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help for cij, ciji
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Binomial confidence intervals for proportions (Jeffreys prior)

cij  [varlist] [weight] [if exp] [in range] [, level(#) total ]

ciji #obs      #succ     [, level(#) ]

by ... : may be used with cij (but not with ciji); see help by.

fweights are allowed with cij; see help weights.

Description

cij computes standard errors and binomial confidence intervals for each
variable in varlist, which should be 0/1 binomial variables.  ciji is the
immediate form of cij, 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 based on the Jeffreys
uninformative prior of a beta distribution with parameters 0.5 and 0.5.

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 take the 0.025 and 0.975 quantiles of the beta
distribution with parameters k + 0.5 and n - k + 0.5.  This Bayesian procedure
has a frequentist interpretation as a continuity-corrected version of the
so-called exact (Clopper-Pearson) confidence interval, produced by ci,
binomial, which takes (in the same example) the 0.025 quantile of beta(k, n - k
+ 1) and the 0.975 quantile of beta(k + 1, n - k). The lower limit if all
values are 0 is taken to be 0 and the upper limit if all values are 1 is taken
to be 1.  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. From a Bayesian point of view,
however, the whole of the posterior distribution is much more fundamental than
any interval derived from it.

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.  The original work on
uninformative priors was by Harold Jeffreys (1946; 1948, Ch.3.9; 1961,
Ch.3.10).  The actuary Wilfred Perks (1947) independently produced very similar
ideas.  Later discussions include Good (1965, esp. pp.18-19), Rubin and
Schenker (1987), Lee (1989, esp. p.93; 1997, esp. pp.88-89), Gelman et al.
(1995, esp. pp.55-56), or Carlin and Louis (1996, esp. pp.50-54; 2000, esp.
pp.42-46).  For more on Jeffreys (1891-1989), see Cook (1990), Lindley (2001)
or Lindley et al. (1991).

The method of calculating beta quantiles used here is based on the fact that if
Y is distributed as beta(a,b) and X is distributed as F(2a,2b), then Y = aX /
(b + aX). See (e.g.) Cramér (1946, pp.241-4) for background or Lee (1989,
p.251; 1997, p.291). In Stata 8, it can be done directly with invibeta().

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

. cij foreign

. ciji 10 1                           (10 binomial events, 1 observed success)

Author

Nicholas J. Cox, University of Durham, U.K.
n.j.cox@durham.ac.uk

Acknowledgements

Alan Feiveson suggested the Cramér reference.  John R. Gleason
increased my interest in this problem.

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.

Carlin, B.P. and Louis, T.A. 1996/2000.  Bayes and empirical Bayes methods for
data analysis.  Boca Raton, FL: Chapman and Hall/CRC (1996: London: Chapman and
Hall.)

Cook, A.H. 1990. Sir Harold Jeffreys.  Biographical Memoirs of Fellows of the
Royal Society 36: 303-333.

Cramér, H. 1946. Mathematical methods of statistics.  Princeton, NJ: Princeton
University Press.

Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B. 1995.  Bayesian data
analysis. London: Chapman and Hall.

Good, I.J. 1965. The estimation of probabilities:  an essay on modern Bayesian
methods. Cambridge, MA: MIT Press.

Jeffreys, H. 1939/1948/1961. Theory of probability.  Oxford: Oxford University
Press.

Jeffreys, H. 1946. An invariant form for the prior probability in estimation
problems.  Proceedings of the Royal Society A 186: 453-461.  Reprinted in
Jeffreys, H. and Jeffreys, B.S. (eds) 1977.  Collected papers of Sir Harold
Jeffreys on geophysics and other sciences.  Volume 6: Mathematics, probability
and miscellaneous other sciences.  London: Gordon and Breach, 403-411.

Lee, P.M. 1989/1997.  Bayesian statistics: an introduction.  London: Edward
Arnold.

Lindley, D.V. 2001. Harold Jeffreys. In Heyde, C.C. and Seneta, E. (eds)
Statisticians of the centuries.  New York: Springer, 402-405.

Lindley, D.V., Bolt, B.A., Huzurbazar, V.S., Jeffreys, B.S., Knopoff, L. 1991.
[articles on Harold Jeffreys] Chance 4(2): 10-26.

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.

Perks, W. 1947. Some observations on inverse probability including a new
indifference rule.  Journal, Institute of Actuaries 73: 285-334.

Rubin, D.M. and Schenker, N. 1987. Logit-based interval estimation for binomial
data using the Jeffreys prior.  Sociological Methodology 17: 131-144.

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.

Also see

Manual:  [R] ci
On-line:  help for ci, bitest, immed

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