  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 Rao's 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 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.
