Confidence limits made easy: interval estimation using a substitution method.

The use of confidence intervals has become standard in the presentation of statistical results in medical journals. Calculation of confidence limits can be straightforward using the normal approximation with an estimate of the standard error, and in particular cases exact solutions can be obtained from published tables. However, for a number of commonly used measures in epidemiology and clinical research, formulae either are not available or are so complex that calculation is tedious. The author describes how an approach to confidence interval estimation which has been used in certain specific instances can be generalized to obtain a simple and easily understood method that has wide applicability. The technique is applicable as long as the measure for which a confidence interval is required can be expressed as a monotonic function of a single parameter for which the confidence limits are available. These known confidence limits are substituted into the expression for the measure--giving the required interval. This approach makes fewer distributional assumptions than the use of the normal approximation and can be more accurate. The author illustrates his technique by calculating confidence intervals for Levin's attributable risk, some measures in population genetics, and the "number needed to be treated" in a clinical trial. Hitherto the calculation of confidence intervals for these measures was quite problematic. The substitution method can provide a practical alternative to the use of complex formulae when performing interval estimation, and even in simpler situations it has major advantages.