American Journal of Epidemiology Vol. 133, No. 11: 1191-1197
Copyright © 1991 by The Johns Hopkins University School of Hygiene and Public Health
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Power and Detectable Risk of Seven Tests for Standardized Mortality Ratios
1Department of Internal Medicine, University of Califor nia Davis, CA
2Department of Obstetrics and Gynecology, University of California Davis, CA
3Department of Biostatistics, University of Washington Seattle, WA
Reprint requests to Dr. Steven Sarnuels, Division of Occupational and Environmental Medicine, ITEH, University of California. Davis, CA 95616.
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-" 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)
) the Z statistic corrected for continuity, 
a statistic based on a square root transformation, 
; and 7) a statistic created by Byar, which, when D is greater than E, is 
. Power differences among these procedures with one-sided 
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.
probability; risk
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