Why the Location of the Whole Interval Matters
An estimated treatment difference or ratio can fall between the equivalence limits even when the result does not establish equivalence. The estimate is only the center of the uncertainty statement: the confidence interval shows which values remain compatible with the data under the procedure used. Classification therefore depends on the entire interval and its location relative to both prespecified limits.
There are three useful outcomes. An interval wholly inside the equivalence region supports established equivalence at the corresponding test level. An interval that overlaps the region's boundary, or spans both sides of it, is inconclusive for equivalence. An interval wholly beyond one limit supports a conclusion in the direction of that limit: the effect is below the lower limit or above the upper limit. These labels describe the evidence from the interval; they do not turn an estimate into proof that a parameter has one exact value.
The limits and parameter must be on the same scale. For a difference, the limits may be expressed in units of the measured endpoint. For a ratio, the interval and limits may be compared on the ratio scale, provided the interval has been correctly back-transformed from the log scale. The examples below use intervals matched to TOST at \(\alpha=0.05\): a 90% confidence interval under the stated normal or \(t\) procedure. A different confidence level does not automatically give the same TOST classification.
Why These Three Classifications Match the Tests
For equivalence, the null hypothesis combines effects at or below the lower limit with effects at or above the upper limit. TOST splits that null into two one-sided tests: one asks whether the effect is greater than \(L\), and the other asks whether it is less than \(U\). Equivalence is established only when both null components are rejected.
For a symmetric normal-based interval with standard error \(SE\), the one-sided critical value at level \(\alpha\) is \(z_{1-\alpha}\). The lower one-sided test rejects its null when the estimate is sufficiently above \(L\); the upper one rejects when the estimate is sufficiently below \(U\). Both conditions must hold:
The endpoints on the left and right are exactly the endpoints of a two-sided \(100(1-2\alpha)\%\) normal confidence interval. Thus both tests reject precisely when that interval lies strictly inside \((L,U)\). With a \(t\) procedure, the same argument uses the relevant \(t\) critical value and degrees of freedom. The interval classification is not a separate rule from TOST; it is a compact way to read the paired decisions.
The same location logic also distinguishes an interval entirely outside the region. For example, if \(C_U<L\), the interval is wholly below the lower equivalence limit, rather than merely failing to fit inside the region. Under a TOST-matched interval, that location provides one-sided evidence that the parameter is below \(L\) at level \(\alpha\). The corresponding statement applies when \(C_L>U\).
Worked Example: A Difference Interval Wholly Inside the Limits
Suppose a hypothetical comparison estimates the mean difference in an exposure endpoint as \(\hat{\theta}=1.2\) units, with \(SE=1.0\) unit and \(df=9\). The prespecified difference limits are \(L=-5\) units and \(U=5\) units. Use a 90% \(t\) interval. Assume the model used to calculate the standard error is appropriate, its observations meet the model's independence requirements, and the \(t\) procedure is valid for this estimate.
For \(df=9\), the 90% two-sided critical value is approximately \(t_{0.95,9}=1.833\). The interval is:
The lower endpoint is greater than the lower limit, since \(-0.633>-5\), and the upper endpoint is less than the upper limit, since \(3.033<5\). The complete interval is therefore inside the equivalence region. Under the stated assumptions and the matching 90% interval, equivalence is established at the corresponding TOST level. The interval does not say that the true difference is zero; it indicates that values compatible with the procedure lie within the prespecified limits.
When the Result Is Inconclusive
An interval can include plausible values inside the equivalence region and also values outside it. That is not evidence that the effect lies beyond the margin, and it is not enough to establish equivalence. The appropriate classification is inconclusive for equivalence. In particular, the point estimate's position cannot resolve the uncertainty represented by the interval.
Worked Example: A Ratio Interval Crossing the Lower Limit
Consider a hypothetical geometric mean ratio estimate of \(0.900\). On the log scale, its standard error is \(0.120\). The ratio equivalence limits are \(0.80\) and \(1.25\). Assume the log-ratio estimate and its standard error satisfy the stated normal approximation. Construct the matched 90% interval using \(z_{0.95}=1.645\), then back-transform.
The log estimate is \(\ln(0.900)=-0.1054\). The interval on the log scale is:
Exponentiating both endpoints gives the ratio interval:
This interval extends below the lower limit \(0.80\) and also includes ratios above \(0.80\). It is not wholly within \((0.80,1.25)\), and it is not wholly below \(0.80\). The result is therefore inconclusive for equivalence, not established equivalence and not evidence that the ratio is entirely below the lower limit. Comparing the back-transformed interval with the original ratio limits is valid because exponentiation is increasing and the interval and limits are now on the same scale.
The width of an interval matters because it determines whether uncertainty reaches across a limit. A narrow interval may fit inside the region or lie entirely beyond one boundary. A wider interval centered at the same estimate may cross a boundary, changing the conclusion to inconclusive. Classification is consequently about both the estimate's location and its precision.
When the Interval Lies Beyond a Margin
A result is different from an inconclusive crossing when the whole interval lies on one side of the equivalence region. If the upper endpoint is below \(L\), every value in the interval is below the lower limit. If the lower endpoint is above \(U\), every value is above the upper limit. These are directional findings relative to the prespecified region, not merely failures to establish equivalence.
