*! version 3.1.6 13jan2026
help tost

Title

tost -- Two one-sided tests for equivalence

tost commands

Miscellaneous
tostmcc Paired z test for equivalence of marginal probabilities in binary data

tostmcci Immediate command for paired z test for equivalence of marginal probabilities in binary data

tostpr One- and two-sample z tests for proportion equivalence

tostpri Immediate command for one- and two-sample z tests for proportion equivalence

tostranksum Two-sample rank sum test for stochastic equivalence

tostregress Linear regression tests for equivalence

tostrrp Test for equivalence of relative risk and unity in paired binary data

tostrrpi Immediate command for test for equivalence of relative risk and unity in paired binary data

tostsignrank Test for the distribution of paired or matched data being equivalent to one that is symmetrical & centered on zero

tostt Mean equivalence t tests

tostti Immediate command for mean equivalence t tests

Description

The tost package provides a suite of commands to perform
two one-sided tests for equivalence corresponding to the ttest &
ttesti, prtest & prtesti, mcc & mcci, regress,
signrank, and ranksum tests for difference, thus addressing
inference about equivalence for a number of paired and unpaired, parametric
and nonparametric study designs and data types. Each command tests a null
hypothesis that samples were drawn from populations different by at least plus
or minus some researcher-defined level of tolerance, which can be defined in
terms of units of the data or rank units (Delta), or (except for tostrrp
and tostrrpi) in units of the test statistic's distribution (epsilon).
Enough evidence rejects this null hypothesis in favor of equivalence within the
tolerance. Equivalence intervals for all tests may be defined symmetrically or
asymmetrically.

These tests are equivalence tests are all more or less based on the t and
z tests following the logic laid out by Schuirmann (1987),
but with variations as detailed in the help files for each command. All the
tost commands are based on a Wald-type test, where some difference in
sample statistics is divided by the standard deviation of that difference:
theta/st.dev. theta. A general test for equivalence null hypothesis takes one
of the following two forms depending on whether equivalence is defined in terms
of Delta, or epsilon:

Ho: |theta| >= Delta,

where the equivalence interval ranges from theta-Delta to theta+Delta.
This translates directly into two one-sided null hypotheses:

Ho1: Delta - theta <= 0; and

Ho2: theta + Delta <= 0

-OR-

Ho: |Z| >= epsilon, (may substitute T for Z in these expressions)

where the equivalence interval ranges from -epsilon to epsilon. This also
translates directly into two one-sided null hypotheses:

Ho1: epsilon - Z <= 0; and

Ho2: Z + epsilon <= 0

When an asymmetric equivalence interval is defined using the the general
negativist null hypothesis becomes:

Ho: theta <= Delta_lower, or theta >= Delta_upper,

where the equivalence interval ranges from theta+Delta_lower to
theta+Delta_upper. This also translates directly into two one-sided null
hypotheses:

Ho1: Delta_upper - theta <= 0; and

Ho2: theta - Delta_lower <= 0

-OR-

Ho: Z <= epsilon_lower, or Z >= epsilon_upper, and

Ho1: epsilon_upper - Z <= 0; and

Ho2: Z - epsilon_lower <= 0

Relevance testing is a term used to describe inference based on the
combined results of a test for difference, and a corresponding test for
equivalence. The relevance() option available to all commands except
tostppr and tostppri provides conclusions from combined results for
a specified alpha(), eqvtype(), and eqvlevel(). Relevance
tests provide four possible conclusions:

Reject Ho for test for difference, and not reject Ho for test for
equivalence: conclude relevant difference.

Not reject Ho for test for difference, and reject Ho for test for
equivalence: conclude equivalence.

Reject Ho for test for difference, and reject Ho for test for
equivalence: conclude trivial difference (i.e. there is evidence for a
difference, but you have stated a priori that differences that small are
not relevant).

Not reject Ho for test for difference, and not reject Ho for test for
equivalence: conclude indeterminate (i.e. underpowered test for these data).

Author

Alexis Dinno

Portland State University

alexis.dinno@pdx.edu

Development of tost is
ongoing. Please contact me with any questions, bug reports or suggestions for
improvement. Fixing bugs will be facilitated by sending along:

(1) a copy of the data (de-labeled or anonymized is fine),

(2) a copy of the command used, and

(3) a copy of the exact output of the command.

Suggested citation

Dinno, A. 2025. tost: Stata software package for performing two one-sided
tests for equivalence. URL:

References

Schuirmann, D. A. 1987. A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability.
Journal of Pharmacokinetics and Biopharmaceutics. 15: 657-680

Also See

Help: tostmcc, tostpr, tostregress, tostrrp, tostranksum, tostsignrank, tostt