# Equivalence of means hypothesis test

An equivalence hypothesis test formally tests if two population means are equivalent, that is, practically the same.

An equality hypothesis test can never prove that the means are equal, it can only ever disprove the null hypothesis of equality. It is therefore of interest when comparing say a new treatment against a placebo, where the null hypothesis (assumption of what is true without evidence to the contrary) is that the treatment has no effect, and you want to prove the treatment produces a useful effect. By contrast, an equivalence hypothesis test is of interest when comparing say a generic treatment to an existing treatment where the aim is to prove that they are equivalent, that is the difference is less than some small negligible effect size. A equivalence hypothesis test therefore constructs the null hypothesis of non-equivalence and the goal is to prove the means are equivalent.

The null hypothesis states that the means are not equivalent, against the alternative
hypothesis that the difference between the means is within the bounds of the equivalence
interval, that is, the effect size is less than some small difference that is considered
practically zero. The hypothesis is tested as a composite of two one-sided t-tests (TOST),
H0_{1} tests the hypothesis that mean difference is less than the lower bound of the
equivalence interval, test H0_{2} that the mean difference is greater than the upper
bounds of the equivalence interval. The p-value is the greater of the two one sided t-test
p-values. When the test p-value is small, you can reject the null hypothesis and conclude the
samples are from populations with practically equivalent means.

**Available in Analyse-it Editions**

Standard edition

Method Validation edition

Quality Control & Improvement edition

Ultimate edition

- What is Analyse-it?
- What's new?
- Administrator's Guide
- User's Guide
- Statistical Reference Guide
- Distribution
- Compare groups
- Calculating univariate descriptive statistics, by group
- Side-by-side univariate plots
- Creating side-by-side univariate plots
- Equality of means/medians hypothesis test
- Equivalence of means hypothesis test
- Tests for means/medians
- Testing equality of means/medians
- Testing equivalence of means
- Difference between means/medians effect size
- Estimators for the difference in means/medians
- Estimating the difference between means/medians
- Multiple comparisons
- Mean-Mean scatter plot
- Multiple comparison procedures
- Comparing multiple means/medians
- Homogeneity of variance hypothesis test
- Tests for homogeneity of variance
- Testing homogeneity of variance
- Study design
- Compare pairs
- Contingency tables
- Correlation and association
- Principal component analysis (PCA)
- Factor analysis (FA)
- Item reliability
- Fit model
- Method comparison / Agreement
- Measurement systems analysis (MSA)
- Reference interval
- Diagnostic performance
- Survival/Reliability
- Control charts
- Process capability
- Pareto analysis
- Study Designs
- Bibliography

Version 6.15

Published 18-Apr-2023