Confidence Interval

confint(test::HypothesisTest, alpha=0.05; tail=:both)

Compute a confidence interval C with coverage 1-alpha.

If tail is :both (default), then a two-sided confidence interval is returned. If tail is :left or :right, then a one-sided confidence interval is returned


Most of the implemented confidence intervals are strongly consistent, that is, the confidence interval with coverage 1-alpha does not contain the test statistic under \(h_0\) if and only if the corresponding test rejects the null hypothesis \(h_0: \theta=\theta_0\):

\[\begin{split}C (x, 1 − \alpha) = \{\theta : p_\theta (x) > \alpha\},\end{split}\]

where \(p_\theta\) is the p-value of the corresponding test.

Confidence Interval for Binomial Proportions

confint(test::BinomialTest, alpha=0.05; tail=:both, method=:clopper_pearson)

Compute a confidence interval with coverage 1-alpha for a binomial proportion using one of the following methods. Possible values for method are:

  • Clopper-Pearson interval :clopper_pearson (default): This interval is based on the binomial distribution. The empirical coverage is never less than the nominal coverage of 1-alpha; it is usually too conservative.
  • Wald interval :wald (normal approximation interval): This interval relies on the standard approximation of the actual binomial distribution by a normal distribution. Coverage can be erratically poor for success probabilities close to zero or one.
  • Wilson score interval :wilson: This interval relies on a normal approximation. In contrast to :wald the standard deviation is not approximated by an empirical estimate resulting in good empirical coverages even for small numbers of draws and extreme success probabilities.
  • Jeffreys interval :jeffrey: Bayesian confidence interval obtained by using a non-informative Jeffreys prior. The interval is very similar to the Wilson interval.
  • Agresti Coull interval :agresti_coull: Simplified version of the Wilson interval; they are centered around the same value. The Agresti Coull interval has higher or equal coverage.


  • Brown, L.D., Cai, T.T., and DasGupta, A. Interval estimation for a binomial proportion. Statistical Science, 16(2):101–117, 2001.

Confidence Interval for Multinomial Proportions

confint(test::PowerDivergenceTest, alpha=0.05; tail=:both, method=:sison_glaz)

Compute a confidence interval with coverage 1-alpha for multinomial proportions using one of the following methods. Possible values for method are:

  • Sison, Glaz intervals :sison_glaz (default):
  • Bootstrap intervals :bootstrap :
  • Quesenberry, Hurst intervals :quesenberry_hurst :
  • Gold intervals :gold (Asymptotic simultaneous intervals):


  • Agresti, Alan. Categorical Data Analysis, 3rd Edition. Wiley, 2013.
  • Sison, C.P and Glaz, J. Simultaneous confidence intervals and sample size determination for multinomial proportions. Journal of the American Statistical Association, 90:366-369, 1995.
  • Quesensberry, C.P. and Hurst, D.C. Large Sample Simultaneous Confidence Intervals for Multinational Proportions. Technometrics, 6:191-195, 1964.
  • Gold, R. Z. Tests Auxiliary to \({\chi^{2}}\) Tests in a Markov Chain. Annals of Mathematical Statistics, 30:56-74, 1963.

Confidence Interval for Fisher exact test

confint(x::FisherExactTest, alpha::Float64=0.05; tail=:both, method=:central)

Compute a confidence interval with coverage 1-alpha by inverting the :central p-value.


  • Gibbons, J.D, Pratt, J.W. P-values: Interpretation and Methodology American Statistican, 29(1):20-25, 1975.
  • Fay, M.P. Supplementary material to Confidence intervals that match Fisher’s exact or Blaker’s exact tests. Biostatistics, 0(0):1-13, 2009.