TOST on a Symmetric Difference Scale

Equivalence for a Difference Parameter

A difference scale is appropriate when the scientific question is naturally about how far apart two treatment means may be in the endpoint’s original units. For example, investigators might prespecify that a treatment difference of more than a certain number of units in a pharmacodynamic response would be important. The equivalence margin must come from scientific or clinical justification, not from the observed estimate or interval.

This tutorial considers symmetric margins: the acceptable difference is equally far below and above zero. The sign convention must be stated, because reversing the order of treatments reverses the sign of the difference. The equivalence conclusion does not change if both the parameter and the margins are reversed consistently.

Definition: Let \(\theta\) be the population difference of interest, defined as treatment T minus treatment R on a specified difference scale. Let \(\hat{\theta}\) be its estimate, and let \(\Delta>0\) be a prespecified equivalence margin in the same units. The symmetric equivalence region is \((-\Delta,\Delta)\). Equivalence means that the parameter lies strictly inside this region: \(-\Delta<\theta<\Delta\).

The TOST hypotheses express the claim that the difference is inside the region against the possibility that it is at or beyond either margin:

H_0:\ \theta\leq-\Delta\ \text{or}\ \theta\geq\Delta \qquad\text{versus}\qquad H_a:\ -\Delta<\theta<\Delta.

The null is a union of two ways equivalence can fail. Therefore, the TOST conclusion requires evidence against both boundary components, as established in the earlier material on the intersection-union logic of TOST. A small estimated difference by itself is not sufficient: the estimate must be precise enough that the plausible values of the parameter are contained within the margins.

The Confidence-Interval Rule

Suppose an estimate has standard error \(SE\) and a \(t\)-based reference distribution with \(df\) degrees of freedom. Under the model used for the estimate, a \(100(1-2\alpha)\%\) two-sided confidence interval is

\hat{\theta}\ \pm\ t_{1-\alpha,df}SE,

where \(t_{1-\alpha,df}\) is the \(1-\alpha\) quantile of the \(t\) distribution with \(df\) degrees of freedom. For example, when each one-sided test uses \(\alpha=0.05\), the matching interval is a 90% interval. The confidence level is \(1-2\alpha\), not \(1-\alpha\), because each end of the two-sided interval has tail area \(\alpha\).

Write the interval as \((L,U)\), where \(L=\hat{\theta}-t_{1-\alpha,df}SE\) and \(U=\hat{\theta}+t_{1-\alpha,df}SE\). The TOST decision is to conclude equivalence at level \(\alpha\) exactly when

L>-\Delta\quad\text{and}\quad U<\Delta.

Thus the entire matching confidence interval must lie inside the equivalence region. This is the confidence-interval rule described in the earlier material on deriving the confidence-interval rule for TOST; here the symmetric margins also give a useful expression in terms of the estimate’s distance from zero.

\left|\hat{\theta}\right|+t_{1-\alpha,df}SE<\Delta.

This expression says that the absolute estimated difference plus the interval’s half-width must be smaller than the margin. It makes the two requirements visible at once: the estimate should be near zero, and its uncertainty should be small relative to the available margin.

Why the Interval Rule Matches the Two Tests

The lower-margin test asks whether the data support \(\theta>-\Delta\); the upper-margin test asks whether they support \(\theta<\Delta\). In a \(t\)-based analysis, the respective one-sided statistics can be written as

t_L=\frac{\hat{\theta}-(-\Delta)}{SE} =\frac{\hat{\theta}+\Delta}{SE}, \qquad t_U=\frac{\hat{\theta}-\Delta}{SE}.

At level \(\alpha\), reject the lower test’s null component when \(t_L>t_{1-\alpha,df}\). Rearranging this inequality gives \(\hat{\theta}-t_{1-\alpha,df}SE>-\Delta\), or \(L>-\Delta\). Reject the upper test’s null component when \(t_U<-t_{1-\alpha,df}\). Rearranging gives \(\hat{\theta}+t_{1-\alpha,df}SE<\Delta\), or \(U<\Delta\). Both rejections are required, so the two tests yield exactly the interval-containment rule.

This justification depends on the confidence interval and one-sided tests using the same estimate, standard error, degrees of freedom, and reference distribution. Mixing, for example, a \(t\)-based interval with normal critical values for the tests can lead to inconsistent decisions.

