to estimate the variance of survey estimates and to build bootstrap confidence intervals using four different techniques, namely asymptotic intervals, bootstrap percentile, basic bootstrap and t-bootstrap intervals for totals, but also ratios, correlation coefficients, medians, and Gini indices. The implementation of t-bootstrap intervals for several bootstrap methods is an original contribution of this dissertation.

In the observational data framework, estimation must account for unmeasured confounders, which affect both exposure and outcome. The reason for this is that if the effect of confounders is not considered, the causal effect estimate will suffer from confounding bias. The econometrics literature addresses this problem by using the instrumental variables method, which enables causal inference when confounding is present by using variables which have the property of being "random" in the posited model and strongly associated with the exposure. Hence, they can serve as instruments to randomize exposure, thus mimicking the setting of randomized control trials—the gold standard to establish causality. With genetic data, the issue can be approached similarly using Mendelian randomization. Mendelian randomization studies apply the instrumental variables approach to genetics by exploiting the natural randomness of allele allocation and using single-nucleotide polymorphisms (SNPs) as instrumental variables to obtain a "quasi-randomized" exposure phenotype. However, the choice of SNPs is not arbitrary; instead, it relies on a set of assumptions for validity, some of which cannot be verified in most empirical settings. When using multiple SNPs as instruments, one might be tempted to use the widely known two-stage least squares estimator, given its simplicity. Yet, the use of this estimator comes at the high price of assuming that all SNPs verify the set of assumptions, whereas in reality, SNPs can be invalid, for instance, due to pleiotropy. To include the possibility that some SNPs may be invalid, Kang et al. (2014) proposed the sisVIVE estimator, which mitigates the effect of invalid SNPs up to a threshold of 50% of invalid SNPs, thereby providing a point estimate of the causal effect. We add to their contribution by exploring a bootstrap method to construct confidence intervals for sisVIVE. Results obtained from simulations and an application to a real dataset from the Montreal Heart Institute's Biobank are presented.

In cardiology, we denote by RR and QT the time lapse of the full cardiac cycle and of going from states Q to T, respectively. An important quantity for cardiologists is QTc, the value of QT when RR is 1 second. To estimate it, many couples of RR and QT are measured, a regression model is fitted and the prediction for RR = 1 is the estimated QTc. A major difficulty is that observed RRs are usually much less than 1, leading to an extrapolation. Preliminary studies seem to suggest that among the six models considered by researchers in cardiology, the most general one (with three parameters) is overparametrized. This research first, demonstrates this overparametrization, and then measures its consequences when that model is used for estimating QTc. The set of all curves from the most general model with QTs and RRs similar to those observed in cardiology is considered. For each of these curves, we found another one sufficiently similar from two of the two parameters models among the six considered. To measure the impact of such an overparametrization, the delta procedure, requiring no distributional assumption, has been developed and applied. In the end, for a given model, the QTc has been estimated from the set of all curves leading to a least squares criterion at most delta = 1% over the minimum. The range of corresponding QTcs from the most general model, when resulting from an extrapolation, is huge. We therefore conclude that this model is not appropriate for estimating QTc.

Most women diagnosed with juvenile idiopathic arthritis (JIA) continue to suffer from arthritis in adulthood. Some of the drugs used to treat arthritis such as corticosteroids and non-steroidal anti-inflammatory drugs (NSAIDs) are not recommended during pregnancy. The objective of this thesis is to estimate the impact of these drugs on adverse birth outcomes in women previously diagnosed with JIA and their baby. Administrative data on drug prescriptions and hospitalizations in a cohort of 1756 women with a history of JIA were used to determine individual histories of drug use for the treatment of arthritis during pregnancy and during the year leading to the pregnancy. Two sub-cohorts of women who suffered from JIA were created : one corresponding to the pregnancy and the other to the pregnancy and the year leading to the pregnancy. The events of interest were : congenital anomalies, neonatal adverse outcomes, maternal adverse outcomes and small for gestational age babies. Proportions of the events ranged between 11,52% and 37,08%. Drugs were modelled in terms of use or duration of use during each of the study periods. Logistic regression models were fitted to measure the association between drugs and each of the events, adjusting for the following potential confounding variables : hypertension before pregnancy, maternal age and graduating from high school. The consumption of corticosteroids was associated with a statistically significant increased risk of congenital anomalies but had no impact on the other adverse events. No statistically significant associations were observed between consumption of NSAIDs and the adverse events of interest.

