January 5, 2006

Confidence Intervals for a Rate

Kevin M. Sullivan, PhD, MPH, MHA cdckms@sph.emory.edu

Minn M. Soe, MD, MCTM, MPH msoe@sph.emory.edu

###### The Person Time module of Open Epi is used to analyze data where the numerator is a count of the events of interest and the denominator is the total person-time over which observations occurred. This method of analysis is frequently used in cohort studies and clinical trials. The idea is that a disease-free population is followed from a baseline. Person-time is the amount of time an individual accumulates until: 1) the study ends; 2) they develop the outcome of interest; or 3) they leave the study for some other reason. Person time is frequently expressed in person-years, although person-hours, days, or months will work just as well.

###### Single Person-Time Rate

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.