Standardized mortality ratio for an estimated number of deaths

The traditional standardized mortality ratio (SMR) compares the mortality rate of a study population with that of a reference population. In order to measure the performance of a surgeon or a group of surgeons in a hospital performing a particular type of surgical operation, a different SMR is used. This SMR compares the observed number of deaths in a sample with an estimated number of deaths usually calculated based on the average performance of a group of surgeons. The estimated number of deaths involved in the new SMR is not a constant but a random variable. This means that all existing results for the traditional SMR may no longer be valid for the new SMR. In this paper, the asymptotic distribution of the SMR based on an estimated number of deaths is derived. We also use the bootstrap procedure to estimate the finite-sample distribution. A simulation study is used to compare both probabilities of type I error and powers of existing confidence intervals and confidence intervals constructed using the asymptotic and bootstrap distributions of SMR. Our study reveals that, in general, existing confidence intervals are conservative in terms of probability of type I error, and the two new confidence intervals are more accurate. To perform a fair power comparison, the coverage probabilities of existing confidence intervals are recalibrated to match that based on the asymptotic distribution of SMR, and then our study shows that the powers of the asymptotic and bootstrap approaches are lower than existing approaches when the odds ratio of death Q is greater than the odds ratio of death under the null hypothesis, , but higher when Q is smaller than . The effect of patients' risk distribution on the SMR is also investigated.