## The Annals of Statistics

### 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

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