TOST With Asymmetric Difference Margins

Why Unequal Difference Margins Need Care

Equivalence is a claim that a population difference is sufficiently small in either direction. “Sufficiently small” need not mean equally far from zero on both sides. A scientific or clinical justification may set a lower acceptable difference at one value and an upper acceptable difference at another. In that case, the equivalence region is off-center or has unequal widths, and each boundary must be assessed on its own terms.

Let \(\delta\) denote the population treatment difference, defined here as treatment minus reference on the original difference scale. Let \(L\) be the prespecified lower margin and \(U\) the prespecified upper margin, with \(L<U\). The estimates and margins must use the same difference definition and units. For example, if \(\delta\) is measured in milligrams, both margins are also in milligrams.

The margins need not be centered at zero, and the distances \(U-\delta\) and \(\delta-L\) are not interchangeable. TOST therefore uses one one-sided test at \(L\) and another at \(U\); both must reject their respective null hypotheses. This is the intersection-union logic of TOST, applied with unequal boundaries.

Model, Assumptions, and Test Calculations

Suppose an analysis produces an estimate \(\hat{\delta}\), its standard error \(SE\), and degrees of freedom \(df\). For the usual t-based procedure, assume the estimate is unbiased for \(\delta\), its standard error is valid under the sampling design and fitted model, and its standardized sampling error follows a t distribution with the stated degrees of freedom. In a small-sample model, this commonly relies on an appropriate normal-error model and on the model’s covariance assumptions. The design and analysis must also correctly represent the units of analysis and any pairing or clustering.

The margins must be chosen before examining the results and justified in context. TOST does not establish that the margins are scientifically acceptable; it assesses the estimate against the supplied margins. If the model or standard error is not credible, the resulting tests and interval are not repaired merely by applying the TOST formulas.

At significance level \(\alpha\), test the lower boundary with \(H_{0L}:\delta\leq L\) against \(H_{aL}:\delta>L\). Then test the upper boundary with \(H_{0U}:\delta\geq U\) against \(H_{aU}:\delta<U\). Using \(t_{1-\alpha,df}\), the \(1-\alpha\) quantile of the t distribution:

Reject the lower-boundary null if \(t_L>t_{1-\alpha,df}\), and reject the upper-boundary null if \(t_U<-t_{1-\alpha,df}\). Equivalently, the one-sided p-values are \(1-F_{t,df}(t_L)\) and \(F_{t,df}(t_U)\), where \(F_{t,df}\) is the t-distribution cumulative distribution function. Declare equivalence only if both p-values are at most \(\alpha\), using the same rejection convention for both tests.

The two statistics have different boundary values in their numerators. Replacing \(L\) and \(U\) with a symmetric pair around zero, or using the same distance from the estimate for both tests, changes the hypotheses and is not a valid shortcut.

Why the Confidence-Interval Rule Works

The two one-sided tests have a direct confidence-interval interpretation. Let \(c=t_{1-\alpha,df}\). The lower test rejects exactly when

The upper test rejects exactly when

Thus both tests reject precisely when the interval from \(\hat{\delta}-cSE\) to \(\hat{\delta}+cSE\) lies inside the equivalence region. This is a two-sided \(100(1-2\alpha)\%\) t confidence interval: its two tails each have probability \(\alpha\). At \(\alpha=0.05\), the matching interval is a 90% confidence interval, not a 95% interval.

This containment rule is valid for unequal margins just as it is for equal-width margins. The interval itself is centered at the estimate; the allowed region need not be centered there or at zero. For decisions exactly at a boundary, apply the test’s stated rejection convention consistently; conclusions near equality can be sensitive to rounding.

Examples Where the Boundaries Matter

Diagnostics, Alternatives, and Limitations

Before interpreting a result, check that the margins were prespecified, use the same sign convention for \(\hat{\delta}\), \(L\), and \(U\), and verify that all three use the same units. Then inspect the interval endpoint nearest each margin. The lower endpoint is compared with \(L\); the upper endpoint is compared with \(U\). A plot can help reveal which boundary is limiting, but it does not replace the numerical test or a check of the model assumptions.

Warning signs include a standard error taken from a model that does not reflect the design, an incorrect degrees-of-freedom value, a sign reversal in the difference definition, or margins selected after seeing the estimate. An interval that extends beyond a boundary means the corresponding one-sided test has not established equivalence. A point estimate inside both margins is not a diagnostic substitute for this interval check.

The main alternative is a symmetric-difference analysis, which uses margins \(-M\) and \(M\). That is appropriate only when a symmetric acceptable range is scientifically justified; imposing it when the prespecified limits are unequal tests a different claim. A conventional test of whether \(\delta=0\) asks whether there is evidence of a difference from zero, not whether the difference is small enough to be equivalent. Likewise, a nonsignificant test of zero difference does not establish equivalence.

TOST is a decision procedure relative to chosen margins, a significance level, and a sampling model. It does not show that every individual response is within the margins, prove exact equality, or compensate for poor precision. When one test fails, report which boundary was not cleared and the interval in context rather than describing the result as proof of non-equivalence.