## The Annals of Statistics

- Ann. Statist.
- Volume 13, Number 3 (1985), 845-874.

### Testing for Independence in a Two-Way Table: New Interpretations of the Chi-Square Statistic

Persi Diaconis and Bradley Efron

#### Abstract

The classical chi-square test for independence in a two-way contingency table often rejects the independence hypothesis at an extremely small significance level, particularly when the sample size is large. This paper proposes some alternative distributions to independence, to help interpret the $\chi^2$ statistic in such situations. The uniform alternative, in which every possible contingency table of the given dimension and sample size receives equal probability, leads to the volume test, as originally suggested in a regression context by H. Hotelling. Exponential family theory is used to generate a class of intermediate alternatives between independence and uniformity, leading to a random effects model for contingency tables.

#### Article information

**Source**

Ann. Statist., Volume 13, Number 3 (1985), 845-874.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349634

**Digital Object Identifier**

doi:10.1214/aos/1176349634

**Mathematical Reviews number (MathSciNet)**

MR803747

**Zentralblatt MATH identifier**

0593.62040

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F05: Asymptotic properties of tests

Secondary: 62G10: Hypothesis testing

**Keywords**

Chi-square test for independence overdispersion volume tests random effects for exponential families

#### Citation

Diaconis, Persi; Efron, Bradley. Testing for Independence in a Two-Way Table: New Interpretations of the Chi-Square Statistic. Ann. Statist. 13 (1985), no. 3, 845--874. doi:10.1214/aos/1176349634. https://projecteuclid.org/euclid.aos/1176349634