Equivalence Testing with TOST

Two one-sided tests for establishing practical equivalence within a margin

Introduction

Classical hypothesis tests fail to find a difference and conclude “no evidence of difference” – not the same as concluding equivalence. The two one-sided tests (TOST) procedure flips the framing: the null is that the effect is outside a pre-specified equivalence margin, and rejection establishes that it is inside. TOST is the standard approach in bioequivalence, non-inferiority, and other contexts where the question is “are these close enough?”.

Prerequisites

Hypothesis testing, confidence intervals.

Theory

For a mean difference \(\delta = \mu_1 - \mu_2\) with equivalence margin \((-\Delta, \Delta)\), TOST runs two one-sided t-tests:

\[H_{01}: \delta \leq -\Delta \quad \text{vs.} \quad H_{11}: \delta > -\Delta\] \[H_{02}: \delta \geq \Delta \quad \text{vs.} \quad H_{12}: \delta < \Delta\]

Equivalence is declared if both one-sided tests reject at level \(\alpha\) (often 0.05). The overall Type I rate is preserved at \(\alpha\) by the intersection-union logic.

Equivalently, a \(1 - 2\alpha\) CI for \(\delta\) entirely inside \((-\Delta, \Delta)\) establishes equivalence at level \(\alpha\).

Non-inferiority is a one-sided analogue: test only \(H_{01}\) with margin \(-\Delta\).

Assumptions

• Standard t-test assumptions (normality or large \(n\), independence).
• Pre-specified equivalence margin \(\Delta\) based on clinical or practical relevance.

R Implementation

library(TOSTER)
set.seed(2026)

# Bioequivalence example: AUC of generic vs. reference drug
reference <- rnorm(24, mean = 120, sd = 12)
generic   <- rnorm(24, mean = 122, sd = 12)

# Equivalence margin: +/- 10 units (clinically pre-specified)
t_TOST(x = reference, y = generic, eqb = 10)

# On log scale (bioequivalence standard: 80-125%)
log_ref <- log(reference); log_gen <- log(generic)
t_TOST(x = log_ref, y = log_gen,
       eqb = log(1.25), eqbound_type = "raw")

Output & Results

Welch Two Sample t-test with equivalence bounds

              estimate df   t    p.value
upper (H02)   -8.76    45.9 -3.58 <.001
lower (H01)    8.76    45.9  3.58 <.001

Equivalence: YES (both one-sided tests reject at alpha = 0.05)

90% CI on the mean difference: (-4.9, 2.2)
Equivalence margin: (-10, 10)

Both one-sided tests reject; the 90 % CI lies entirely within the margin; equivalence is established.

Interpretation

“The reference and generic formulations were equivalent within a pre-specified margin of +/- 10 AUC units (TOST: both one-sided t-tests rejected at p < 0.001; 90 % CI for the mean difference -4.9 to 2.2, within -10 to 10).”

Practical Tips

• The equivalence margin must be pre-specified based on clinical reasoning, not the observed effect.
• Use a 90 % CI, not 95 %, for the standard TOST at \(\alpha = 0.05\).
• Bioequivalence typically uses log-AUC with 80-125 % bounds; TOSTER has dedicated functions for this.
• A non-significant classical t-test does NOT establish equivalence.
• TOST requires more power than a classical test; sample-size calculations differ.