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PROPCI
A Program that Calculates Confidence Intervals for a Proportion
Version 1.3, 8/June/1990
(c) 1988, 1989, 1990


by
Kevin M. Sullivan
Division of Nutrition
Center for Chronic Disease Prevention and Health Promotion
Centers for Disease Control
1600 Clifton Road NE, MS A08
Atlanta, GA 30333

This program was developed to calculate confidence intervals for a
proportion by use of the following methods: the normal approximation to
the binomial, the normal approximation with a correction factor, the
method by Wilson, the quadratic method, the exact binomial method, and
the mid-p (Miettinen) method. Either 90, 95, or 99 percent confidence
intervals can be calculated. The user inputs the number of individuals
who have the event of interest (x) and the sample size (n). The point
estimate is x/n. The standard error (SE) of the normal approximation to
the binomial(1,2) is:
____
SE = \/pq/n

where
p=x/n
q=1-p
n=denominator

The 95% confidence interval for the normal approximation is:

p + Z * SE

where
Z = Z value, e.g., for 95% two-sided CI this is 1.96
SE = the standard error calculated above

The lower and upper confidence limits for a normal approximation with
a correction factor(2) are:

p - Z * SE - 1/(2n)

and

p + Z * SE + 1/(2n)

The normal approximation to the binomial can produce estimates
outside of the 0-100 percent limits. PROPCI will provide the results of
the normal approximation calculations even if the estimates are outside
the limits, although most authors truncate the estimates to the limits.
One suggested criterion for determining when the normal approximation is
inappropriate is when npq<5.(2) A more correct approximation for the
confidence interval for a proportion is calculated using the quadratic
method.(1) This quadratic formula includes a correction factor. The
formula for the lower bound of the quadratic method is:
____________________________
2 | 2
(2np + Z - 1) - Z \| Z - (2 + 1/n) + 4p(nq + 1)
_________________________________________________
2
2(n + Z )

and the upper bound is:
____________________________
2 | 2
(2np + Z + 1) + Z \| Z + (2 - 1/n) + 4p(nq - 1)
_________________________________________________
2
2(n + Z )

Another approximate method is by Wilson.(3) It appears that the method
by Wilson is a quadratic equation without the correction factor. The
formula for the lower bound is:
_ _____________ _
| 2 | 2 |
n | x Z | x(n-x) Z |
______ | ___ + __ | ______ - ___ |
2 | | 3 2 |
n+Z |_ n 2n \| n 4n _|

and the upper bound is:
_ _____________ _
| 2 | 2 |
n | x Z | x(n-x) Z |
______ | ___ + __ | ______ + ___ |
2 | | 3 2 |
n+Z |_ n 2n \| n 4n _|

The exact binomial confidence interval is calculated by using
formulas as described by Rosner2 and Rothman.(3) The formulas for the
lower and upper limits for a two-sided 95% confidence interval (i.e.,
.025 in each tail) are:
n
___ n! k n-k
.025 = \ -------- p (1-p )
/__ k!(n-k)! 1 1
k=x

x
___ n! k n-k
.025 = \ -------- p (1-p )
/__ k!(n-k)! 2 2
k=0

Exact mid-p (Miettinen) confidence intervals are calculated by using
formulas as described by Rothman.(3) The formula for the lower and upper
limits for a two-sided 95% confidence interval (i.e., .025 in each tail)
are:
n
1 n! x n-x ___ n! k n-k
.025 = - * -------- p (1-p ) + \ -------- p (1-p )
2 x!(n-x)! 1 1 /__ k!(n-k)! 1 1
k=x+1

x-1
1 n! x n-x ___ n! k n-k
.025 = - * -------- p (1-p ) + \ -------- p (1-p )
2 x!(n-x)! 2 2 /__ k!(n-k)! 2 2
k=0

For each proportion, the normal approximation (with and without
correction factor), Wilson, and quadratic confidence intervals are
automatically provided. If npq<5, a message is provided near the bottom
of the screen warning users that the normal approximation may not be
appropriate. Next, the user is then prompted as to whether they would
like to have exact confidence intervals calculated (the default is
"no"). Both the exact binomial and mid-p formulas require iterative
solutions to determine the value of lower and upper confidence limits
and therefore are not automatically performed. A fast method to
determine the exact confidence intervals using the F-distribution is
used when the denominator is less than 300.(3) Finally, the user is
asked whether they would like to perform another calculation or return
to DOS.
Which confidence interval method should you use? My opinion is that
among the approximate methods (i.e., normal approximation, Wilson, and
quadratic), the quadratic provides the best estimate of the exact
binomial confidence interval. If the data are sparse, then use one of
the exact methods (exact binomial or mid-p).

EXAMPLE

In this example from Rothman,3 x=10 and n=11 with 90% confidence
intervals.

+--------------------------------------------------------------------+
| 01/09/90 ** PROPCI 1.2 ** |
+--------------------------------------------------------------------+

Numerator: 10 / Denominator: 11

Enter two-sided confidence level (90, 95, or 99%): 90

The point estimate is: 90.909%

Confidence Interval Method Std Error 90% CI
Normal Approx. to the Binomial 8.668 76.650, 105.168
Normal Approx. with Correction Factor 8.668 72.105, 109.713
Wilson Method 67.719, 97.945
Quadratic Method 62.330, 99.372
Exact Binomial 63.564, 99.535
Miettinen Limits (Mid-p) 67.759, 99.090

**The normal approximation may not be valid for this example**
Would you like to do another? (Y/N) Y



DIFFERENCES BETWEEN VERSION 1.3 AND PREVIOUS VERSIONS

Version 1.2 implements the F-distribution method to arrive at the
exact confidence limits. This dramatically reduces the computation time
involved compared to other iterative procedures and produces the exact
same results.

Because of some problems with the F-distribution method, in version
1.3 the F-distribution method is used only when the denominator is less
than 300; the longer iterative method is used with larger numbers.

DISTRIBUTION CONDITIONS
NON-WARRANTY. PROPCI is provided "as is" and without any warranty
expressed or implied. The user assumes all risks of the use of PROPCI.
PROPCI may not run on your particular hardware/software configuration.
We bear no responsibility for any mishap or economic loss resulting
therefrom the use of this software.
COPYRIGHT CONDITIONS. You may make and distribute copies of PROPCI
provided that there is no material gain involved.
USE AT YOUR OWN RISK. All risk of loss of any kind due to the use of
PROPCI is with you, the user. You are responsible for all mishaps, even
if the program proves to be defective. This program makes certain
assumptions about the data. These assumptions affect the validity of
conclusions made based on the output from this program.

Please acknowledge PROPCI in any manuscript that uses its
calculations.

REFERENCES

1. Fleiss JL. Statistical Methods for Rates and Proportions, 2nd Ed.
John Wiley & Sons, New York, 1981.
2. Rosner B. Fundamentals of Biostatistics. Duxbury Press, Boston,
1982.
3. 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.