Non-Inferiority and Equivalence
A two-sample equivalence test for independent means using Schuirmann’s two one-sided tests (TOST). Enter summary statistics and pre-specified equivalence limits to determine whether the confidence interval for the mean difference lies completely within the equivalence region. Runs entirely in your browser.
The calculator implements the two one-sided tests (TOST) procedure for equivalence of two independent means, using the difference in means as the parameter of interest and a common pooled variance. The equivalence region is the interval from the lower equivalence limit ΔL to the upper equivalence limit ΔU.
Each null hypothesis is tested at the specified one-sided significance level α. Equivalence is concluded only if both null hypotheses are rejected.
For the equal-variance two-sample t-test, the pooled standard deviation and standard error are
With D = x̄1 − x̄2, the two TOST statistics are tL = (D − ΔL)/SE and tU = (D − ΔU)/SE, with n1 + n2 − 2 degrees of freedom. The lower-side p-value is P(T ≥ tL), while the upper-side p-value is P(T ≤ tU).
The TOST decision is equivalent to checking whether the two-sided 100(1−2α)% confidence interval for the mean difference is contained completely inside the equivalence limits. With α = 0.05, this is a 90% confidence interval.
A numerical verification example uses Group 1 mean = 5.2, SD = 1.5, n = 50; Group 2 mean = 5.0, SD = 1.4, n = 50; equivalence limits = −0.5 and +0.5; and α = 0.05. The expected calculations are: pooled SD = 1.4508618, SE = 0.2901724, df = 98, mean difference = 0.2000, tL = 2.4123593 with p = 0.0088550, tU = −1.0338683 with p = 0.1518712, TOST p = 0.1518712, and 90% CI = [−0.2818461, 0.6818461]. Because the interval extends above +0.5, equivalence is not established.
Schuirmann, D. J. (1987). A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. Journal of Pharmacokinetics and Biopharmaceutics, 15, 657–680.
Phillips, K. F. (1990). Power of the two one-sided tests procedure in bioequivalence. International Journal of Clinical Pharmacology, Therapy and Toxicology, 28, 261–266.
NCSS, LLC. PASS User’s Guide II, Chapter 460: Equivalence Tests for Two Means using Differences. PASS documents the parallel-group TOST procedure and its calculation/validation examples. The current PASS documentation lists the corresponding procedure under Equivalence → Means → Two Independent Means.
Statistical Solutions / nQuery Advisor. nQuery Advisor User’s Guide, Chapter 11: Demonstrating Equivalence. The guide describes the two-group TOST for differences in means and identifies the MTE1tg procedure.