Worked Decisions on the Difference Scale

Worked Example: Paired Differences Clearly Inside the Margins

In a hypothetical crossover study, each of 25 participants has a paired difference in a specified response, calculated as treatment T minus treatment R. The differences have sample mean \(\bar{x}=0.18\) units and sample standard deviation \(s=0.55\) units. Investigators prespecified symmetric margins of \(\Delta=0.50\) units and use a 5% one-sided significance level. Assess equivalence using a paired \(t\) interval.

Check the analysis conditions. The analysis treats each participant’s paired difference as one independent observation across participants. For a paired \(t\) interval, the differences should be reasonably compatible with a normal distribution, particularly with this sample size, and should not be dominated by extreme observations. Assume these checks are satisfactory and the paired design and endpoint definition support the stated estimand.

The estimate is \(\hat{\theta}=\bar{x}=0.18\) units. Its standard error is

SE=\frac{s}{\sqrt{n}} =\frac{0.55}{\sqrt{25}} =\frac{0.55}{5} =0.11\text{ units}.

Here \(df=n-1=24\), and \(t_{0.95,24}\approx1.711\). The matching 90% interval is

0.18\pm1.711(0.11) =0.18\pm0.1882 =(-0.0082,\ 0.3682)\text{ units}.

Both endpoints are strictly inside \((-0.50,0.50)\) units. The TOST procedure therefore supports equivalence at the 5% level for this difference parameter and these prespecified margins. The conclusion concerns the specified response and scale; it does not establish that all treatment effects are identical.

Worked Example: An Estimate Inside the Margins Is Not Enough

In a hypothetical analysis of another difference parameter, a model estimates \(\hat{\theta}=-0.35\) units with \(SE=0.20\) units and \(df=20\). The symmetric margins are \(\Delta=0.60\) units, and each one-sided test uses \(\alpha=0.05\). Assume the model-based estimate and standard error are valid for the prespecified contrast, and use the corresponding \(t\) reference distribution.

The point estimate is inside the margins because \(-0.60<-0.35<0.60\). That fact alone does not decide equivalence. With \(t_{0.95,20}\approx1.725\), the interval half-width is

1.725(0.20)=0.345\text{ units}.

The matching 90% interval is

-0.35\pm0.345=(-0.695,\ -0.005)\text{ units}.

The lower endpoint, \(-0.695\) units, is below the lower margin of \(-0.60\) units. The interval is not fully contained in the equivalence region, so equivalence is not established at the stated level. The result does not prove that the difference is outside the margins; it means the analysis has not ruled out differences at or beyond the lower margin with the required confidence.

Worked Example: A Narrow Interval Near a Margin

Consider a hypothetical model-based difference estimate of \(\hat{\theta}=0.10\) units, with \(SE=0.12\) units and \(df=30\). Suppose the prespecified margins are \(\Delta=0.35\) units. Assume the model and its standard error are appropriate for this contrast, and that the endpoint’s difference scale and margin have been justified independently of these results.

Using \(t_{0.95,30}\approx1.697\), the half-width is

1.697(0.12)=0.2036\text{ units}.

The matching 90% interval is

0.10\pm0.2036=(-0.1036,\ 0.3036)\text{ units}.

The upper endpoint is close to, but below, the upper margin of \(0.35\) units, and the lower endpoint is above \(-0.35\) units. The interval is contained in the margins, so equivalence is supported at the 5% level. This close result also shows why rounding should not be used to make the decision: retain enough digits to compare the calculated endpoints with the margins.

Assumptions, Diagnostics, and Interpretation

The interval rule is only as credible as the estimate and its uncertainty. Before making the equivalence decision, verify the following:

The parameter and sign are defined. State which treatment is subtracted from which, what population or model contrast is targeted, and the endpoint’s units.
The margins were justified and prespecified. Symmetry is a substantive choice: it says that departures of equal magnitude in either direction have the same relevance for this decision.
The analysis model is appropriate. For paired \(t\) inference, check independence between participants and the distribution of paired differences. For a model-based contrast, check that the model’s assumptions and standard-error calculation support its stated reference distribution.
The confidence interval matches the tests. Use the same estimate, standard error, degrees of freedom, and critical-value convention as the TOST procedure.
The result is not being decided by careless rounding. Compare unrounded interval endpoints with the margins, then report suitably rounded values and units.

