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

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, δ, to Pearson chi-square statistic, X² (or the likelihood ratio statistic, G²) that takes account of the survey design. It was also shown that δ 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 δ 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 δ 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.