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First, where possible, standard errors for Y were derived from confidence intervals reported by original investigators. In this way, variance reduction techniques such as matching of cases and controls are accurately reflected in the estimated standard error. In particular, if the reported odds ratio or RR is R, and the upper limit of its 95 percent confidence interval is U, the standard error was estimated as  s = ln(U/R)/1.96. For studies in which no confidence interval was calculated by the original authors, the committee calculated an RR as appropriate and an associated confidence interval using standard methods for 2 x 2 tables (Fleiss, 1981, pp. 61-67) and applied the same procedure described above to estimate the standard error.

Second, 50 and 80 percent power levels for the RR were calculated as follows. Under the null hypotheses of no association, the expected value of Y, E(Y), is 0.0 and the critical point for a two-sided test with   a = 0.05 is 1.96s. If E(Y) were equal to 1.96s, there would be a 50 percent chance that the test would detect an elevated risk; thus, the RR for which there is 50 percent power was calculated as e1.96s. To achieve 80 percent power, E(Y) must be 0.84s above the critical point, or 2.80s. Thus, the RR for which there is 80 percent power was calculated as e2.80s.

To illustrate this approach, Table D-1 shows the results of these calculations for the SONIC study relating DPT use to afebrile seizures (Gale et al., 1990; see also Table 4-5). The estimated RRs range from 0.5 to 0.8, and the upper confidence limits range from 1.1 to 1.5. On the basis of these results, the power calculations show that the SONIC study had a 50 percent chance of detecting an RR for afebrile seizures within 7 days of 3.0 and an 80 percent chance of detecting an RR within 7 days of 4.8. Thus, a relatively large increase in the risk of afebrile seizures could have gone unde-

TABLE D-1 Power Calculations for the SONIC Study Relating DPT Use to Afebrile Seizures




Time Period


Upper CIa




Within 7 days






Within 14 days






Within 28 days






aCI, Confidence interval.
bs , Standard error.
c"Power" denotes the probability that a statistical test based on a sample of the same size as the one in the study cited would find a statistically significant increased risk (with alpha = 0.05), given that the true RR in the population being studied is the number stated in the table. The numbers tabulated are the RRs such that the powers are 50 and 80 percent, respectively.

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