# Confidence Interval for Binomial Proportions

The binomial test (e.g. under the NONPARAMETRIC STATISTICS menu in SPSS) may be used to test if a proportion equals a particular constant (usually 0.5) however it does not provide a confidence interval for the magnitude of this difference. (See also the Graduate Statistics Talk on Categorical Data Analysis).

Newcombe (1998) suggests using alternatives to the usual Wald procedure to obtain confidence intervals for a proportion, particularly for proportions less than 0.2 or greater than 0.8. He suggests the exact binomial method and Wilsonâ€™s (1927) method provide slightly better coverage. Confidence intervals for both Wald, Wilson and Agresti-Coull methods may be computed by using this spreadsheet. The latter two are recommended for a single proportion by Brown, Cai and DasGupta (2001).

The reason for this poor coverage of confidence interval using the Wald statistic for binomial proportions relates to the shape of the binomial proportions. They can't go above 0 or 1 so if you plot the range from min to max it is an S shaped (sinusoidal) function that curves sharply towards and asymptotes at 0 or 1. However the portion in the middle is roughly linear and can be approximated by any linear regression model (e.g. a "logistic function") as assumed by the Wald statistic. The range 0.2 to 0.8 is a rough guide. The variance also is a function of the mean and is more stable in this region.

A limited range of CIs (Pearson-Clopper, Jeffreys and Wald) can be obtained in SPSS using the binomial test as below:

incomp |
count |

1 |
23 |

2 |
12 |

WEIGHT BY COUNT. NPTESTS /ONESAMPLE TEST (incomp) BINOMIAL(TESTVALUE=0.5 JEFFREYS SUCCESSCATEGORICAL=FIRST SUCCESSCONTINUOUS=CUTPOINT(MIDPOINT)) /MISSING SCOPE=ANALYSIS USERMISSING=EXCLUDE /CRITERIA ALPHA=0.05 CILEVEL=95. WEIGHT OFF.

You can also obtain confidence intervals for the difference in two proportions here. Formulae used for the difference in proportions in 2x2 tables are described here.

Confidence interval for differences in two independent binomial proportions

Confidence interval for differences in two paired binomial proportions

Confidence interval for a proportion with a correction for the size of the target population

References

Brown LD, Cai TT and DasGupta A (2001) Interval estimation for a binomial proportion. *Statistical Science* **16** 101-133.

Newcombe RG. (1998) Two sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine 1998;17:857-872

Newcombe RG (2012) Confidence intervals for proportions and related measures of effect size. Chapman and Hall:London. (This book also contains details of web links to easy-to-use EXCEL programs located on the CRC press website which will work out the confidence intervals mentioned in the text).

Wilson EB (1927) Probable inference, the law of succession, and statistical inference. J Am Stat Assoc 22, 209-212.

Last updated on 12 February, 2008