# Equality of proportions hypothesis test

A hypothesis test formally tests if the proportions in two or more populations are equal.

When one variable is an explanatory variable (X, fixed) and the other a response variable (Y, random), the hypothesis of interest is whether the populations have the same or different proportions in each category.

- For two samples, the null hypothesis states that the parameter of interest is equal to the hypothesized value, against the alternative hypothesis it is not equal to (or less than, or greater than) the hypothesized value.
You can formulate the hypotheses in terms of the parameter of interest: odds ratio = 1, the ratio of proportions = 1, or the difference of proportions = 0 depending on the desired effect size estimate.

- For more than two samples, the null hypothesis states that the proportions in each category are equal for all populations, against the alternative hypothesis that the proportions in a category are not equal for at least 2 populations.

When the test p-value is small, you can reject the null hypothesis and conclude that the populations differ in the proportions in at least one category.

Tests for contingency tables larger than 2 x 2 are omnibus tests and do not tell you which groups differ from each other or in which categories. You should use the mosaic plot to examine the association, or partition the contingency table into a series of 2 x 2 sub-tables and test each table.

**Related tasks**

**Available in Analyse-it Editions**

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- What is Analyse-it?
- What's new?
- Administrator's Guide
- User's Guide
- Statistical Reference Guide
- Distribution
- Compare groups
- Compare pairs
- Contingency tables
- Contingency table
- Creating a contingency table
- Creating a contingency table (related data)
- Grouped frequency plot
- Effect size
- Estimators
- Estimating the odds ratio
- Estimating the odds ratio (related data)
- Relative risk
- Inferences about equality of proportions
- Equality of proportions hypothesis test
- Exact and asymptotic p-values
- Wald, Score, Likelihood ratio
- Tests for equality of proportions (independent samples)
- Testing equality of proportions (independent samples)
- Tests for equality of proportions (related samples)
- Testing equality of proportions (related samples)
- Inferences about independence
- Mosaic plot
- Creating a mosaic plot
- Study design
- 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