Extensions to Canonical Correlation Analysis and Principal Components Analysis with Applicatoins to Survival and Brain Imaging Data

Canonical correlation analysis (CCA) is a method for finding a low dimension representation of the linear associations between two sets of variables. Likewise principal components analysis (PCA) is a tool for finding a low dimensional representation of a single set of variables. The solution to CCA is an eigendecomposition involving the joint covariance or correlation matrix of both sets of variables and the solution to PCA is an eigendecomposition involving the covariance or correlation matrix of the single set of variables. We extend CCA and PCA using robust or non-standard estimators of the covariance or correlation matrix. First we extend CCA using a robust correlation estimator based on transformations of Kendall's tau rank correlation coefficient. We show that the CCA estimates using this robust correlation estimator are consistent and asymptotically normal. We also define a bootstrap based testing procedure for identifying informative canonical directions. Simulations show that this robust estimator performs better than standard CCA for data from heavy tailed and skewed distributions. We apply this method to brain white matter structure data from diffusion tensor imaging (DTI) and executive function (EF) test scores in six-year-old children to show that lateralization of white matter brain structure is correlated with higher EF scores. Next we define PCA for the multivariate survival setting where failure time data can be right censored. We estimate the covariance and correlation matrices of the counting processes defined by the failure times and their associated martingales. We use eigendecomposition of these covariance and correlation matrix estimates to obtain principal direction estimates. These estimates are consistent and asymptotically normal. We apply this method to a data set from a clinical trial for patients with pancreatic cancer and are able to define medically relevant groupings of adverse events. Finally we extend robust CCA to the multi-set and high dimensional setting in which there are more than two sets of variables and one or more of the sets is high-dimensional. We use the same robust correlation estimate using transformations of Kendall's tau. We also use cross-validation for dimension reduction and testing procedures. Unlike existing methods this cross-validation testing procedure is valid when data come from a heavy tailed elliptical distribution. We extend our analysis of DTI and EF data to include brain gray matter volume data from 88 different brain regions to further investigate the association between brain structure and EF test scores.Doctor of Philosoph