Equivalence Test for Two Means (TOST)

Methodology

The Two One-Sided Tests (TOST) procedure tests the null hypothesis of non-equivalence against the alternative hypothesis that the true difference between two independent means lies within prespecified lower and upper equivalence limits. This is the approach described by Schuirmann (1987) and implemented for two-group mean differences in this method and documentation.

Hypotheses

Here, δ = μ₁ − μ₂. The lower one-sided test evaluates whether the difference is greater than the lower equivalence limit, while the upper one-sided test evaluates whether the difference is less than the upper equivalence limit.

Welch TOST (default)

The lower-test p-value is P(T_df ≥ t_L), and the upper-test p-value is P(T_df ≤ t_U). The TOST p-value is the larger of the two one-sided p-values. The corresponding confidence interval is a 100(1−2α)% interval; at α = 0.05 this is a 90% confidence interval.

Pooled TOST (optional)

When the pooled equal-variance option is selected, the same two one-sided statistics are evaluated using the pooled standard deviation and Student t degrees of freedom.

Confidence-Interval Interpretation

TOST equivalence at level α is equivalent to the entire 100(1−2α)% confidence interval for the mean difference falling inside the prespecified equivalence interval. This calculator displays that interval alongside the two one-sided tests so the numerical decision can be audited directly.

Validation Example

The default values reproduce the independent-groups sleep-data example documented in the TOSTER 2025 vignette: Group 1 mean = 0.75, Group 2 mean = 2.33, equivalence bounds = ±0.5, n = 10 per group, and sample SDs approximately 1.79 and 2.00. With Welch's test, the expected TOST results are approximately t_L = −1.272, p_L = 0.890; t_U = −2.451, p_U = 0.0124; TOST p = 0.890; and the 90% CI for the mean difference is approximately [−3.053, −0.107]. Because this interval is not wholly contained within [−0.5, 0.5], equivalence is not established in this example.

References

1. 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(6), 657–680. PubMed.
2. Lakens, D. (2017). Equivalence Tests: A Practical Primer for t Tests, Correlations, and Meta-Analyses. Social Psychological and Personality Science, 8(4), 355–362. DOI.
3. the software. Two-Sample T-Tests for Equivalence Allowing Unequal Variance. the relevant methodological literature. The procedure uses Schuirmann's TOST approach for the lower and upper equivalence limits.
4. Statistical Solutions. Advisor User's Guide, Chapter 11, “Demonstrating Equivalence.” The MTE1tg procedure implements two one-sided t-tests for equivalence of means in a two-group design.
5. Caldwell, A. R. (2025). An Introduction to t_TOST. TOSTER package vignette. The independent-groups example and summary-statistics results are used as the numerical validation target for the default Welch calculation.