Means: One and Two Groups
A one-sided test for whether a treatment mean is no worse than a reference mean by more than a prespecified non-inferiority margin. Uses the two-sample t-test with an equal-variance option or Welch's t-test when equal variances are not assumed.
This calculator implements the difference-in-means non-inferiority framework documented by NCSS/PASS. For higher-is-better outcomes, the hypotheses are H0: μ1 − μ2 ≤ −Δ versus H1: μ1 − μ2 > −Δ, where Δ is the positive non-inferiority margin. For lower-is-better outcomes, the hypotheses are H0: μ1 − μ2 ≥ Δ versus H1: μ1 − μ2 < Δ.
For higher-is-better outcomes, the null boundary is −Δ, so the statistic is the observed difference minus −Δ, divided by its standard error. For lower-is-better outcomes, the null boundary is +&Delta.
With equal variances, the pooled standard deviation is used:
When equal variances are not assumed, Welch's standard error and the Welch–Satterthwaite degrees of freedom are used instead:
For higher-is-better outcomes, non-inferiority is concluded when the one-sided p-value is below α, equivalently when the lower one-sided 100(1−α)% confidence bound for μ1−μ2 is greater than −Δ. For lower-is-better outcomes, the corresponding upper confidence bound must be less than +Δ.
A non-inferiority test is directional. The margin must be specified before interpreting the result and should represent the largest clinically acceptable loss in the favorable direction. A statistically significant result means the observed data provide evidence against the null boundary; it does not by itself establish that the treatment is superior.
NCSS, LLC. PASS User's Guide II, Chapter 450: Non-Inferiority & Superiority Tests for Two Means using Differences. NCSS, LLC. The chapter describes the two-sample t-test and Welch's t-test formulations for non-inferiority tests of two independent means and gives validation examples.
Statistical Solutions, Ltd. nQuery Advisor User's Guide, Chapter 11, “Demonstrating Equivalence.” The manual describes one-sided non-inferiority testing for differences in means using a two-group t-test and provides worked clinical-trial examples.