In the fields of neuroimaging and genetics a key goal is testing the association of a single outcome with a very high-dimensional imaging or genetic variable. Oftentimes summary measures of the high-dimensional variable are created to sequentially test and localize the association with the outcome. In some cases, the results for summary measures are significant, but subsequent tests used to localize differences are underpowered and do not identify regions associated with the outcome. We propose a generalization of Rao's score test based on maximizing the score statistic in a linear subspace of the parameter space. If the test rejects the null, then we provide methods to localize signal in the high-dimensional space by projecting the scores to the subspace where the score test was performed. This allows for inference in the high-dimensional space to be performed on the same degrees of freedom as the score test, effectively reducing the number of comparisons. We illustrate the method by analyzing a subset of the Alzheimer's Disease Neuroimaging Initiative dataset. Results suggest cortical thinning of the frontal and temporal lobes may be a useful biological marker of Alzheimer’s risk. Simulation results demonstrate the test has competitive power relative to others commonly used.



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