Two One-Sided Tests (TOST) declare two means equivalent when their difference falls entirely within ±Δ, so a significant result means the difference is small enough to be practically zero.
A standard t-test answers "are the means different?" — failing to reject H₀ is not evidence of equivalence (absence of evidence is not evidence of absence). TOST flips the framing: H₀ is that the difference exceeds the equivalence margin ±Δ, H₁ is that it's inside. Rejecting H₀ ⇒ the difference is, with high confidence, smaller than Δ.
Operationally, TOST runs two one-sided t-tests against −Δ and +Δ; if both reject at α (so max p < α), we conclude equivalence at level α. The companion (1 − 2α) CI of the mean difference contained entirely within ±Δ is the equivalent visual statement.
Picking Δ is the hard part — it must come from clinical or scientific judgement, not from the data. Common defaults: 0.5 SD for behavioural data, 20% of the reference mean for bioequivalence (the FDA cut-off for plasma concentrations).
Two numeric columns + equivalence margin Δ + α (default 0.05).
Verdict (equivalent / not), per-side t and p, max(p), (1 − 2α) CI for the mean difference.
Two complementary outcomes can co-occur: not-different (t-test p > 0.05) AND not-equivalent (TOST p > 0.05) — i.e. you have too little data to claim either. The honest report is the CI itself.
The (1 − 2α) CI (90% by convention for α = 0.05) is the right visual: if it lies entirely between −Δ and +Δ, the two means are equivalent at level α.