One-Sided Test Statistics at Equivalence Boundaries

Why the Boundary Changes the Direction

In an equivalence analysis, the parameter estimate is compared with two boundaries: a lower margin $L$ and an upper margin $U$, where $L<U$. The lower-boundary test asks whether the parameter is greater than $L$; the upper-boundary test asks whether it is less than $U$. These are both one-sided questions, but their alternatives point in opposite directions.

A signed test statistic makes that direction visible. Throughout this tutorial, the numerator is the estimate minus the boundary being tested. With this convention, a positive statistic points above a boundary and a negative statistic points below it. Consequently, the lower-boundary alternative is supported by a statistic far enough in the positive direction, whereas the upper-boundary alternative is supported by one far enough in the negative direction.

Definition: Let $\hat{\theta}$ estimate a parameter $\theta$, and let $SE$ be its standard error. The signed statistics for the lower and upper boundaries are $$ t_L=\frac{\hat{\theta}-L}{SE}, \qquad t_U=\frac{\hat{\theta}-U}{SE}. $$ For a t procedure, these statistics are compared with a t reference distribution using the appropriate degrees of freedom $df$. The sign is part of the statistic’s interpretation: $t_L>0$ points above $L$, while $t_U<0$ points below $U$.

The symbol $t$ in these expressions denotes a standardized statistic; the same signed formulas can also be used in a procedure with a different specified reference distribution. The essential construction is the signed distance from the estimate to the boundary, measured in standard-error units. The units of $\hat{\theta}$, $L$, $U$, and $SE$ must agree, so each statistic is dimensionless.

Hypotheses and the Sign Convention

The lower-boundary test has $H_0:\theta\leq L$ against $H_a:\theta>L$. Under the null, the parameter is at or below the lower margin. An estimate above $L$ gives a positive numerator, and increasingly positive values of $t_L$ indicate increasingly strong evidence in the alternative’s direction.

The upper-boundary test has $H_0:\theta\geq U$ against $H_a:\theta<U$. Here, an estimate below $U$ gives a negative numerator. Therefore, increasingly negative values of $t_U$ indicate increasingly strong evidence in the alternative’s direction. The direction is not reversed by the word “upper”: under the signed estimate-minus-boundary convention, the upper test statistic remains $(\hat{\theta}-U)/SE$.

Boundary test	Null and alternative	Statistic	Direction supporting the alternative
Lower boundary	$H_0:\theta\leq L$; $H_a:\theta>L$	$t_L=(\hat{\theta}-L)/SE$	Large positive values
Upper boundary	$H_0:\theta\geq U$; $H_a:\theta<U$	$t_U=(\hat{\theta}-U)/SE$	Large negative magnitudes

These directions follow from the alternatives, not from a rule that positive statistics are always favorable. For a given estimate between the margins, $t_L$ is positive and $t_U$ is negative. The two statistics are calculated from the same estimate and standard error but use different boundary values.

Why Standardizing at the Boundary Works

A test statistic measures how far the observed estimate is from the value specified by the null boundary, relative to its sampling uncertainty. For the lower test, the boundary value is $L$; for the upper test, it is $U$. The following justification also explains why the rejection directions differ.

Center at the boundary.
For the lower test, subtract $L$ from $\hat{\theta}$; for the upper test, subtract $U$. A numerator of zero means the estimate equals the tested boundary.

Scale by uncertainty.
Divide the difference by $SE$, which is positive. Division by a positive quantity changes the magnitude but not the sign.

Match the tail to the alternative.
$H_a:\theta>L$ concerns values above $L$, so evidence is in the upper tail of $t_L$. $H_a:\theta<U$ concerns values below $U$, so evidence is in the lower tail of $t_U$.

