American Journal of Epidemiology
Copyright O 1998 by The Johns Hopkins University School of Hygiene and PubDc Health
All rights reserved
Vol. 147, No. 8
Printed in USA.
Confidence Limits Made Easy: Interval Estimation Using a Substitution
Leslie E. Daly
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 assump-
tions 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.
Am J Epidemiol 1998; 147:783-90.
binomial distribution; confidence intervals; epidemiologic methods; Poisson distribution; statistics
Confidence intervals are now required by most med-
ical journals for the presentation of statistical results.
A confidence interval is a range of likely values for an
unknown population parameter at a given confidence
level. The endpoints of this range are called the con-
A number of different methods can be used to
estimate confidence limits. Exact limits can be ob-
tained using published tables (1-3) or appropriate soft-
ware (4, 5) for a single proportion, percentage, or risk
(binomial limits), as well as for a count (Poisson
limits). However, the most commonly used method of
calculating confidence limits involves the normal ap-
proximation, in which a multiple of the standard error
(SE) is added to and subtracted from the sample value
for the measure. For 95 percent confidence limits, the
general expression is
statistic ± 1.96 SE(statistic),
Received for publication October 13, 1994, and In final form
October 3, 1997.
Abbreviations: LAR, Levin's attributable risk; NNT, number
needed to be treated; RR, relative risk; SE, standard error.
From the Department of Public Health Medicine and Epidemiol-
ogy, University College Dublin, Earlsfort Terrace, Dublin 2, Ireland.
(Reprint requests to Prof. Leslie E. Daly at this address).
where SE(statistic) is the standard error of the relevant
quantity and 1.96 is the appropriate percentile of the
normal distribution. Confidence limit estimation is
relatively straightforward using this approach, and
methods for use with single means, proportions, or
counts, for differences between these, and for relative
risk-type measures are well known (1, 2, 6). Several
commonly used standard error formulae are given in
Although the normal approximation (expression 1)
can often be used directly for confidence interval
estimation, sometimes it must be used on a transfor-
mation of the measure of interest For instance, 95
percent confidence limits for the relative risk (RR) can
be based on the limits for loge RR:
where SE(loge RR) is the standard error of the natural
logarithm of RR (expression A2 in the Appendix).
Transforming back to the original scale, the exponen-
tial of these limits gives the limits for the relative risk
itself. A similar approach can also be used for the odds
ratio. It is important to realize that it is the actual limits
of the transformed quantity that must be back-
transformed. When the limits are transformed in this
by guest on July 8, 2011
Another major advantage of the method is that no
distributional assumptions are necessary for the sam-
pling distribution of the measure for which the confi-
dence limits are required. If exact confidence limits
are known for the underlying parameter (as in the
binomial or Poisson cases), the limits for a function of
the parameter will also be exact. Thus, there is a
distinct advantage to the substitution method even
when an alternative exists using known standard error
For measures that are a function of a single param-
eter, a Taylor series expansion is often used for inter-
val estimation (6, pp. 91-2). The standard error of the
function of a parameter f(x) is given by
SE[/«] ~ ±
where the derivative of/is evaluated at the mean value
Not only is this standard error an approximation but
the additional assumption of normality is required to
derive the confidence limits for the function using
expression 1. The standard error formula for the gene
frequency (expression 11) can be derived in this way.
The requirements for valid use of the Taylor series
expansion method are also more stringent than those
for the substitution method, in that the functional
relation must be strictly monotonically increasing (or
decreasing) and must have a nonzero first derivative.
(A strictly monotonic function requires that the func-
tion always changes as the parameter changes.) Thus,
the substitution method can and should always be used
instead of a Taylor series expansion.
For the end user, the general approach of the sub-
stitution method and its lack of reliance on complex
formulae make it clearer what the confidence limits
are measuring. It is particularly suitable for "hand"
calculations when specialized computer software is
not available. The substitution method should be
adopted as a practical alternative to complex formulae
when performing interval estimation. Even in simple
cases, the inherent accuracy of the method suggests
that it should replace some standard approaches.
The author thanks Dr. Douglas G. Altaian of the Imperial
Cancer Research Fund (Oxford, England) and Prof.
Marcello Pagano of the Department of Biostatistics, Har-
vard School of Public Health (Brookline, Massachusetts),
for valuable suggestions and encouragement
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Some common standard error formulae which were
employed in the derivation of results presented in this
paper are shown below.
For a binomial proportion (p) in a sample size of n,
the standard error is calculated as
For the natural logarithm of the relative risk (log^
RR) (a, b, c, and d are table entries—see table 1 in the
text), the standard error is calculated as
a + b c c + d'
For the difference between two risks (Rl — R^) in
sample sizes of nl and n?, the standard error is calcu-
Am J Epidemiol Vol. 147, No. 8, 1998
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