  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.
