In the analysis of binary response data from many types of large studies, the data are likely to have arisen from multiple centers, resulting in a within-center correlation for the response. Such correlation, or clustering, occurs when outcomes within centers tend to be more similar to each other than to outcomes in other centers. In studies where there is also variability among centers with respect to the exposure of interest, analysis of the exposure-outcome association may be confounded, even after accounting for within-center correlations. We apply several analytic methods to compare the risk of major complications associated with two strategies, staged and combined procedures, for performing percutaneous transluminal coronary angioplasty (PTCA), a mechanical means of relieving blockage of blood vessels due to atherosclerosis. Combined procedures are used in some centers as a cost-cutting strategy. We performed a number of population-averaged and cluster-specific (conditional) analyses, which (a) make no adjustments for center effects of any kind; (b) make adjustments for the effect of center on only the response; or (c) make adjustments for both the effect of center on the response and the relationship between center and exposure. The method used for this third approach decomposes the procedure type variable into within-center and among-center components, resulting in two odds ratio estimates. The naive analysis, ignoring clusters, gave a highly significant effect of procedure type (OR = 1.6). Population average models gave marginally to very nonsignificant estimates of the OR for treatment type ranging from 1.6 to 1.2 with adjustment only for the effect of centers on response. These results depended on the assumed correlation structure. Conditional (cluster-specific) models and other methods that decomposed the treatment type variable into among- and within-center components all found no within-center effect of procedure type (OR = 1.02, consistently) and a considerable among-center effect. This among-center variability in outcomes was related to the proportion of patients who receive combined procedures and was found even when conditioned on procedure type (within-center) and other patient- and center-level covariates. This example illustrates the importance of addressing the potential for center effects to confound an outcome-exposure association when average exposure varies across clusters. While conditional approaches provide estimates of the within-cluster effect, they do not provide information about among-center effects. We recommend using the decomposition approach, as it provides both types of estimates.