POWER AND DETECTABLE RISK OF 7 TESTS FOR STANDARDIZED MORTALITY RATIOS

Power and minimum detectable risk are calculated for seven one-sided tests of standardized mortality ratios with Poisson-distributed events. Each test contrasts the number of observed deaths (D) with the number expected (E). Three tests use exact Poisson probabilities: 1) the exact test, which computes a p value as the probability of equaling or exceeding the number of observed events; 2) the optimal randomized exact test which, although not used in practice, serves as a standard for the other statistics; and 3) the exact "mid-p" procedure, which counts only one-half the probability of the observed event. The remaining four tests use normal approximations to the Poisson ("Z statistics"): 4) Z = (D - E)/square-root E; 5) the Z statistic corrected for continuity, Z = ((D - E) - 0.5)/square-root E; 6) a statistic based on a square root transformation, Z = 2(square-root D - square-root E); and 7) a statistic created by Byar, which, when D is greater than E, is Z = square-root 9D {1 - 1/(9D) - cube-root D/E}. Power differences among these procedures with one-sided-alpha of 0.05, 0.025, and 0.01 are small as long as four or more events are expected. If fewer than four events are expected, the uncorrected Z has unacceptably high type I error. Simple approximations to the power and detectable risk of these tests are evaluated and prove satisfactory. Differences in minimum detectable risk, actual and approximated, are slight for E of 2.0 or more.