When confidence intervals for risk-adjusted rates are based on large-sample approximations, with truncation in case the lower or upper limit falls outside the range of 0-100%, you know there is Trouble in River City. This paper presents an exact calculation of confidence intervals for risk-adjusted rates, applied to the evaluation of hospital performance. From subjects with known probabilities of an event, the method takes the inverse of the binomial distribution generalized to the case of unequal probabilities. Like the logistic model, the confidence interval is calculated assuming that a group effect is additive in the logit domain, causing a shift of the probabilities as a group. When the exact upper tail probability for the observed events is α/2, the mean of the shifted probabilities is the lower (1-α) % confidence limit on the underlying rate. This generalization of the Clopper-Pearson confidence interval method compares favorably with intervals based on the Poisson and normal distributions. With mid-P adjustment, the resulting intervals have coverage probability close to the nominal probability
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