A previous study of 12 procedures of varying complexity in 1,498 hospitals identified a strong negative curvilinear relationship between the volume of a particular operation and postoperative mortality. The current study uses multiple regression techniques to explore the role of other potentially important variable and alternative interpretations of the volume-mortality relationship. The dependent variable is the difference between the hospital's actual death rate and its expected death rate based upon the riskiness of its case mix. The inclusion of other variables, such as size of hospital, teaching status, geographic location and cost, improves the fit of the regression, but does not diminish the importance of volume. There is no evidence that volume accumulated over 2 years is a better measure than volume in 1 year. Experience and volume of related operations are important in some cases but not others. Several likely alternative explanations for the observed relationship were not supported: larger hospitals and those with more house staff had outcomes that were worse than expected. Large geographic differences in mortality rates remain unexplained. A simultaneous-equation model is used to test whether higher volume leads to better outcomes or better outcomes lead to higher volumes. Both causal paths are supported, but their relative importance varies with the procedure in ways that are consistent with anticipated referral patterns.