A confidence interval that is too wide to fit inside the margins indicates insufficient precision for an equivalence conclusion. It may reflect a small sample, substantial variability, or a narrow margin; the interval alone does not identify which cause dominates. Inspect the data and model diagnostics to assess whether the standard error is credible, and report the estimate and interval rather than only the binary decision.

How This Approach Differs From Alternatives

Approach	What it answers	Why it differs from symmetric-margin TOST
TOST with symmetric difference margins	Is the difference sufficiently constrained to lie within \((-\Delta,\Delta)\)?	Requires the matching interval to fit inside both margins.
Test of a zero difference	Is there evidence that the difference is nonzero?	Failure to find a difference does not show that the difference is small enough for equivalence.
Point-estimate comparison	Does the observed estimate fall within the margins?	Ignores uncertainty, so it can appear favorable even when the interval crosses a margin.

At the same one-sided level \(\alpha\), TOST and the matching \(100(1-2\alpha)\%\) interval are two presentations of the same decision, not competing methods. Other interval levels can be useful descriptively, but they do not automatically encode the same TOST significance level.

Common Mistakes and Limitations

Wrong reasoning: “The estimate is inside the margins, so the treatments are equivalent.” Correction: The interval must also fit inside the margins; uncertainty can extend beyond either limit.
Wrong reasoning: “A nonsignificant test of zero difference proves equivalence.” Correction: That test addresses whether the data show a difference from zero, not whether they rule out differences large enough to matter.
Wrong reasoning: “A 95% interval is always the TOST interval at a 5% level.” Correction: For two one-sided tests each at \(\alpha=0.05\), the matching two-sided interval is 90%.
Wrong reasoning: “The margins can be chosen to include the calculated interval.” Correction: Margins define what differences are acceptable and must be justified independently of the observed result.

The conclusion is specific to the chosen difference parameter, endpoint, population, analysis, and margin. A symmetric difference scale is not automatically suitable for every pharmacokinetic endpoint: the scientific meaning of an absolute difference must support the scale. This tutorial addresses symmetric difference margins only; a parameter whose acceptable limits are unequal above and below zero requires a different formulation.

Key takeaway: For symmetric margins \((-\Delta,\Delta)\), conclude equivalence at one-sided level \(\alpha\) only when the matching \(100(1-2\alpha)\%\) confidence interval for the difference lies wholly inside those margins. In equivalent form, the absolute estimate plus the interval half-width must be less than \(\Delta\).

Check Your Understanding

Answer each question using the stated difference scale and significance level. Solutions follow the questions.

A difference estimate is \(0.04\) mg/L, with a matching 90% confidence interval of \((-0.12,0.20)\) mg/L and symmetric margins of \((-0.25,0.25)\) mg/L. Is equivalence supported at one-sided \(\alpha=0.05\)?
An estimate is \(0.02\) units and lies inside margins of \((-0.10,0.10)\) units. Its matching interval is \((-0.11,0.08)\) units. What is the TOST conclusion, and why?
For one-sided \(\alpha=0.025\), what two-sided confidence level corresponds to the interval rule?
In a paired analysis with \(n=16\), the paired differences have \(\bar{x}=0.10\) units and \(s=0.40\) units. With margins \((-0.50,0.50)\) units, calculate the paired \(t\) interval using \(t_{0.95,15}=1.753\) and decide whether it establishes equivalence.

Solutions. (1) Yes. The 90% interval is wholly inside \((-0.25,0.25)\) mg/L, so it supports equivalence at the stated level. (2) Equivalence is not established: the lower endpoint, \(-0.11\) units, falls below the lower margin of \(-0.10\) units, even though the estimate is inside the margins. (3) The matching confidence level is \(1-2(0.025)=0.95\), or 95%. (4) The standard error is \(SE=0.40/\sqrt{16}=0.10\) units; the half-width is \(1.753(0.10)=0.1753\) units. The interval is \(0.10\pm0.1753=(-0.0753,0.2753)\) units. Both endpoints are inside \((-0.50,0.50)\), so equivalence is supported at one-sided \(\alpha=0.05\), provided the paired-difference assumptions are adequate.