Nallamothu et al1 concluded that differences in in-hospital mortality rates between low- and high-volume centers rose as the expected risk of in-hospital death increased. However, the predicted risk for each patient was derived from a risk prediction model that excluded volume as an explanatory variable. The coefficient for any risk factor that is correlated with volume in this risk prediction model is subject to omitted variable bias.2 For example, the authors noted that patients undergoing coronary artery bypass grafting at low-volume centers were older on average. Therefore, the described risk prediction model will mistakenly assign some of the lower mortality associated with high volume to the age variable because patients at high-volume hospitals are younger on average, and the association between volume and mortality is negative. The risk prediction model therefore systematically underestimates the association between risk factors and mortality for characteristics more prevalent in low-volume hospitals and overestimates these effects for characteristics more prevalent in high-volume hospitals. In turn, a regression analysis including volume and this measure of predicted risk as explanatory variables will produce a biased estimate of the association between volume and mortality. Analyses of the relation between volume and mortality by surgical risk strata will also be confounded. Although it is plausible that differences in outcomes between low- and high-volume health care providers will be greater for high-risk patients, the reported analyses cannot be used to draw this conclusion.
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