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

Abstract

Summary 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.

The Use of Functional Data Analysis to Evaluate Activity in a Spontaneous Model of Degenerative Joint Disease Associated Pain in Cats

“…Finally, we investigated whether the way the activity varies (the population distributions) is the same in Normal cats and DJD cats; for this we used the functional Anderson-Darling testing procedure of Pomann et al [55]. We found significant evidence against this null hypothesis for both weekends (p = 0.013) and weekdays (p = 0.010).…”

“…Null distributions were based on N = 10,000 simulations. Further, we formally assessed whether the population distributions of activity and intensity profiles for the two groups of DJD cats were the same using the Anderson-Darling testing procedure proposed by Pomann et al [55]. Results are discussed in the results section; they were supportive of separating weekend and weekday activity and pooling data from the two groups of cats with DJD into one DJD group.…”

Additive Function-on-Function Regression

“…This choice of orthogonal basis also allows us to formulate the mean model for the conditional response profile, given scalar/vector covariates, based on mean models for the conditional FPC scores given the covariates: $\mathrm{normalE}false[Y_{i}false(tfalse)false|X_{i}false(·false)false]=true{\sum }_{k=1}^K{\varphi }_{k}false(tfalse)\mathrm{normalE}false[{\xi }_{ik}false|X_{i}false(·false)false]$, where $\mathrm{normalE}false[{\xi }_{ik}false|X_{i}false(·false)false]={{\displaystyle \int }}_{{\mathrm{scriptT}}_X}{G}_{k}false\{X_{i}false(sfalse),sfalse\}ds$, G k (·, ·) are unknown bivariate functions and ξ ik are the FPC scores of response. The representation is novel and extends ideas of Aston et al (2010) and Pomann et al (2015) to the case of a functional covariate. Also, it is related to Müller and Yao (2008) for $\mathrm{normalE}false[{\xi }_{ik}false|X_{i}false(·false)false]=true{\sum }_{m=1}^M{f}_{km}false({\xi }_{im}^{X}false)$, where f km (·) are unknown smooth functions for m = 1, …, M and $k=1,\dots ,K,false\{{\xi }_{i1}^X,\dots ,{\xi }_{iM}^{X}false\}$ are the FPC scores of the functional covariate X i (·), and M is a finite truncation.…”

Longitudinal functional data analysis

“…From this viewpoint, the model representation is optimal. The idea of using the eigenbasis of the pooled covariance can be related to Jiang and Wang () and Pomann et al (), who considered independent functional data.…”