Under a t procedure, suppose $\hat{\theta}$ is normally distributed around $\theta$, and its standard error is estimated in a way that yields the stated t reference distribution under the boundary equality. At $\theta=L$, the lower statistic is centered at the boundary and has a central t distribution with the procedure’s $df$. At a value $\theta<L$, its distribution shifts toward smaller values, not toward the upper-tail rejection region. Thus the boundary value is the least favorable case for controlling the size of the lower test across $H_0:\theta\leq L$. The corresponding argument for the upper test is reversed: at $\theta=U$ its statistic has the central reference distribution, while $\theta>U$ shifts it upward, away from the lower-tail rejection region.

This justification depends on the model supplying a valid estimate and standard error, and on the reference distribution and degrees of freedom matching that model. For example, a crossover-model contrast must use the uncertainty estimate and degrees of freedom appropriate to the fitted analysis. A statistic computed with an unrelated standard error may have a plausible-looking sign but an invalid scale and reference distribution.

Worked Example: The Estimate Lies Between the Boundaries

Suppose a hypothetical analysis estimates a treatment contrast as $\hat{\theta}=7.4$ units, with lower margin $L=2$, upper margin $U=10$, and $SE=2.0$ units. Assume the fitted procedure supplies $df=18$ and its t-reference assumptions are appropriate. Calculate and interpret both signed statistics.

For the lower boundary:

t_L=\frac{\hat{\theta}-L}{SE} =\frac{7.4-2}{2.0} =\frac{5.4}{2.0} =2.70.

The positive value points above $L$, in the direction of $H_a:\theta>L$. It says the estimate is 2.70 standard errors above the lower margin.

For the upper boundary:

t_U=\frac{\hat{\theta}-U}{SE} =\frac{7.4-10}{2.0} =\frac{-2.6}{2.0} =-1.30.

The negative value points below $U$, in the direction of $H_a:\theta<U$. The estimate is 1.30 standard errors below the upper margin. These are directional interpretations of the two statistics; neither statistic by itself summarizes the result of both component tests.

Worked Examples at and Beyond a Boundary

Worked Example: An Estimate Equal to the Lower Margin

For a hypothetical parameter measured in units, suppose $L=5$, $U=15$, $\hat{\theta}=5$, and $SE=1.6$ units. Assume the model’s t procedure is appropriate and reports $df=22$. Calculate the statistics.

At the lower boundary:

t_L=\frac{5-5}{1.6}=0.

A zero statistic means the estimate is exactly at $L$; it does not point above that boundary. At the upper boundary:

t_U=\frac{5-15}{1.6} =\frac{-10}{1.6} =-6.25.

This statistic points strongly in the direction of $\theta<U$, because the estimate is far below $U$ relative to its standard error. The example illustrates why a favorable direction for one boundary does not settle the question at the other boundary.

Worked Example: An Estimate Above the Upper Margin

In a second hypothetical analysis, let $L=10$, $U=20$, $\hat{\theta}=22$, and $SE=3.0$ units. Suppose $df=14$ is appropriate under the analysis model. Calculate the signed statistics and identify their directions.

For the lower boundary:

t_L=\frac{22-10}{3.0} =\frac{12}{3.0} =4.00.

This is positive and points above $L$, in the lower test’s alternative direction. For the upper boundary:

t_U=\frac{22-20}{3.0} =\frac{2}{3.0} =0.667.

The upper statistic is positive, so it points above $U$, opposite to the upper test’s alternative $\theta<U$. The arithmetic is not a sign error: because the estimate exceeds the upper margin, its estimate-minus-boundary difference is positive. A statistic’s sign must always be interpreted against the particular alternative being tested.

Assumptions, Checks, and Common Errors

The formulas are simple, but valid inference requires more than arithmetic. Before interpreting the statistics, check the following conditions:

Correct target and margins. $\hat{\theta}$, $L$, and $U$ must refer to the same parameter and scale, and the margins must be the prespecified boundaries for that parameter.
Valid standard error. $SE$ must quantify uncertainty in $\hat{\theta}$ under the fitted design and model. It must be positive and expressed in the same units as the estimate.
Appropriate reference distribution. Use the distribution and $df$ justified by the analysis procedure. The signed formulas alone do not establish a t distribution or determine degrees of freedom.
Model adequacy. Dependence, influential observations, or poorly fitting residual assumptions can make the reported standard error or reference distribution unreliable. Inspect the diagnostics appropriate to the analysis rather than relying on the sign of the statistic.

