[PhD thesis of FCP.] Nowadays, genetics studies large amounts of very diverse variables. Mathematical statistics has evolved in parallel to its applications, with much recent interest high-dimensional settings. In the genetics of human common disease, a number of relevant problems can be formulated as tests of independence. We show how defining adequate premetric structures on the support spaces of the genetic data allows for novel approaches to such testing. This yields a solid theoretical framework, which reflects the underlying biology, and allows for computationally-efficient implementations. For each problem, we provide mathematical results, simulations and the application to real data.

Inferring linear relationships lies at the heart of many empirical investigations. A measure of linear dependence should correctly evaluate the strength of the relationship as well as qualify whether it is meaningful for the population. Pearson's correlation coefficient (PCC), the \textit{de-facto} measure for bivariate relationships, is known to lack in both regards. The estimated strength $r$ maybe wrong due to limited sample size, and nonnormality of data. In the context of statistical significance testing, erroneous interpretation of a $p$-value as posterior probability leads to Type I errors -- a general issue with significance testing that extends to PCC. Such errors are exacerbated when testing multiple hypotheses simultaneously. To tackle these issues, we propose a machine-learning-based predictive data calibration method which essentially conditions the data samples on the expected linear relationship. Calculating PCC using calibrated data yields a calibrated $p$-value that can be interpreted as posterior probability together with a calibrated $r$ estimate, a desired outcome not provided by other methods. Furthermore, the ensuing independent interpretation of each test might eliminate the need for multiple testing correction. We provide empirical evidence favouring the proposed method using several simulations and application to real-world data.

Recently, the binary expansion testing framework was introduced to test the independence of two continuous random variables by utilizing symmetry statistics that are complete sufficient statistics for dependence. We develop a new test by an ensemble method that uses the sum of squared symmetry statistics and distance correlation. Simulation studies suggest that this method improves the power while preserving the clear interpretation of the binary expansion testing. We extend this method to tests of independence of random vectors in arbitrary dimension. By random projections, the proposed binary expansion randomized ensemble test transforms the multivariate independence testing problem into a univariate problem. Simulation studies and data example analyses show that the proposed method provides relatively robust performance compared with existing methods.

We consider support recovery in the quadratic logistic regression setting - where the target depends on both p linear terms $x_i$ and up to $p^2$ quadratic terms $x_i x_j$. Quadratic terms enable prediction/modeling of higher-order effects between features and the target, but when incorporated naively may involve solving a very large regression problem. We consider the sparse case, where at most $s$ terms (linear or quadratic) are non-zero, and provide a new faster algorithm. It involves (a) identifying the weak support (i.e. all relevant variables) and (b) standard logistic regression optimization only on these chosen variables. The first step relies on a novel insight about correlation tests in the presence of non-linearity, and takes $O(pn)$ time for $n$ samples - giving potentially huge computational gains over the naive approach. Motivated by insights from the boolean case, we propose a non-linear correlation test for non-binary finite support case that involves hashing a variable and then correlating with the output variable. We also provide experimental results to demonstrate the effectiveness of our methods.