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. Often, 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. Here, we propose a generalization of Raos score test based on projecting the score statistic onto a linear subspace of a high-dimensional parameter space. In addition, 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. Simulation results demonstrate the test has competitive power relative to others commonly used. We illustrate the method by analyzing a subset of the Alzheimers Disease Neuroimaging Initiative dataset. Results suggest cortical thinning of the frontal and temporal lobes may be a useful biological marker of Alzheimers risk.
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