## The Annals of Statistics

- Ann. Statist.
- Volume 12, Number 1 (1984), 46-60.

### On Chi-Squared Tests for Multiway Contingency Tables with Cell Proportions Estimated from Survey Data

#### Abstract

The impact of survey design on standard multinomial-based methods for a multiway contingency table is studied, under nested loglinear models. The asymptotic null distribution of the Pearson chi-squared test statistic, $X^2$ (or the likelihood ratio test statistic, $G^2$) is obtained as a weighted sum of independent $\chi^2_1$ random variables, and the weights are then related to the familiar design effects (deffs) used by survey samplers. A simple correction to $X^2$ (or $G^2$) is also obtained which requires the knowledge of only the cell deffs and the deffs for collapsed tables (marginals), whenever the model admits a direct solution of likelihood equations under multinomial sampling. Finally, an example on the relative performance of $X^2$ and some corrected $X^2$ statistics in a three-way table is given, using some data from the Canada Health Survey, 1978-1979.

#### Article information

**Source**

Ann. Statist., Volume 12, Number 1 (1984), 46-60.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346391

**Digital Object Identifier**

doi:10.1214/aos/1176346391

**Mathematical Reviews number (MathSciNet)**

MR733498

**Zentralblatt MATH identifier**

0622.62059

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62D05: Sampling theory, sample surveys

Secondary: 62H15: Hypothesis testing

**Keywords**

Multiway contingency tables chi-squared and loglikelihood ratio tests effect of survey design Wald statistic

#### Citation

Rao, J. N. K.; Scott, A. J. On Chi-Squared Tests for Multiway Contingency Tables with Cell Proportions Estimated from Survey Data. Ann. Statist. 12 (1984), no. 1, 46--60. doi:10.1214/aos/1176346391. https://projecteuclid.org/euclid.aos/1176346391