Proportions: One and Two Groups
A two one-sided tests (TOST) calculator for comparing two independent proportions on the absolute difference scale. Uses the unpooled large-sample Z-test, reports both one-sided p-values, and displays the corresponding two-sided confidence interval. Runs entirely in your browser.
The TOST procedure tests whether the difference between two independent proportions lies entirely inside a prespecified equivalence interval. Let D = p1 − p2, with lower bound L and upper bound U. The null hypothesis is that the true difference is at or beyond either equivalence boundary; the alternative is that the true difference lies between the boundaries.
This calculator uses the unpooled large-sample Z formulation. The estimated proportions are p̂1 = x1/n1 and p̂2 = x2/n2. Their difference is D̂ = p̂1 − p̂2.
Equivalence is concluded when both p-values are less than alpha. Equivalently, the two-sided 100(1−2α)% confidence interval for the difference lies entirely within the equivalence bounds.
The displayed TOST confidence interval uses the same unpooled standard error and a critical value of z1−α:
For the default alpha = 0.05, this is a 90% confidence interval. This is the interval associated with the TOST decision, not the conventional 95% interval used for an ordinary two-sided superiority test.
A statistically significant TOST result does not mean that the two observed proportions are exactly equal. It means that the data provide evidence that the true difference is contained within the prespecified interval of clinically acceptable differences. The equivalence margins should therefore be specified from clinical or scientific considerations before examining the observed results.
Tunes da Silva, G., Logan, B. R., & Klein, J. P. (2008). Methods for Equivalence and Noninferiority Testing. Biology of Blood and Marrow Transplantation, 15(1 Suppl), 120–127.
Yin, G. (2012). Clinical Trial Design: Bayesian and Frequentist Adaptive Methods. Hoboken, NJ: John Wiley & Sons.
The TOSTER documentation provides a directly comparable TOST implementation for two independent proportions using the normal confidence-interval approach.