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Efficient algorithms for the sensitivities of the Pearson correlation coefficient and its statistical significance to online data
Methodology · Submitted 23 May 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Further, we show that the resulting optimization problems yield elegant closed-form solutions that can be accurately computed by linear- and constant-time algorithms.
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The abstract, sentence by sentence
Reliably measuring the collinearity of bivariate data is crucial in statistics, particularly for time-series analysis or ongoing studies in which incoming observations can significantly impact current collinearity estimates. Leveraging identities from Welford's online algorithm for sample variance, we develop a rigorous theoretical framework for analyzing the maximal change to the Pearson correlation coefficient and its p-value that can be induced by additional data. Further, we show that the resulting optimization problems yield elegant closed-form solutions that can be accurately computed by linear- and constant-time algorithms. Our work not only creates new theoretical avenues for robust correlation measures, but also has broad practical implications for disciplines that span econometrics, operations research, clinical trials, climatology, differential privacy, and bioinformatics. Software implementations of our algorithms in Cython-wrapped C are made available at https://github.com/marc-harary/sensitivity for reproducibility, practical deployment, and future theoretical development.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Reliably measuring the collinearity of bivariate data is crucial in statistics, particularly for time-series analysis or ongoing studies in which incoming observations can significantly impact current collinearity estimates. | background 1.00 | 0.05 | 0.04 | asserted 1.00 |
| s1 | Leveraging identities from Welford's online algorithm for sample variance, we develop a rigorous theoretical framework for analyzing the maximal change to the Pearson correlation coefficient and its p-value that can be induced by additional data. | method 0.50 | 0.94 | 0.17 | asserted 1.00 |
| s2 | Further, we show that the resulting optimization problems yield elegant closed-form solutions that can be accurately computed by linear- and constant-time algorithms. | result 0.74 | 0.95 | 0.32 | asserted 1.00 |
| s3 | Our work not only creates new theoretical avenues for robust correlation measures, but also has broad practical implications for disciplines that span econometrics, operations research, clinical trials, climatology, differential privacy, and bioinformatics. | interpretation 0.99 | 0.49 | 0.22 | asserted 0.94 |
| s4 | Software implementations of our algorithms in Cython-wrapped C are made available at https://github.com/marc-harary/sensitivity for reproducibility, practical deployment, and future theoretical development. | contribution 0.81 | 0.90 | 0.05 | asserted 1.00 |
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The paper
Original abstract
Reliably measuring the collinearity of bivariate data is crucial in statistics, particularly for time-series analysis or ongoing studies in which incoming observations can significantly impact current collinearity estimates. Leveraging identities from Welford's online algorithm for sample variance, we develop a rigorous theoretical framework for analyzing the maximal change to the Pearson correlation coefficient and its p-value that can be induced by additional data. Further, we show that the resulting optimization problems yield elegant closed-form solutions that can be accurately computed by linear- and constant-time algorithms. Our work not only creates new theoretical avenues for robust correlation measures, but also has broad practical implications for disciplines that span econometrics, operations research, clinical trials, climatology, differential privacy, and bioinformatics. Software implementations of our algorithms in Cython-wrapped C are made available at https://github.com/marc-harary/sensitivity for reproducibility, practical deployment, and future theoretical development.
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