Penalized regression methods for interaction and mixed-effects models with applications to genomic and brain imaging data

In high-dimensional (HD) data, where the number of covariates (??) greatly exceeds the number of observations (??), estimation can benefit from the bet-on-sparsity principle, i.e., only a small number of predictors are relevant in the response. This assumption can lead to more interpretable models, improved predictive accuracy, and algorithms that are computationally efficient. In genomic and brain imaging studies, where the sample sizes are particularly small due to high data collection costs, we must often assume a sparse model because there isn't enough information to estimate ?? parameters. For these reasons, penalized regression methods such as the lasso and group-lasso have generated substantial interest since they can set model coefficients exactly to zero. In the penalized regression framework, many approaches have been developed for main effects. However, there is a need for developing interaction and mixed-effects models. Indeed, accurate capture of interactions may hold the potential to better understand biological phenomena and improve prediction accuracy since they may reflect important modulation of a biological system by an external factor. Furthermore, penalized mixed-effects models that account for correlations due to groupings of observations can improve sensitivity and specificity. This thesis is composed primarily of three manuscripts. In the first manuscript, we propose a method called sail for detecting non-linear interactions that automatically enforces the strong heredity property using both the ℓ₁ and ℓ₂ penalty functions. We describe a blockwise coordinate descent procedure for solving the objective function and provide performance metrics on both simulated and real data. The second manuscript develops a general penalized mixed effects model framework to account for correlations in genetic data due to relatedness called ggmix. Our method can accommodate several sparsity-inducing penalties such as the lasso, elastic net and group lasso and also readily handles prior annotation information in the form of weights. Our algorithm has theoretical guarantees of convergence and we again assess its performance in both simulated and real data. The third manuscript describes a novel strategy called eclust for dimension reduction that leverages the effects of an exposure variable with broad impact on HD measures. With eclust, we found improved prediction and variable selection performance compared to methods that do not consider the exposure in the clustering step, or to methods that use the original data as features. We further illustrate this modeling framework through the analysis of three data sets from very different fields, each with HD data, a binary exposure, and a phenotype of interest. We provide efficient implementations of all our algorithms in freely available and open source software.

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