Significance tests for functional data with complex dependence structure

Abstract

We propose an L2-norm based global testing procedure for the null hypothesis that multiple group mean functions are equal, for functional data with complex dependence structure. Specifically, we consider the setting of functional data with a multilevel structure of the form groups–clusters or subjects–units, where the unit-level profiles are spatially correlated within the cluster, and the cluster-level data are independent. Orthogonal series expansions are used to approximate the group mean functions and the test statistic is estimated using the basis coefficients. The asymptotic null distribution of the test statistic is developed, under mild regularity conditions. To our knowledge this is the first work that studies hypothesis testing, when data have such complex multilevel functional and spatial structure. Two small-sample alternatives, including a novel block bootstrap for functional data, are proposed, and their performance is examined in simulation studies. The paper concludes with an illustration of a motivating experiment.

Simple fixed-effects inference for complex functional models

“…Crainiceanu et al (2012) discussed bootstrap-based inferential methods for the difference in the mean profiles. Staicu et al (2014a) proposed likelihood-ratio type testing procedure, while Staicu et al (2014b) considered L 2 norm-based testing procedures for testing the null hypothesis that multiple group mean functions are equal. Horváth et al (2013) developed inference for the mean function of a functional time series.…”

Testing Gait with Ankle-Foot Orthoses in Children with Cerebral Palsy by Using Functional Mixed-Effects Analysis of Variance

“…Tests for functional mixed-effects models have only been considered in a few reports 33 , 34 . The new test generalizes previous work on functional F tests 3 , 35 for independent curves, to repeated measurements in which the curves are correlated.…”

Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis

“…() proposed an F ‐test for independent processes; Staicu et al . () considered an L 2 ‐norm‐based global testing procedure for dependent processes; and Crainiceanu et al . () introduced bootstrap‐based procedures using joint confidence intervals.…”

“…We consider a wider spectrum of null hypotheses, which includes the hypothesis that the means of two functional processes are the same. Several recent methodological developments address this problem: Fan & Lin (1998) developed an adjusted Neyman testing procedure for independent stationary linear Gaussian processes; Cuevas et al (2004) proposed an F -test for independent processes; Staicu et al (2012) considered an L 2 -norm-based global testing procedure for dependent processes; and Crainiceanu et al (2012) introduced bootstrap-based procedures using joint confidence intervals. Our pLRT procedure has the advantage that it is applicable to independent or dependent samples of curves with both dense and sparse sampling designs.…”