Identifying associations between the shape properties of brain regions, measured from magnetic resonance imaging (MRI), and numerical measures of neurodegenerative disease burden can clarify whether disease processes lead to distinctive spatial patterns of brain atrophy. However, prior methods for identifying such associations between shape and clinical variables either failed to summarize shape patterns into a concise set of summary measurements, or risked failing to discover such associations by extracting summary shape features blinded to the clinical variables. We present a method that overcomes these limitations by directly searching for a small set of linear shape features--shape regression components--that simultaneously account for a large amount of population shape variability and are highly correlated with a numerical clinical variable of interest. When applied to hippocampi of 299 Alzheimer's Disease Neuroimaging Initiative (ADNI) participants, the method identified correlations between hippocampal atrophy and markers of AD pathology and cogniton that were stronger than, and covered a more extended spatial region than, those identified by competing approaches.