Statistical and Computational Approaches for Data Integration and Constrained Variable Selection in Large Datasets

Abstract

With the number of covariates, sample size, and heterogeneity in datasets continuously increasing, the incorporation of prior domain knowledge or the addition of structural constraints in a model represents an attractive means to perform informed variable selection on high numbers of potential predictors. The growing complexity of individual datasets has been accompanied by their increasing availability, as researchers nowadays can access ever-expanding biobanks and other large clinical datasets. Integration of external datasets can increase the generalizability of locally-gathered data, but these datasets can be affected by context-specific confounders, necessitating weighted integration methods to differentiate datasets of variable quality.

In Chapter 2, we present a method to perform weighted data integration based on minimizing the local data leave-one-out cross-validation (LOOCV) error, under the assumption that the local data is generated from the set of unknown true parameters. We demonstrate how the optimization of the LOOCV error for various models can be written as functions of external dataset weights. Furthermore, we develop an accompanying reduced space approach that reduces the weighted integration of any number of external datasets to a two-parameter optimization. The utility of the weighted data integration method in comparison to existing methods is shown through extensive simulation work mimicking heterogeneous clinical data, as well as in two real-world examples. The first examines kidney transplant patients from the Scientific Registry of Transplant Recipients and the second looks at the genomic data of bladder cancer patients from The Cancer Genome Atlas. Ongoing work on calculating standard error estimates and developing significance testing under a false discovery rate framework is also presented.

In Chapter 3, we devise a fast solution to the equality-constrained lasso problem with a two-stage algorithm: first obtaining candidate covariates subsets of increasing size from unconstrained lasso problems and then leveraging an efficient alternating direction method of multipliers (ADMM) algorithm. Our “candidate subset approach” produces the same solution path as solving the constrained lasso over the entire predictor space, and in simulation studies, our approach is over an order of magnitude faster than existing methods. The ability to solve the equality-constrained lasso with multiple constraints and with a large number of potential predictors is demonstrated in a microbiome regression analysis and a myeloma survival analysis, neither of which could be solved by naively fitting the constrained lasso on all predictors.

In Chapter 4, we aim to extend the candidate subset approach for constrained variable selection to accommodate different penalty functions and inequality constraints. Despite its desirable selection properties, it is well-known that the lasso is biased for large regression coefficients; to address this shortcoming, we consider our approach with two non-convex penalty functions, SCAD and MCP. Furthermore, we also consider the approach with inequality constraints and dual equality/inequality constraints, which greatly increases the number of potential applications. We demonstrate that the properties of the candidate subset approach, in terms of its speed and producing the same solution over the whole predictor space, additionally hold for these two extensions.