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

Description

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

Usage

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  ciBinomN(half.width, p.hat.or.p1.hat = 0.5, p2.hat = 0.4, 
    conf.level = 0.95, sample.type = "one.sample", ratio = 1, 
    ci.method = "score", correct = TRUE, warn = TRUE, 
    n.or.n1.min = 2, n.or.n1.max = 10000, 
    tol.half.width = 5e-04, tol.p.hat = 5e-04, 
    tol = 1e-7, maxiter = 1000)

Arguments

half.width

numeric vector of (positive) half-widths. 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.

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 p2.hat or ratio is supplied.

ratio

numeric vector indicating the ratio of sample size in group 2 to sample size in group 1 (n_2/n_1). The default value is ratio=1. All values of ratio must be greater than or equal to 1. This argument is ignored if
sample.type="one.sample".

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.

n.or.n1.min

integer indicating the minimum allowed value for
n (sample.type="one.sample") or
n_1 (sample.type="two.sample").
The default value is n.or.n1.min=2.

n.or.n1.max

integer indicating the maximum allowed value for
n (sample.type="one.sample") or
n_1 (sample.type="two.sample").
The default value is n.or.n1.max=10000.

tol.half.width

numeric scalar indicating the tolerance to use for the half width for the search algorithm. The sample sizes are computed so that the actual half width is less than or equal to half.width + tol.half.width. The default value is tol.half.width=5e-04.

tol.p.hat

numeric scalar indicating the tolerance to use for the estimated proportion(s) for the search algorithm. For the one-sample case, the sample sizes are computed so that the absolute value of the difference between the user supplied value of p.hat.or.p1.hat and the actual estimated proportion is less than or equal to tol.p.hat. For the two-sample case, the sample sizes are computed so that the absolute value of the difference between the user supplied value of p.hat.or.p1.hat and the actual estimated proportion for group 1 is less than or equal to tol.p.hat, and the absolute value of the difference between the user supplied value of p2.hat and the actual estimated proportion for group 2 is less than or equal to tol.p.hat. The default value is tol.p.hat=0.005.

tol

positive scalar indicating the tolerance to use for the search algorithm (passed to uniroot). The default value is tol=1e-7.

maxiter

integer indicating the maximum number of iterations to use for the search algorithm (passed to uniroot). The default value is maxiter=1000.

Details

If the arguments half.width, p.hat.or.p1.hat, p2.hat, conf.level and ratio are not all the same length, they are replicated to be the same length as the length of the longest argument.

For the one-sample case, the arguments p.hat.or.p1.hat, tol.p.hat, half.width, and
tol.half.width must satisfy:
(p.hat.or.p1.hat + tol.p.hat + half.width + tol.half.width) <= 1,
and
(p.hat.or.p1.hat - tol.p.hat - half.width - tol.half.width) >= 0.

For the two-sample case, the arguments p.hat.or.p1.hat, p2.hat, tol.p.hat,
half.width, and tol.half.width must satisfy:
((p.hat.or.p1.hat + tol.p.hat) - (p2.hat - tol.p.hat) + half.width + tol.half.width) <= 1, and
((p.hat.or.p1.hat - tol.p.hat) - (p2.hat + tol.p.hat) - half.width - tol.half.width) >= -1.

The function ciBinomN uses the search algorithm in the function uniroot to call the function ciBinomHalfWidth to find the values of n (sample.type="one.sample") or n_1 and n_2
(sample.type="two.sample") that satisfy the requirements for the half-width, estimated proportions, and confidence level. See the Details section of the help file for ciBinomHalfWidth for more information.

Value

a list with information about the sample sizes, estimated proportions, and half-widths.

