January
5, 2006
Confidence Intervals for a
Rate
Kevin M. Sullivan, PhD, MPH, MHA cdckms@sph.emory.edu
For a single rate (also known as “incidence
rate”), the numerator is the number of cases of the “disease,” and the
denominator is the sum of person-years (or days, weeks, months) of exposure for
all individuals prior to onset of the disease.
The person-time variable represents the sum of the number of time units
in which individuals were under study and disease-free. It should include units
for those who never developed disease and those who were lost to follow-up
after a defined period.
This module calculates various confidence intervals for a rate. First, the user is prompted to enter a numerator and denominator value:
The output from the example above is as follows:
The observed rate is 2 per 10 person-time units. Five different methods are used to calculate the confidence interval around this point estimate: Mid-P exact test, Fisher’s exact test, normal approximation, Byar approximation, and the Rothman/Greenland method. Of the five methods, the Mid-P exact test is generally the preferred method.
For confidence limit estimates < 0.0, the value 0.0 is shown. All confidence intervals calculated are two-sided and depend on the current setting of user’s choice (90%, 95%, 99%, 99.9% or 99.99%). Formulas for the methods are provided in the following section.
Formulae
The notation for the formulae is:
a = the observed numerator
PT = is the observed denominator in person-time units
rate = a/PT
= the two-sided Z
value (eg. Z=1.96 for a 95% confidence interval).
Exact Tests (Mid-P
and Fisher)
The limits for ‘a’ with 100(1-α) percent confidence are
the iterative solutions and
.
Computing iterative solutions and
is below………
Mid-P exact test (see Rothman and Boice):
Lower bound:
Upper bound:
Fisher’s exact test (see Rothman and Boice):
Lower bound:
Upper bound:
Therefore, the exact
lower and upper limits for single person-time rate equal to “a/PT”
would be
, respectively.
Normal Approximation:
Byar Method (see Rothman and Boice):
Lower bound:
Upper bound:
Rothman Greenland Method:
Lower bound:
Upper bound:
References
Rosner B. Fundamentals of Biostatistics, 5th Edition. Duxbury Press, 2000.
Rothman KJ, Boice JD Jr: Epidemiologic analysis with a programmable calculator. NIH Pub No. 79-1649. Bethesda , MD : National Institutes of Health, 1979;31-32.
Rothman KJ, Greenland S. Modern Epidemiology, 2nd Edition. Lippincott-Raven Publishers, Philadelphia , 1998.
Update
The formulae for Mid-P and Fisher’s exact tests were added to the existing single person-time module on December 14, 2005.