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\ldot, 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\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 δ\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 δ\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.