Our aim in this thesis is to conduct the goodness-of-fit test based on empirical characteristic functions proposed by Jiménez-Gamero et al. (2009) in the case of the bivariate Poisson distribution. We first evaluate the test’s behaviour in the case of the univariate Poisson distribution and find that the estimated type I error probabilities are close to the nominal values. Next, we extend it to the bivariate case and calculate and compare its power with the dispersion index test for the bivariate Poisson, Crockett’s Quick test for the bivariate Poisson and the two test families proposed by Novoa-Muñoz et Jiménez-Gamero (2014). Simulation results show that the probability of type I error is close to the claimed level and that it is generally less powerful than other tests. We also discovered that the dispersion index test should be bilateral whereas it rejects for large values only. Finally, the p-value of all these tests is calculated on a real dataset from soccer. The p-value of the test is 0,009 and we reject the hypothesis that the data come from a Poisson bivariate while the tests proposed by Novoa-Muñoz et Jiménez-Gamero (2014) leads to a different conclusion.

Following an internship with the company Hatch, we have access to datasets that are composed of wind speed time series measured at different sites accross the world and over several years. The wind speed engineers from Hatch are using these datasets jointly with Environment Canada databases in order to ascertain the wind energy potential of these sites and to know whether it is worth installing wind turbines there. For a few years, some companies are also offering mesoscale simulations of wind speed based on different environmental characteristics from the site we want to evaluate. We would like to know if it is worth paying for those mesoscale datasets and if they can be used to provide better estimations of the wind energy potential. Among other things, these data could be used to provide a better estimation of the long term mean wind speed. Since we already possess measured datasets, we will also use them to test, with statistical methods, the methodology currently used and the different steps leading to an estimation of the wind energy production. First of all, we will see what are the different methods that could be used to extrapolate wind speed to a wind turbine’s height and we will evaluate those methods with the mean squared extrapolation error. Also, we will study wind distribution modelling by the Weibull distribution and consider its variability over time. Finally, cross-validation and block bootstrap will be used to see whether we should use mesoscale data instead of wind data from Environment Canada or whether it would even be beneficial to use both kind of data to predict wind speed. In summary, the whole methodology used by wind speed engineers to estimate the energy production will be tested from a statistical point of view and we will attempt to propose changes in this methodology that could improve the estimation of the wind speed annual energy production.

The aim of this thesis is to study the bootstrap variance estimators of a statistic based on imputed survey data. Applying a bootstrap method designed for complete survey data (full response) in the presence of imputed values and treating them as true observations may lead to underestimation of the variance. In this context, Shao and Sitter (1996) introduced a bootstrap procedure in which the variable under study and the response status are bootstrapped together and bootstrap non-respondents are imputed using the imputation method applied on the original sample. The resulting bootstrap variance estimator is valid when the sampling fraction is small. In Chapter 1, we begin by doing a survey of the existing bootstrap methods for (complete and imputed) survey data and, for the first time in the literature, present them in a unified framework. In Chapter 2, we introduce a new bootstrap procedure to estimate the variance under the non-response model approach when the uniform non-response mechanism is assumed. Using only information about the response rate, unlike Shao and Sitter (1996) which requires the individual response status, the bootstrap response status is generated for each selected bootstrap sample leading to a valid bootstrap variance estimator even for non-negligible sampling fractions. In Chapter 3, we investigate pseudo-population bootstrap approaches and we consider a more general class of non-response mechanisms. We develop two pseudo-population bootstrap procedures to estimate the variance of an imputed estimator with respect to the non-response model and the imputation model approaches. These procedures are also valid even for non-negligible sampling fractions.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.