Power and detectable risk of seven 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 = magnitude of D-E/square root of E; 5) the Z statistic corrected for continuity, Z = (magnitude of D-E)-0.5)/square root of E; 6) a statistic based on a square root transformation, Z = 2(square root of D-square root of E); and 7) a statistic created by Byar, which, when D is greater than E, is Z = square root of 9D[1-1/(9D)-3 square root of 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.