Non-Inferiority and Equivalence
One-sided two-sample t-test for determining whether a treatment mean is non-inferior to a reference mean by more than a prespecified margin. Supports outcomes where either higher or lower values are better and reports the test statistic, p-value, one-sided confidence bound, and non-inferiority conclusion.
This calculator implements the two-sample equal-variance t-test for a non-inferiority comparison of two independent means. this method describes this procedure as a parallel two-group design with a continuous normally distributed outcome and equal population variances. The treatment is population 1 and the reference is population 2.
When higher values are better, the non-inferiority hypotheses are H0: μ1 − μ2 ≤ −Δ versus H1: μ1 − μ2 > −Δ, where Δ is the positive non-inferiority margin. When lower values are better, the direction is reversed: H0: μ1 − μ2 ≥ Δ versus H1: μ1 − μ2 < Δ. These are the directional formulations specified in the this method documentation.
For higher-is-better outcomes, the null boundary is −Δ, so the statistic is (d + Δ) / SE(d). For lower-is-better outcomes, the null boundary is +Δ, so the statistic is (d − Δ) / SE(d), with the corresponding lower-tailed p-value.
The pooled standard error and degrees of freedom follow the equal-variance two-sample t-test formulation documented in the statistical literature. The null hypothesis is rejected when the one-sided p-value is below the specified alpha level.
For higher-is-better outcomes, the calculator reports the one-sided lower confidence bound for the treatment-reference difference: d − t1−α,dfSE(d). Non-inferiority is concluded when this lower bound is greater than −Δ. For lower-is-better outcomes, the corresponding one-sided upper confidence bound is d + t1−α,dfSE(d), and non-inferiority is concluded when it is less than +Δ.
this method gives a published validation example based on Chow, Shao, Wang, and Lokhnygina (2018): δ = 0, NIM = 0.05, common SD = 0.1, one-sided α = 0.05, with 51 subjects per group producing the documented sample-size result for 80% power.
For a fixed observed-data verification of this implementation, using those same parameters with observed means of 0 in both groups gives: n1 = n2 = 51, X̄1 = X̄2 = 0, s1 = s2 = 0.1, Δ = 0.05, and α = 0.05. The implementation should return t = 2.524876, df = 100, one-sided p = 0.006573, and a lower 95% confidence bound of approximately −0.032878. Because the lower bound is above the −0.05 non-inferiority boundary, the test concludes non-inferiority.
the software, LLC. Two-Sample T-Tests for Non-Inferiority Assuming Equal Variance, this method Sample Size Software, Chapter 450. The documentation specifies the hypotheses, pooled-variance t statistic, degrees of freedom, and non-inferiority examples used as the basis for this calculator.
Chow, S.C., Shao, J., Wang, H., & Lokhnygina, Y. (2018). Sample Size Calculations in Clinical Research, 3rd Edition. Taylor & Francis/CRC, Boca Raton, Florida. Note that this reference as the source for the non-inferiority sample-size example used in its validation documentation.
Elashoff, J.D. (2007). Advisor Version 7.0 User's Guide. Statistical Solutions Ltd. The documentation identifies MTE0 as the two-group t-test of equivalence in means used for one-sided non-inferiority testing of a difference in means; current procedure listing identifies MTE0 as “Non-Inferiority t-test for Two Means.”