Weighted Adaptive Methods for Multivariate Response Models with an HIV/Neurocognitive Application

Multivariate response models are being used increasingly more in almost all fields with the necessary employment of inferential methods such as Canonical Correlation Analysis (CCA). This requires the estimation of the number of uncorrelated canonical relationships between the two sets, or, equivalently so, determining the rank of the coefficient estimator in the multivariate response model.One way to do this is by the Rank Selection Criterion (RSC) by Bunea et al. with the assumption the error matrix has independent constant variance entries. While this assumption is necessary to show their strong theoretical results, in practical application, some flexibility is required. That is, such assumption cannot always be safely made. What is developed here are the theoretics that parallel Bunea et al.'s work with the addition of a "decorrelator" weight matrix. One choice for the weight matrix is the residual covariance, but this introduces many issues in practice. A computationally more convenient weight matrix is the sample response covariance. When such a weight matrix is chosen, CCA is directly accessible by this weighted version of RSC giving rise to an Adaptive CCA (ACCA) with principal proofs for the large sample setting. However, particular considerations are required for the high-dimensional setting, where similar theoretics do not hold. What is offered instead are extensive empirical simulations that reveal that using the sample response covariance still provides good rank recovery and estimation of the coefficient matrix, and hence, also provides good estimation of the number of canonical relationships and variates. It is argued precisely why other versions of the residual covariance, including a regularized version, are poor choices in the high-dimensional setting. Another approach to avoid these issues is to employ some type of variable selection methodology first before applying ACCA. Truly, any group selection method may be applied prior to ACCA as variable selection in the multivariate response model is the same as group selection in the univariate response model and thus completely eliminates these high-dimensional concerns. To offer a practical application of these ideas, ACCA is applied to a "large sample'" neurocognitive dataset. Then, a high-dimensional dataset is generated to which Group LASSO will be first utilized before ACCA. This provides a unique perspective into the relationships between cognitive deficiencies in HIV-positive patients and the extensive, available neuroimaging measures.