Clustered binary data occur commonly in both the biomedical and health sciences. In this paper, we consider logistic regression models for multivariate binary responses, where the association between the responses is largely regarded as a nuisance characteristic of the data. In particular, we consider the estimator based on independence estimating equations (IEE), which assumes that the responses are independent. This estimator has been shown to be nearly efficient when compared with maximum likelihood (ML) and generalized estimating equations (GEE) in a variety of settings. The purpose of this paper is to highlight a circumstance where assuming independence can lead to quite substantial losses of efficiency. In particular, when the covariate design includes within-cluster covariates, assuming independence can lead to a considerable loss of efficiency in estimating the regression parameters associated with those covariates.