ciBinomHalfWidth: Half-Width of Confidence Interval for Binomial Proportion or Difference Between Two Proportions

Description

Compute the half-width of a confidence interval for a binomial proportion or the difference between two proportions, given the sample size(s), estimated proportion(s), and confidence level.

Usage

ciBinomHalfWidth(n.or.n1, p.hat.or.p1.hat = 0.5, 
    n2 = n.or.n1, p2.hat = 0.4, conf.level = 0.95, 
    sample.type = "one.sample", ci.method = "score", 
    correct = TRUE, warn = TRUE)

Arguments

n.or.n1

numeric vector of sample sizes. When sample.type="one.sample", n.or.n1 denotes $n$, the number of observations in the single sample. When sample.type="two.sample", n.or.n1 denotes $n_1$, the n

p.hat.or.p1.hat

numeric vector of estimated proportions. When sample.type="one.sample", p.hat.or.p1.hat denotes the estimated value of $p$, the probability of success. When sample.type="two.sample", p.h

numeric vector of sample sizes for group 2. The default value is the value of n.or.n1. This argument is ignored when sample.type="one.sample". Missing (NA), undefined (NaN), and infinite (

p2.hat

numeric vector of estimated proportions for group 2. This argument is ignored when sample.type="one.sample". Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not a

conf.level

numeric vector of numbers between 0 and 1 indicating the confidence level associated with the confidence interval(s). The default value is conf.level=0.95.

sample.type

character string indicating whether this is a one-sample or two-sample confidence interval. When sample.type="one.sample", the computed half-width is based on a confidence interval for a single proportion. When sample.type="t

ci.method

character string indicating which method to use to construct the confidence interval. Possible values are "score" (the default), "exact", "adjusted Wald", and "Wald" (the "Wald" method

correct

logical scalar indicating whether to use the continuity correction when ci.method="score" or ci.method="Wald". The default value is correct=TRUE.

warn

logical scalar indicating whether to issue a warning when ci.method="Wald" for cases when the normal approximation to the binomial distribution probably is not accurate. The default value is warn=TRUE.

Value

a list with information about the half-widths, sample sizes, and estimated proportions. One-Sample Case (sample.type="one.sample"). When sample.type="one.sample", the function ciBinomHalfWidth returns a list with these components:
half.widththe half-width(s) of the confidence interval(s)
nthe sample size(s) associated with the confidence interval(s)
p.hatthe estimated proportion(s)
methodthe method used to construct the confidence interval(s)
Two-Sample Case (sample.type="two.sample"). When sample.type="two.sample", the function ciBinomHalfWidth returns a list with these components:
half.widththe half-width(s) of the confidence interval(s)
n1the sample size(s) for group 1 associated with the confidence interval(s)
p1.hatthe estimated proportion(s) for group 1
n2the sample size(s) for group 2 associated with the confidence interval(s)
p2.hatthe estimated proportion(s) for group 2
methodthe method used to construct the confidence interval(s)

Details

If the arguments n.or.n1, p.hat.or.p1.hat, n2, p2.hat, and conf.level are not all the same length, they are replicated to be the same length as the length of the longest argument. The values of p.hat.or.p1.hat and p2.hat are automatically adjusted to the closest legitimate values, given the user-supplied values of n.or.n1 and n2. For example, if n.or.n1=5, legitimate values for p.hat.or.p1.hat are 0, 0.2, 0.4, 0.6, 0.8 and 1. In this case, if the user supplies p.hat.or.p1.hat=0.45, then p.hat.or.p1.hat is reset to p.hat.or.p1.hat=0.4, and if the user supplies p.hat.or.p1.hat=0.55, then p.hat.or.p1.hat is reset to p.hat.or.p1.hat=0.6. In cases where the two closest legitimate values are equal distance from the user-suppled value of p.hat.or.p1.hat or p2.hat, the value closest to 0.5 is chosen since that will tend to yield the wider confidence interval. One-Sample Case (sample.type="one.sample"). [object Object],[object Object],[object Object],[object Object] Two-Sample Case (sample.type="two.sample"). [object Object],[object Object],[object Object]

