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Sparse linear regression methods for high-dimensional data commonly assumethat residuals have constant variance, which can be violated in practice. Forexample, Aphasia Quotient (AQ) is a critical measure of language impairment andinforms treatment decisions, but it is challenging to measure in strokepatients. It is of interest to use high-resolution T2 neuroimages of braindamage to predict AQ. However, sparse regression models show marked evidence ofheteroscedastic error even after transformations are applied. This violation ofthe 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 errorassumption but can enforce restrictive prior assumptions on parameters, andmany are computationally infeasible in the high-dimensional setting. This paperproposes estimating high-dimensional heteroscedastic linear regression modelsusing a heteroscedastic partitioned empirical Bayes Expectation ConditionalMaximization (H-PROBE) algorithm. H-PROBE is a computationally efficientmaximum a posteriori estimation approach that requires minimal priorassumptions and can incorporate covariates hypothesized to impactheterogeneity. We apply this method by using high-dimensional neuroimages topredict and provide PIs for AQ that accurately quantify predictive uncertainty.Our analysis demonstrates that H-PROBE can provide narrower PI widths thanstandard methods without sacrificing coverage. Narrower PIs are clinicallyimportant for determining the risk of moderate to severe aphasia. Additionally,through extensive simulation studies, we exhibit that H-PROBE results insuperior prediction, variable selection, and predictive inference compared toalternative methods.

Quantifying predictive uncertainty of aphasia severity in stroke patients with sparse heteroscedastic Bayesian high-dimensional regression