Binomial confidence intervals for proportions (Wilson score)
ciw [varlist] [weight] [if exp] [in range] [, level(#) total ]
ciwi #obs #succ [, level(#) ]
by ... : may be used with ciw (but not with ciwi); see help by.
fweights are allowed with ciw; see help weights.
Description
ciw computes standard errors and binomial confidence intervals for each variable in varlist, which should be 0/1 binomial variables. ciwi is the immediate form of ciw, for which specify the number of observations and the number of successes. See help immed for more on immediate commands. With both commands confidence intervals are calculated as Wilson score intervals.
Remarks
Suppose we observe n events and record k successes. Here as usual "success" is conventional terminology for whatever is coded 1. For a 95% confidence interval, for example, we then solve the quadratic equation in p
(k / n - p) / sqrt[p (1 - p) / n] = +/-z
where z is invnorm(0.975). See Wilson (1927). This contribution is of historical interest as an anticipation of Neyman's subsequent formulation of confidence intervals. It is equivalent to adding z^2 observations to a sample, half of them successes and half failures. (At a 95% level, z^2 / 2 = 1.921 to 3 d.p. and so is nearly 2.) Among other properties, note that this interval is typically less conservative than the exact interval, so that coverage probabilities are on average close to the nominal confidence level.
See Brown et al. (2001) for a much fuller discussion and an entry to the literature. Brown et al. (2002) provide supporting technical background to that paper. Among many references, Agresti (2002, pp.14-21), Agresti and Coull (1998), Newcombe (1998, 2001) and Vollset (1993) provide clear and helpful context. Williams (2001, Ch.6) provides a lively alternative treatment of confidence intervals for one-parameter models. For more on the American mathematician, physicist and statistician Edwin Bidwell Wilson (1879-1964), see Stigler (1997). He should not be confused with the American biologist Edmund Beecher Wilson (1856-1939), who worked on cytology, embryology and heredity, nor with the American physical chemist Edgar Bright Wilson (1908-1992); the writings of the last include statistical topics (Wilson 1952).
Note that this is close to, but not identical to, a procedure implemented by Gleason (1999).
Options
level(#) specifies the confidence level, in percent, for confidence intervals; see help level.
total is for use with the by ... : prefix; it requests that, in addition to output for each by-group, output be added for all groups combined.
Examples
. ciw foreign
. ciwi 10 1 (10 binomial events, 1 observed success)
References
Agresti, A. 2002. Categorical data analysis. Hoboken, NJ: John Wiley.
Agresti, A. and Coull, B.A. 1998. Approximate is better than "exact" for interval estimation of binomial proportions. American Statistician 52: 119-126.
Brown, L.D., Cai, T.T., DasGupta, A. 2001. Interval estimation for a binomial proportion. Statistical Science 16: 101-133.
Brown, L.D., Cai, T.T., DasGupta, A. 2002. Confidence intervals for a binomial proportion and asymptotic expansions. Annals of Statistics 30: 160-201.
Gleason, J.R. 1999. Improved confidence intervals for binomial proportions. Stata Technical Bulletin 52: 16-18 (STB Reprints 9: 208-211).
Newcombe, R.G. 1998. Two-sided confidence intervals for the single proportion: comparison of seven methods. Statistics in Medicine 17: 857-872.
Newcombe, R.G. 2001. Logit confidence intervals and the inverse sinh transformation. American Statistician 55: 200-202.
Stigler, S.M. 1997. Wilson, Edwin Bidwell. In Johnson, N.L. and Kotz, S. (eds) Leading personalities in statistical sciences: from the seventeenth century to the present. New York: John Wiley, 344-346.
Vollset, S.E. 1993. Confidence intervals for a binomial proportion. Statistics in Medicine 12: 809-824.
Williams, D. 2001. Weighing the odds: a course in probability and statistics. Cambridge: Cambridge University Press.
Wilson, Edgar Bright. 1952. An introduction to scientific research. New York: McGraw-Hill.
Wilson, Edwin Bidwell. 1927. Probable inference, the law of succession, and statistical inference. Journal, American Statistical Association 22: 209-212.
Acknowledgements
John R. Gleason increased my interest in this problem. Richard Williams found a bug in ciwi.
Also see
Manual: [R] ci On-line: help for ci, bitest, immed