References

Agresti, A., and B.A. Coull. (1998). Approximate is Better than "Exact" for Interval Estimation of Binomial Proportions. The American Statistician, 52(2), 119--126. Agresti, A., and B. Caffo. (2000). Simple and Effective Confidence Intervals for Proportions and Differences of Proportions Result from Adding Two Successes and Two Failures. The American Statistician, 54(4), 280--288. Berthouex, P.M., and L.C. Brown. (1994). Statistics for Environmental Engineers. Lewis Publishers, Boca Raton, FL, Chapters 2 and 15. Cochran, W.G. (1977). Sampling Techniques. John Wiley and Sons, New York, Chapter 3. Fisher, R.A., and F. Yates. (1963). Statistical Tables for Biological, Agricultural, and Medical Research. 6th edition. Hafner, New York, 146pp. Fleiss, J. L. (1981). Statistical Methods for Rates and Proportions. Second Edition. John Wiley and Sons, New York, Chapters 1-2. Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York, NY, Chapter 11. Millard, S.P., and Neerchal, N.K. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, Florida. Newcombe, R.G. (1998a). Two-Sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods. Statistics in Medicine, 17, 857--872. Newcombe, R.G. (1998b). Interval Estimation for the Difference Between Independent Proportions: Comparison of Eleven Methods. Statistics in Medicine, 17, 873--890. Ott, .W.R. (1995). Environmental Statistics and Data Analysis. Lewis Publishers, Boca Raton, FL, Chapter 4. USEPA. (1989b). Statistical Analysis of Ground-Water Monitoring Data at RCRA Facilities, Interim Final Guidance. EPA/530-SW-89-026. Office of Solid Waste, U.S. Environmental Protection Agency, Washington, D.C. USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C. p.6-38. Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ, Chapter 24.

Examples

Run this code

# Look at how the half-width of a one-sample confidence interval 
  # decreases with sample size:

  ciBinomHalfWidth(n.or.n1 = c(10, 50, 100, 500))
  #$half.width
  #[1] 0.26340691 0.13355486 0.09616847 0.04365873
  #
  #$n
  #[1]  10  50 100 500
  #
  #$p.hat
  #[1] 0.5 0.5 0.5 0.5
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"

  #----------------------------------------------------------------

  # Look at how the half-width of a one-sample confidence interval 
  # tends to decrease as the estimated value of p decreases below 
  # 0.5 or increases above 0.5:

  seq(0.2, 0.8, by = 0.1) 
  #[1] 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

  ciBinomHalfWidth(n.or.n1 = 30, p.hat = seq(0.2, 0.8, by = 0.1)) 
  #$half.width
  #[1] 0.1536299 0.1707256 0.1801322 0.1684587 0.1801322 0.1707256 
  #[7] 0.1536299
  #
  #$n
  #[1] 30 30 30 30 30 30 30
  #
  #$p.hat
  #[1] 0.2 0.3 0.4 0.5 0.6 0.7 0.8
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"

  #----------------------------------------------------------------

  # Look at how the half-width of a one-sample confidence interval 
  # increases with increasing confidence level:

  ciBinomHalfWidth(n.or.n1 = 20, conf.level = c(0.8, 0.9, 0.95, 0.99)) 
  #$half.width
  #[1] 0.1377380 0.1725962 0.2007020 0.2495523
  #
  #$n
  #[1] 20 20 20 20
  #
  #$p.hat
  #[1] 0.5 0.5 0.5 0.5
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"

  #----------------------------------------------------------------

  # Compare the half-widths for a one-sample 
  # confidence interval based on the different methods:

  ciBinomHalfWidth(n.or.n1 = 30, ci.method = "score")$half.width
  #[1] 0.1684587

  ciBinomHalfWidth(n.or.n1 = 30, ci.method = "exact")$half.width
  #[1] 0.1870297
 
  ciBinomHalfWidth(n.or.n1 = 30, ci.method = "adjusted Wald")$half.width
  #[1] 0.1684587

  ciBinomHalfWidth(n.or.n1 = 30, ci.method = "Wald")$half.width
  #[1] 0.1955861

  #----------------------------------------------------------------

  # Look at how the half-width of a two-sample 
  # confidence interval decreases with increasing 
  # sample sizes:

  ciBinomHalfWidth(n.or.n1 = c(10, 50, 100, 500), sample.type = "two")
  #$half.width
  #[1] 0.53385652 0.21402654 0.14719748 0.06335658
  #
  #$n1
  #[1]  10  50 100 500
  #
  #$p1.hat
  #[1] 0.5 0.5 0.5 0.5
  #
  #$n2
  #[1]  10  50 100 500
  #
  #$p2.hat
  #[1] 0.4 0.4 0.4 0.4
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"

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