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

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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

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  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 number of observations from group 1.
Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.

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.hat.or.p1.hat denotes the estimated value of p_1, the probability of “success” in group 1.
Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.

n2

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 (Inf, -Inf) values are not allowed.

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 allowed.

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="two.sample", the computed half-width is based on a confidence interval for the difference between two proportions. The default value is sample.type="one.sample" unless the argument n2 or p2.hat is supplied.

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 is never recommended but is included for historical purposes). The exact method is only available for the one-sample case, i.e., when sample.type="one.sample".

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.

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").

ci.method="score"

The confidence interval for p based on the score method was developed by Wilson (1927) and is discussed by Newcombe (1998a), Agresti and Coull (1998), and Agresti and Caffo (2000). When ci=TRUE and ci.method="score", the function ebinom calls the R function prop.test to compute the confidence interval. This method has been shown to provide the best performance (in terms of actual coverage matching assumed coverage) of all the methods provided here, although unlike the exact method, the actual coverage can fall below the assumed coverage.

ci.method="exact"

The confidence interval for p based on the exact (Clopper-Pearson) method is discussed by Newcombe (1998a), Agresti and Coull (1998), and Zar (2010, pp.543-547). This is the method used in the R function binom.test. This method ensures the actual coverage is greater than or equal to the assumed coverage.

ci.method="Wald"

The confidence interval for p based on the Wald method (with or without a correction for continuity) is the usual “normal approximation” method and is discussed by Newcombe (1998a), Agresti and Coull (1998), Agresti and Caffo (2000), and Zar (2010, pp.543-547). This method is never recommended but is included for historical purposes.

ci.method="adjusted Wald"

The confidence interval for p based on the adjusted Wald method is discussed by Agresti and Coull (1998), Agresti and Caffo (2000), and Zar (2010, pp.543-547). This is a simple modification of the Wald method and performs surpringly well.

Two-Sample Case (sample.type="two.sample").

ci.method="score"

This method is presented in Newcombe (1998b) and is based on the score method developed by Wilson (1927) for the one-sample case. This is the method used by the R function prop.test. In a comparison of 11 methods, Newcombe (1998b) showed this method performs remarkably well.

ci.method="Wald"

The confidence interval for the difference between two proportions based on the Wald method (with or without a correction for continuity) is the usual “normal approximation” method and is discussed by Newcombe (1998b), Agresti and Caffo (2000), and Zar (2010, pp.549-552). This method is not recommended but is included for historical purposes.

ci.method="adjusted Wald"

This method is discussed by Agresti and Caffo (2000), and Zar (2010, pp.549-552). This is a simple modification of the Wald method and performs surpringly well.

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.width

the half-width(s) of the confidence interval(s)

n

the sample size(s) associated with the confidence interval(s)

p.hat

the estimated proportion(s)

method

the 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.width

the half-width(s) of the confidence interval(s)

n1

the sample size(s) for group 1 associated with the confidence interval(s)

p1.hat

the estimated proportion(s) for group 1

n2

the sample size(s) for group 2 associated with the confidence interval(s)

p2.hat

the estimated proportion(s) for group 2

method

the method used to construct the confidence interval(s)

Note

The binomial distribution is used to model processes with binary (Yes-No, Success-Failure, Heads-Tails, etc.) outcomes. It is assumed that the outcome of any one trial is independent of any other trial, and that the probability of “success”, p, is the same on each trial. A binomial discrete random variable X is the number of “successes” in n independent trials. A special case of the binomial distribution occurs when n=1, in which case X is also called a Bernoulli random variable.

In the context of environmental statistics, the binomial distribution is sometimes used to model the proportion of times a chemical concentration exceeds a set standard in a given period of time (e.g., Gilbert, 1987, p.143), or to compare the proportion of detects in a compliance well vs. a background well (e.g., USEPA, 1989b, Chapter 8, p.3-7). (However, USEPA 2009, p.8-27 recommends using the Wilcoxon rank sum test (wilcox.test) instead of comparing proportions.)

In the course of designing a sampling program, an environmental scientist may wish to determine the relationship between sample size, confidence level, and half-width if one of the objectives of the sampling program is to produce confidence intervals. The functions ciBinomHalfWidth, ciBinomN, and plotCiBinomDesign can be used to investigate these relationships for the case of binomial proportions.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

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.

See Also

ciBinomN, plotCiBinomDesign, ebinom, binom.test, prop.test.

Examples

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  # 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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