Worked Example: A Difference Interval Wholly Above the Upper Limit
In a hypothetical study, suppose the estimated mean difference is \(8.0\) mg/L, with \(SE=1.2\) mg/L and \(df=15\). The prespecified equivalence limits are \(-5\) mg/L and \(5\) mg/L. Assume the \(t\)-interval model and its independence and standard-error conditions are suitable. For a 90% interval, use \(t_{0.95,15}=1.753\).
The interval is:
The lower endpoint, \(5.896\) mg/L, exceeds the upper equivalence limit of \(5\) mg/L. The entire interval is above the equivalence region. The classification is evidence that the difference lies beyond the upper margin, not an inconclusive crossing and not equivalence. In context, the interval procedure places the difference above the prespecified upper limit; it does not establish that the difference equals \(8.0\) mg/L exactly.
Reading the Interval Systematically
A reliable classification can be made by checking the two endpoints in order. This avoids letting the point estimate or a single endpoint dominate the decision.
Identify the parameter, its scale, and the prespecified lower and upper limits. Confirm \(L<U\).
Check that the interval uses the appropriate model, standard error, degrees of freedom if needed, and confidence level for the TOST decision.
If \(C_L>L\) and \(C_U<U\), classify the result as equivalent. If \(C_U<L\), classify it as below the lower margin; if \(C_L>U\), classify it as above the upper margin.
If the interval intersects the equivalence region but is not wholly within it, classify the result as inconclusive for equivalence. State which limit it crosses when relevant.
For the strict cases above, the categories can be summarized as follows. Exact equality between an endpoint and a margin is a boundary case; it should not be silently rounded into one of these strict patterns.
| Interval location | Classification | What it supports |
|---|---|---|
| \(L<C_L\) and \(C_U<U\) | Equivalent | The interval is wholly within the prespecified region. |
| \(C_U<L\) | Beyond the lower margin | The interval is wholly below the equivalence region. |
| \(C_L>U\) | Beyond the upper margin | The interval is wholly above the equivalence region. |
| Interval overlaps the region and extends outside it | Inconclusive for equivalence | Equivalence is not established, and the interval is not wholly beyond a margin. |
Diagnostics, Alternatives, and Common Pitfalls
Before interpreting a classification, check that the interval and margins are comparable. A ratio interval should be compared with ratio limits, not with log limits; alternatively, both may be put on the log scale. A difference interval should use the same units as its difference margins. Verify that the confidence level matches the TOST significance level and that the standard error, degrees of freedom, and model assumptions used to construct the interval are appropriate. If an endpoint is very close to a limit, retain adequate numerical precision and state the endpoint and limit rather than relying on a rounded label.
Reading the interval and conducting both one-sided tests are alternative presentations of the same TOST decision when they use the same model, scale, and matched confidence level. The tests display each one-sided comparison directly; the interval makes the combined location of uncertainty easy to see. Neither presentation repairs a misspecified model or inappropriate standard error. A point-estimate-only comparison is not an alternative inferential procedure: it leaves out uncertainty and can label a result equivalent even when the interval crosses a limit.
- Pitfall: “The estimate is inside, so the result is equivalent.” An estimate inside the limits is not enough. Check whether both confidence limits are inside them.
- Pitfall: “The interval crosses a margin, so the result is non-equivalent.” A crossing interval includes values on both sides of that margin. Unless the entire interval lies beyond a limit, the location does not support that directional classification; the result is inconclusive for equivalence.
- Pitfall: “Anything that does not establish equivalence proves non-equivalence.” These are different conclusions. An interval overlapping the region and extending outside it establishes neither equivalence nor that the effect lies wholly beyond a margin.
- Pitfall: “Any reported confidence interval gives the TOST classification.” The interval must be constructed with the confidence level and procedure corresponding to the tests. A different interval may give a different location and does not automatically reproduce the TOST decision.
The classification is useful because it separates a positive equivalence conclusion from uncertainty and from a result clearly outside the acceptable region. In reporting, include the estimate, interval, limits, scale, and classification. For an inconclusive result, identify the crossed limit; for a result beyond a margin, state the direction. These details make the conclusion interpretable without implying more certainty than the interval provides.
Check Your Understanding
Classify each interval using the stated limits. Assume the intervals are valid and matched to the TOST procedure, and use strict inequalities for these questions.
- For limits \(-2\) and \(2\) units, classify the interval \([-1.1,1.6]\) units.
- For ratio limits \(0.80\) and \(1.25\), classify \([0.77,1.03]\).
- For limits \(-4\) and \(4\) mg/L, classify \([4.6,7.2]\) mg/L.
- An estimate is inside the equivalence limits, but its interval extends beyond the upper limit while still overlapping the equivalence region. What is the classification, and why?
Solutions. (1) Equivalent: \(-1.1>-2\) and \(1.6<2\), so the full interval is inside the limits. (2) Inconclusive for equivalence: the interval crosses the lower limit \(0.80\), but is not wholly below it. (3) Beyond the upper margin: the lower endpoint \(4.6\) exceeds \(4\), so the entire interval is above the equivalence region. (4) Inconclusive for equivalence: an inside point estimate does not make the full interval fit inside, and an interval that overlaps the region is not wholly beyond the upper limit.