[Paper Review] General multiple tests for functional data

Merle Munko, Marc Ditzhaus+3 more|arXiv (Cornell University)|Jun 27, 2023

Optimal Experimental Design Methods4 citations

TL;DR

This paper proposes a general framework for multiple hypothesis testing in functional data analysis, leveraging functional linear models and bootstrap-based inference to test multiple contrasts simultaneously. It establishes asymptotic validity under weak moment conditions and demonstrates superior empirical power across diverse simulation settings, particularly under heteroscedasticity and non-normality.

ABSTRACT

While there exists several inferential methods for analyzing functional data in factorial designs, there is a lack of statistical tests that are valid (i) in general designs, (ii) under non-restrictive assumptions on the data generating process and (iii) allow for coherent post-hoc analyses. In particular, most existing methods assume Gaussianity or equal covariance functions across groups (homoscedasticity) and are only applicable for specific study designs that do not allow for evaluation of interactions. Moreover, all available strategies are only designed for testing global hypotheses and do not directly allow a more in-depth analysis of multiple local hypotheses. To address the first two problems (i)-(ii), we propose flexible integral-type test statistics that are applicable in general factorial designs under minimal assumptions on the data generating process. In particular, we neither postulate homoscedasticity nor Gaussianity. To approximate the statistics' null distribution, we adopt a resampling approach and validate it methodologically. Finally, we use our flexible testing framework to (iii) infer several local null hypotheses simultaneously. To allow for powerful data analysis, we thereby take the complex dependencies of the different local test statistics into account. In extensive simulations we confirm that the new methods are flexibly applicable. Two illustrate data analyses complete our study. The new testing procedures are implemented in the R package multiFANOVA, which will be available on CRAN soon.

Motivation & Objective

• Address the lack of unified methods for multiple testing in functional data, especially under complex dependence and heteroscedasticity.
• Develop a general testing procedure that allows simultaneous inference on multiple contrasts in functional linear models.
• Ensure theoretical validity through asymptotic normality and consistency of the parametric bootstrap under minimal moment conditions.
• Evaluate performance across diverse scenarios, including non-normal errors and unequal variances, to ensure robustness in real-world applications.

Proposed method

• Formulate multiple contrast tests in a functional linear model framework using a Hilbert space-valued central limit theorem for triangular arrays.
• Apply a parametric bootstrap approach to approximate the null distribution of test statistics, ensuring consistency under weak moment conditions.
• Incorporate a scaling function to stabilize test statistics under heteroscedasticity, improving empirical size and power.
• Use Moore-Penrose inverses and spectral norm bounds to control the behavior of covariance estimators in high-dimensional functional settings.
• Implement nonparametric bootstrap procedures (pooled and groupwise) for competitor tests, ensuring robust critical value estimation.
• Apply Bonferroni correction to multiple p-values from random projections in the CAFB-test to stabilize inference under high-dimensional projections.

Experimental results

Research questions

• RQ1 Can a unified framework be developed for multiple testing in functional data that maintains correct size and high power under general moment conditions?
• RQ2 How does the inclusion of a scaling function affect the empirical size and power of functional multiple tests under heteroscedasticity?
• RQ3 How do the proposed tests compare in size and power to existing competitors (e.g., Fmax, GPF, L2b, Fb, CAFB) across various error distributions and variance structures?
• RQ4 What is the theoretical justification for the consistency of the parametric bootstrap in this functional multiple testing context?
• RQ5 To what extent does the method remain robust under non-normal errors and non-identical variances in functional data?

Key findings

• The proposed method achieves empirical rejection rates close to nominal levels (e.g., ~5% for α=0.05) across all simulation settings, including heteroscedastic and non-normal error structures.
• Under alternative A6 with Tukey contrasts and scaling, the test achieved empirical power of 91.65% (homoscedastic) and 85.00% (heteroscedastic), outperforming competitors in most scenarios.
• The parametric bootstrap consistently maintained correct size across all configurations, with Type I error rates within acceptable bounds (e.g., 4.80–6.70% for α=0.05).
• The inclusion of a scaling function significantly improved power in heteroscedastic settings, increasing median power by up to 15 percentage points compared to uncorrected versions.
• The CAFB-test with 30 random projections and Bonferroni correction showed robustness but lower power (e.g., ~32.7% under A6) compared to the proposed method.
• The method demonstrated strong performance across all tested alternatives, with empirical power exceeding 85% for 4 out of 6 alternatives under both homoscedastic and heteroscedastic conditions.