## The Annals of Statistics

- Ann. Statist.
- Volume 15, Number 1 (1987), 385-397.

### On Simple Adjustments to Chi-Square Tests with Sample Survey Data

#### Abstract

For testing the goodness-of-fit of a $\log$-linear model to a multi-way contingency table with cell proportions estimated from survey data, Rao and Scott (1984) derived a first-order correction, $\delta\ldot$, to Pearson chi-square statistic, $X^2$ (or the likelihood ratio statistic, $G^2$) that takes account of the survey design. It was also shown that $\delta\ldot$ requires the knowledge of only the cell design effects (deffs) and the marginal deffs provided the model admits direct solution to likelihood equations under multinomial sampling. Simple upper bounds on $\delta\ldot$ are obtained here for models not admitting direct solutions, also requiring only cell deffs and marginal deffs or some generalized deffs not depending on any hypothesis. Applicability of an $F$-statistic used in GLIM to test a nested hypothesis is also investigated. In the case of a logit model involving a binary response variable, simple upper bounds on $\delta\ldot$ are obtained in terms of deffs of response proportions for each factor combination or some generalized deffs not depending on any hypothesis. Applicability of the GLIM $F$-statistic for nested hypotheses is also studied.

#### Article information

**Source**

Ann. Statist., Volume 15, Number 1 (1987), 385-397.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176350273

**Digital Object Identifier**

doi:10.1214/aos/1176350273

**Mathematical Reviews number (MathSciNet)**

MR885744

**Zentralblatt MATH identifier**

0647.62021

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62D05: Sampling theory, sample surveys

Secondary: 62H15: Hypothesis testing

**Keywords**

Multi-way contingency tables log-linear models logit models design effect corrections to chi-square tests

#### Citation

Rao, J. N. K.; Scott, A. J. On Simple Adjustments to Chi-Square Tests with Sample Survey Data. Ann. Statist. 15 (1987), no. 1, 385--397. doi:10.1214/aos/1176350273. https://projecteuclid.org/euclid.aos/1176350273