A Two Sample Distribution-Free Test for Functional Data with Application to a Diffusion Tensor Imaging Study of Multiple Sclerosis.

Motivated by an imaging study, this paper develops a nonparametric testing procedure for testing the null hypothesis that two samples of curves observed at discrete grids and with noise have the same underlying distribution. The objective is to formally compare white matter tract profiles between healthy individuals and multiple sclerosis patients, as assessed by conventional diffusion tensor imaging measures. We propose to decompose the curves using functional principal component analysis of a mixture process, which we refer to as marginal functional principal component analysis. This approach reduces the dimension of the testing problem in a way that enables the use of traditional nonparametric univariate testing procedures. The procedure is computationally efficient and accommodates different sampling designs. Numerical studies are presented to validate the size and power properties of the test in many realistic scenarios. In these cases, the proposed test has been found to be more powerful than its primary competitor. Application to the diffusion tensor imaging data reveals that all the tracts studied are associated with multiple sclerosis and the choice of the diffusion tensor image measurement is important when assessing axonal disruption.