One-Sample Case (sample.type="one.sample").
When sample.type="one.sample", the function ciBinomN returns a list with these components:

n

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

p.hat

the estimated proportion(s)

half.width

the half-width(s) of the confidence interval(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 ciBinomN returns a list with these components:

n1

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

n2

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

p1.hat

the estimated proportion(s) for group 1

p2.hat

the estimated proportion(s) for group 2

half.width

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

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

ciBinomHalfWidth, uniroot, plotCiBinomDesign, ebinom,
binom.test, prop.test.

Examples

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

  ciBinomN(half.width = c(0.1, 0.05, 0.03))
  #$n
  #[1]   92  374 1030
  #
  #$p.hat
  #[1] 0.5 0.5 0.5
  #
  #$half.width
  #[1] 0.10010168 0.05041541 0.03047833
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"

  #----------

  # Note that the required sample size decreases if we are less 
  # stringent about how much the confidence interval width can 
  # deviate from the supplied value of the 'half.width' argument:

  ciBinomN(half.width = c(0.1, 0.05, 0.03), tol.half.width = 0.005)
  #$n
  #[1]  84 314 782
  #
  #$p.hat
  #[1] 0.5 0.5 0.5
  #
  #$half.width
  #[1] 0.10456066 0.05496837 0.03495833
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"

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

  # Look at how the required sample size for 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 

  ciBinomN(half.width = 0.1, p.hat = seq(0.2, 0.8, by = 0.1)) 
  #$n
  #[1]  70  90 100  92 100  90  70
  #
  #$p.hat
  #[1] 0.2 0.3 0.4 0.5 0.6 0.7 0.8
  #
  #$half.width
  #[1] 0.09931015 0.09839843 0.09910818 0.10010168 0.09910818 0.09839843
  #[7] 0.09931015
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"

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

  # Look at how the required sample size for a one-sample 
  # confidence interval increases with increasing confidence level:

  ciBinomN(half.width = 0.05, conf.level = c(0.8, 0.9, 0.95, 0.99)) 
  #$n
  #[1] 160 264 374 644
  #
  #$p.hat
  #[1] 0.5 0.5 0.5 0.5
  #
  #$half.width
  #[1] 0.05039976 0.05035948 0.05041541 0.05049152
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"

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

  # Compare required sample size for a one-sample 
  # confidence interval based on the different methods:

  ciBinomN(half.width = 0.05, ci.method = "score")
  #$n
  #[1] 374
  #
  #$p.hat
  #[1] 0.5
  #
  #$half.width
  #[1] 0.05041541
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"

  ciBinomN(half.width = 0.05, ci.method = "exact")
  #$n
  #[1] 394
  #
  #$p.hat
  #[1] 0.5
  #
  #$half.width
  #[1] 0.05047916
  #
  #$method
  #[1] "Exact"

  ciBinomN(half.width = 0.05, ci.method = "adjusted Wald")
  #$n
  #[1] 374
  #
  #$p.hat
  #[1] 0.5
  #
  #$half.width
  #[1] 0.05041541
  #
  #$method
  #[1] "Adjusted Wald normal approximation"

  ciBinomN(half.width = 0.05, ci.method = "Wald")
  #$n
  #[1] 398
  #
  #$p.hat
  #[1] 0.5
  #
  #$half.width
  #[1] 0.05037834
  #
  #$method
  #[1] "Wald normal approximation, with continuity correction"

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

  ## Not run: 
  # Look at how the required sample size of a two-sample 
  # confidence interval increases with decreasing 
  # required half-width:

  ciBinomN(half.width = c(0.1, 0.05, 0.03), sample.type = "two")  
  #$n1
  #[1]  210  778 2089
  #
  #$n2
  #[1]  210  778 2089
  #
  #$p1.hat
  #[1] 0.5000000 0.5000000 0.4997607
  #
  #$p2.hat
  #[1] 0.4000000 0.3997429 0.4001915
  #
  #$half.width
  #[1] 0.09943716 0.05047044 0.03049753
  #
  #$method
  #[1] "Score normal approximation, with continuity correction"
  
## End(Not run)

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