Sign and signed rank tests for paired functions

TL;DR

This work develops nonparametric tests for paired functional data by introducing two functional sign tests and a signed doubly ranked test within a unified framework. The authors formalize paired curves via $D_i(s)$ with a null $H_0: \Lambda = 0$ and provide two score-based approaches and a magnitude-aware, doubly-ranked extension that culminates in Wilcoxon-type tests on subject-level summaries. Through simulations, they demonstrate that the signed doubly ranked test achieves superior type I error control and power, across varying correlation structures and preprocessing schemes (e.g., FPCA FACE, FPCA SC). They illustrate practical use by reanalyzing an in-flight heart-rate study, where heart rate shows a significant difference under flight conditions, while HRV metrics do not, underscoring the methods’ relevance for functional data in biomedical settings. Overall, the paper advances nonparametric paired functional testing and offers robust tools for primary analysis and exploratory data analysis in functional studies with real-world applications.

Abstract

Simple nonparametric tests for paired functional data are an understudied area, despite recent advances in similar tests for other types of functional data. While the sign test has received limited treatment, the signed rank-type test has not previously been examined. The aim of the present work is to develop and evaluate these types of tests for functional data. We derive a simple, theoretical framework for both sign and signed rank tests for pairs of functions. In particular, we demonstrate that doubly ranked testing -- a newly developed framework for testing hypotheses involving functional data -- is a useful conduit for examining hypotheses regarding pairs o,f functions. We briefly examine the operating characteristics of all derived tests. We also use the described approaches to re-analyze pairs of functions from a randomized crossover study of heart health during simulated flight.

Figures (6)

• Figure 1: Preprocessed and differenced data from Meyer2019 (flight condition $-$ control condition). rMSSD (root mean square of successive RR interval differences), SDNN (the standard deviation of normal-to-normal intervals), HF (high frequency power), and LF (low frequency power) all measure heart rate variability.
• Figure 2: Power curves for functional sign and signed doubly ranked tests under $\Delta_1(s)$ and $\rho = 0.75$. The top row contains results under the multivariate normal, bottom row under $t_2$. Curves for the signed doubly ranked tests are in green, while curves for the functional sign tests are in blue (integral) and gold (sufficient statistic). Solid curves are for when $n = 60$, dashed curves for when $n = 30$, and dotted-dashed curves for when $n = 15$.
• Figure 3: Power curves for functional sign and signed doubly ranked tests under $\Delta_2(s)$ and $\rho = 0.75$. The top row contains results under the multivariate normal, bottom row under $t_2$. Curves for the signed doubly ranked tests are in green, while curves for the functional sign tests are in blue (integral) and gold (sufficient statistic). Solid curves are for when $n = 60$, dashed curves for when $n = 30$, and dotted-dashed curves for when $n = 15$.
• Figure 4: Power curves for functional sign and signed doubly ranked tests under $\Delta_1(s)$ and $\rho = 0.50$. The top row contains results under the multivariate normal, bottom row under $t_2$. Curves for the signed doubly ranked tests are in green, while curves for the functional sign tests are in blue (integral) and gold (sufficient statistic). Solid curves are for when $n = 60$, dashed curves for when $n = 30$, and dotted-dashed curves for when $n = 15$.
• Figure 5: Power curves for functional sign and signed doubly ranked tests under $\Delta_2(s)$ and $\rho = 0.50$. The top row contains results under the multivariate normal, bottom row under $t_2$. Curves for the signed doubly ranked tests are in green, while curves for the functional sign tests are in blue (integral) and gold (sufficient statistic). Solid curves are for when $n = 60$, dashed curves for when $n = 30$, and dotted-dashed curves for when $n = 15$.

Theorems & Definitions (12)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.1
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Proposition 4.5
• proof