Skip to contents

Selected confidence intervals for the single binomial or Poisson rate.

Source: R/rateci.R
rateci.Rd

Confidence intervals for the single binomial or Poisson rate. This convenience wrapper function produces a selection of alternative methods. The first three are recommended for achieving 1-sided and 2-sided coverage probability close to the nominal levels (see Laud 2017 and Laud 2018):

  • SCAS (skewness-corrected asymptotic score)

  • Jeffreys

  • mid-p (two versions, using exact calculation or approximation via Beta/Gamma distribution, see p.115 of Brown et al.)) The following more approximate methods are included for users wishing to use a more established or commonly used method:

  • Wilson score

  • Agresti-Coull

  • Wald (strongly advise this is not used for any purpose but included for reference)

All methods can be made more conservative with a 'continuity adjustment', which may either be specified as TRUE, or an intermediate 'compromise' value between 0 and 0.5 may be selected. When cc is TRUE or 0.5, the mid-p method becomes the 'exact' Clopper-Pearson interval (or Garwood for Poisson rates), and the output also includes the slightly less conservative Blaker interval. Hence the full list of conservative methods produced (when cc is TRUE) is:

  • SCAS_cc

  • Jeffreys_cc

  • Clopper-Pearson 'exact' (two identical versions, using exact calculation or approximation via Beta/Gamma distribution))

  • Wilson_cc

  • Wald_cc (strongly advise this is not used for any purpose but included for reference)

  • Blaker 'exact'

Note that Brown et al's Beta formulation perfectly matches the exact interval when cc is TRUE (i.e. for Clopper-Pearson) but not when cc is FALSE (for mid-p) All methods except Agresti-Coull have equivalent formulae for the Poisson distribution: Garwood for Clopper-Pearson, Rao score for Wilson score. Jeffreys has a Poisson equivalent using the Gamma distribution. e.g. See Brown et al. 2003, Swift 2009 and Laud 2017. The formulation for the approximate mid-p interval using Gamma distribution for a Poisson rate has been deduced by the package author from the corresponding formulae from Brown et al., and has not (to the best of my knowledge) been published.

This function is vectorised in x, n.

Usage

rateci(
  x,
  n,
  distrib = "bin",
  level = 0.95,
  std_est = TRUE,
  cc = FALSE,
  precis = 8
)

Arguments

x

Numeric vector of number of events.

n

Numeric vector of sample size (for binomial rate) or exposure times (for Poisson rate).

distrib

Character string indicating distribution assumed for the input data: "bin" = binomial (default), "poi" = Poisson.

level

Number specifying confidence level (between 0 and 1, default 0.95).

std_est

logical, specifying if the crude point estimate for the proportion value x/n should be returned (TRUE, default) or the method-specific alternative point estimate consistent with a 0% confidence interval (FALSE).

cc

Number or logical (default FALSE) specifying continuity adjustment.

precis

Number (default 8) specifying precision (i.e. number of decimal places) to be used in root-finding subroutine for the exact confidence interval. (Note all other methods use closed-form calculations so are not affected.)

Value

A list containing, for each method, a matrix containing lower and upper confidence limits and point estimate of p for each value of x and n. Methods shown depend on the cc parameter, which specifies whether the continuity adjustment is applied to the SCAS and Jeffreys methods. The corresponding 'exact' method is Clopper-Pearson/Garwood if cc = TRUE and mid-p if cc = FALSE. An additional output object 'estimates' is provided for a side-by-side comparison of all methods. These are grouped depending on the cc argument (if cc = TRUE then the continuity-adjusted and exact strictly conservative methods are included) The last list item contains details of the function call.

References

Laud PJ. Equal-tailed confidence intervals for comparison of rates. Pharmaceutical Statistics 2017; 16:334-348. (Appendix A.4)

Brown LD, Cai TT and DasGupta A. Interval estimation for a binomial proportion. Statistical Science 2001; 16(2):101-133.

Garwood F. Fiducial limits for the Poisson distribution. Biometrika 1936; 28(3-4):437, doi:10.1093/biomet/28.3-4.437.

Author

Pete Laud, p.j.laud@sheffield.ac.uk