A useful arithmetic check is to compare each statistic’s sign with the raw estimate’s position relative to its boundary. If $\hat{\theta}>L$, then $t_L$ must be positive; if $\hat{\theta}<U$, then $t_U$ must be negative. If either check fails, verify the subtraction order, units, and boundary labels. If $\hat{\theta}$ equals a boundary, the corresponding statistic must be zero.

A common mistake is to write the upper statistic as $(U-\hat{\theta})/SE$ and then interpret it as though it followed the estimate-minus-boundary convention. That expression can be used with an explicitly reversed sign convention, but it is not the signed statistic defined here. Mixing conventions reverses signs and can lead to using the wrong tail. The correction is to keep the numerator $\hat{\theta}-U$ and recognize that the upper alternative is supported by negative, not positive, values.

Another mistake is to conclude that equivalence has been shown because one statistic points in a favorable direction or is numerically large. Each statistic addresses only its own one-sided boundary question. The TOST conclusion depends on both component tests; direction alone is not a rejection decision. The p-values and the rule for combining the component results are separate from constructing and interpreting these signed statistics.

The main alternative presentation is to reverse the sign of the upper statistic, using $(U-\hat{\theta})/SE$, so that positive values point toward the upper alternative. This is mathematically equivalent if the reference tail and all interpretations are reversed consistently. The estimate-minus-boundary convention used here has the practical advantage that both formulas share the same subtraction order. Confidence-interval comparisons provide another way to express boundary evidence, but they do not change the signed-statistic definitions. In all presentations, consistency between numerator, tail, and alternative is essential.

Key takeaway: With estimate-minus-boundary numerators, $t_L=(\hat{\theta}-L)/SE$ is oriented toward the lower-test alternative when positive, while $t_U=(\hat{\theta}-U)/SE$ is oriented toward the upper-test alternative when negative. Always interpret each sign relative to its own boundary and alternative.

Check Your Understanding

For each question, use the signed estimate-minus-boundary convention. The solutions follow the questions.

If $\hat{\theta}=12$, $L=8$, and $SE=2$, what is $t_L$, and which direction does it indicate?
If $\hat{\theta}=12$, $U=16$, and $SE=2$, what is $t_U$, and which direction does it indicate?
Suppose $\hat{\theta}=4$, $L=6$, and $SE=0.5$. What is $t_L$, and why is its sign consistent with the estimate’s position?
Why does a positive $t_U$ point opposite to the upper-test alternative?
Name two checks needed before interpreting a calculated boundary statistic inferentially.

Solutions. (1) $t_L=(12-8)/2=2$, a positive value pointing above $L$. (2) $t_U=(12-16)/2=-2$, a negative value pointing below $U$. (3) $t_L=(4-6)/0.5=-4$. The estimate is below $L$, so the estimate-minus-boundary numerator is negative. (4) The upper alternative is $\theta<U$, while a positive $\hat{\theta}-U$ means the estimate is above $U$. (5) Answers include confirming that the estimate and boundary use the same parameter and scale, that the standard error is appropriate and in matching units, and that the reference distribution and degrees of freedom match the model.

Boundary test	Null and alternative	Statistic	Direction supporting the alternative
Lower boundary	\(H_0:\theta\leq L\); \(H_a:\theta>L\)	\(t_L=(\hat{\theta}-L)/SE\)	Large positive values
Upper boundary	\(H_0:\theta\geq U\); \(H_a:\theta<U\)	\(t_U=(\hat{\theta}-U)/SE\)	Large negative magnitudes