On Bangdiwala's B Coefficient and Its Variance

Bangdiwala's B is a lesser-known agreement statistic that measures concordance between two raters geometrically rather than through the usual chance-correction approach — and until now it hasn't had a validated way to compute a confidence interval around it.

Originally proposed by Bangdiwala in 1985 and later revisited by Bangdiwala and Shankar, the B coefficient compares the area of exact agreement in a rater-by-rater table to the area one would expect based on the raters' marginal totals. It has seen some use in medical agreement studies, but a proper variance estimator — the piece needed to build confidence intervals or run hypothesis tests — hasn't been widely established.

This post proposes a variance formula for B and checks whether it actually behaves the way a variance estimator should. The test is a Monte Carlo simulation: start with a population of 10,000 rated subjects where the true value of B is known, then repeatedly draw samples of varying sizes, compute B and its proposed variance for each sample, and build a 95% confidence interval each time.

What the simulation showed

Across 4,000 simulated samples at each of eleven sample sizes (from 25 up to 350 subjects), the resulting confidence intervals captured the true population value close to the target 95% rate. Even at a sample size as small as 25, coverage was already around 92%; from about 75 subjects onward, coverage settled into the 94%–95.5% range.

That's the practical takeaway: the proposed variance estimator for Bangdiwala's B holds up well across a realistic range of study sizes, which means researchers who prefer B's geometric interpretation of agreement over a chance-corrected coefficient now have a defensible way to report its precision.