The main research topic in this dissertation is the development of the closure method of multiple testing procedures. Considering a general procedure that allows the underlying test statistics as well as the associated parameters to be dependent, we first propose a step-down procedure controlling the FWER, which is defined as the probability of committing at least one false discovery. Holm (1979) first proposed a step-down procedure for multiple hypothesis testing with a control of the familywise error rate (FWER) under any kind of dependence. Under the normal distributional setup, Seneta and Chen (2005) sharpened the Holm procedure by taking into account the correlations between the test statistics. In this dissertation, the Seneta-Chen procedure is further modified yielding a more powerful FWER controlling procedure. We then advance our research and propose another step-down procedure to control the generalized FWER (k-FWER), which is defined as the probability of making at least k false discoveries. We compare our proposed k-FWER procedure with the Lehmann and Romano (2005) procedure. The proposed k-FWER procedure is more powerful, particularly when there is a strong dependence in the tests. When the proportion of true null hypotheses is expected to be small, the traditional tests are usually conservative by a factor associated with pi0, which is the proportion of true null hypotheses among all null hypotheses. Under independence, two procedures controlling the FWER and the k-FWER are proposed in this dissertation. Simulations are carried out to show that our procedures often provide much better FWER or k-FWER control and power than the traditional procedures. / Statistics

Data resulting from eye-tracking experiments allows researchers to analyze the decision making process as study participants consider alternative items prior to the ultimate end point selection. The aim of such an analysis is to extract the underlying cognitive decision making process that develops throughout the experiment. Resulting data can be difficult to analyze, however, as eye-tracking curves have very high autocorrelation values which consists of measurements that are milliseconds apart, as mandated by the nature of eye movements. We propose an analytic approach to eye-tracking data that tests for statistically significant differences at every time point along the curve while calculating an appropriate familywise error rate correction which is based upon an autoregressive correlation assumption of the test statistics. Our technique has been implemented in the R package BDOTS with various extensions relevant to the real-world analysis of highly correlated nonlinear data. A popular alternative approach to analyzing eye-tracking data is to fit mixed models to the area under the curve. Through simulation studies we provide evidence for the benefit of using information criterion measures in selection of the random effects structure and make an argument against current industry-standard approaches such as sequential likelihood ratio tests or always using a maximal random effects structure.

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The importance of multiplicity adjustment has gained wide recognition in modern scientific research. Without it, there will be too many spurious results and reproducibility becomes an issue; with it, if overtly conservative, discoveries will be made more difficult. In the current literature on repeated testing of multiple hypotheses, Bonferroni-based methods are still the main vehicle carrying the bulk of multiplicity adjustment. There is room for power improvement by suitably utilizing both hypothesis-wise and analysis- wise dependencies. This research will contribute to the development of a natural group-sequential extension of the classical stepwise multiple testing procedures, such as Dunnett’s stepdown and Hochberg’s step-up procedures. It is shown that the proposed group-sequential procedures strongly control the familywise error rate while being more powerful than the recently developed class of group-sequential Bonferroni-Holm’s procedures. Particularly in this research, a convexity property is discovered for the distribution of the maxima of pairwise null P-values with the underlying test statistics having distributions such as bivariate normal, t, Gamma, F, or Archimedean copulas. Such property renders itself for an immediate use in improving Holm’s procedure by incorporating pairwise dependencies of P-values. The improved Holm’s procedure, as all stepdown multiple testing procedures, can also be naturally extended to group-sequential setting. / Statistics

The central theme of this thesis is the construction of simultaneous confidence regions (SCR) corresponding to stepwise multiple comparison procedures (MCP). The first chapter is devoted to the theory of multiple comparisons, including the class of closed testing procedures which contains every MCP that strongly con- trols the familywise error rate. The second chapter is concerned with the gene- ral principle of construction of SCR corresponding to closed testing procedures. These general results are used in the third and the forth chapter for deriving the SCR corresponding to a subclass of closed testing procedures which are based on weighted Bonferroni tests. The SCR corresponding to the Holm, the Holm(W), the fixed-sequence and the fallback MCP are derived explicitly. The theoretical results are numerically illustrated on a bioequivalence study. In the fifth chapter we briefly discuss the SCR corresponding to the Hommel, the Hochberg and the step-down Dunnett MCP.

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Considérons un vecteur gaussien Y de loi N (m,sigma²Idn) et X une matrice de dimension n x p avec Y observé, m inconnu, Sigma et X connus. Dans le cadre du modèle linéaire, m est supposé être une combinaison linéaire des colonnes de X. En petite dimension, lorsque n ≥ p et que ker (X) = 0, il existe alors un unique paramètre Beta* tel que m = X Beta* ; on peut alors réécrire Y sous la forme Y = X Beta* + Epsilon. Dans le cadre du modèle linéaire gaussien en petite dimension, nous construisons une nouvelle procédure de tests multiples contrôlant le FWER pour tester les hypothèses nulles Beta*i = 0 pour i appartient à [[1,p]]. Cette procédure est appliquée en métabolomique au travers du programme ASICS qui est disponible en ligne. ASICS permet d'identifier et de quantifier les métabolites via l'analyse des spectres RMN. En grande dimension, lorsque n < p on a ker (X) ≠ 0, ainsi le paramètre Beta* décrit précédemment n'est pas unique. Dans le cas non bruité lorsque Sigma = 0, impliquant que Y = m, nous montrons que les solutions du système linéaire d'équations Y = X Beta avant un nombre de composantes non nulles minimales s'obtiennent via la minimisation de la "norme" lAlpha avec Alpha suffisamment petit. / Let Y be a Gaussian vector distributed according to N (m,sigma²Idn) and X a matrix of dimension n x p with Y observed, m unknown, sigma and X known. In the linear model, m is assumed to be a linear combination of the columns of X In small dimension, when n ≥ p and ker (X) = 0, there exists a unique parameter Beta* such that m = X Beta*; then we can rewrite Y = Beta* + Epsilon. In the small-dimensional linear Gaussian model framework, we construct a new multiple testing procedure controlling the FWER to test the null hypotheses Beta*i = 0 for i belongs to [[1,p]]. This procedure is applied in metabolomics through the freeware ASICS available online. ASICS allows to identify and to qualify metabolites via the analyse of RMN spectra. In high dimension, when n < p we have ker (X) ≠ 0 consequently the parameter Beta* described above is no longer unique. In the noiseless case when Sigma = 0, implying thus Y = m, we show that the solutions of the linear system of equation Y = X Beta having a minimal number of non-zero components are obtained via the lalpha with alpha small enough.