  Sparse linear regression methods for high-dimensional data commonly assume
that residuals have constant variance, which can be violated in practice. For
example, Aphasia Quotient (AQ) is a critical measure of language impairment and
informs treatment decisions, but it is challenging to measure in stroke
patients. It is of interest to use high-resolution T2 neuroimages of brain
damage to predict AQ. However, sparse regression models show marked evidence of
heteroscedastic error even after transformations are applied. This violation of
the homoscedasticity assumption can lead to bias in estimated coefficients,
prediction intervals (PI) with improper length, and increased type I errors.
Bayesian heteroscedastic linear regression models relax the homoscedastic error
assumption but can enforce restrictive prior assumptions on parameters, and
many are computationally infeasible in the high-dimensional setting. This paper
proposes estimating high-dimensional heteroscedastic linear regression models
using a heteroscedastic partitioned empirical Bayes Expectation Conditional
Maximization (H-PROBE) algorithm. H-PROBE is a computationally efficient
maximum a posteriori estimation approach that requires minimal prior
assumptions and can incorporate covariates hypothesized to impact
heterogeneity. We apply this method by using high-dimensional neuroimages to
predict and provide PIs for AQ that accurately quantify predictive uncertainty.
Our analysis demonstrates that H-PROBE can provide narrower PI widths than
standard methods without sacrificing coverage. Narrower PIs are clinically
important for determining the risk of moderate to severe aphasia. Additionally,
through extensive simulation studies, we exhibit that H-PROBE results in
superior prediction, variable selection, and predictive inference compared to
alternative methods.
