Estimate Quantiles of a Binomial Distribution

Description

Estimate quantiles of a binomial distribution.

Usage

1
 eqbinom(x, size = NULL, p = 0.5, method = "mle/mme/mvue", digits = 0)

Arguments

x

numeric or logical vector of observations, or an object resulting from a call to an estimating function that assumes a binomial distribution (e.g., ebinom). If x is a vector of observations, then when size is not supplied, x must be a numeric vector of 0s (“failures”) and 1s (“successes”), or else a logical vector of FALSE values (“failures”) and TRUE values (“successes”). When size is supplied, x must be a non-negative integer containing the number of “successes” out of the number of trials indicated by size. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed.

size

positive integer indicating the of number of trials; size must be at least as large as the value of x.

p

numeric vector of probabilities for which quantiles will be estimated. All values of p must be between 0 and 1. The default value is p=0.5.

method

character string specifying the method of estimation. The only possible value is "mle/mme/mvue" (maximum likelihood, method of moments, and minimum variance unbiased). See the DETAILS section of the help file for ebinom for more information.

digits

an integer indicating the number of decimal places to round to when printing out the value of 100*p. The default value is digits=0.

Details

The function eqbinom returns estimated quantiles as well as estimates of the prob parameter.

Quantiles are estimated by 1) estimating the prob parameter by calling ebinom, and then 2) calling the function qbinom and using the estimated value for prob.

Value

If x is a numeric vector, eqbinom returns a list of class "estimate" containing the estimated quantile(s) and other information. See estimate.object for details.

If x is the result of calling an estimation function, eqbinom returns a list whose class is the same as x. The list contains the same components as x, as well as components called quantiles and quantile.method.

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). The binomial distribution is also used to compute an upper bound on the overall Type I error rate for deciding whether a facility or location is in compliance with some set standard. Assume the null hypothesis is that the facility is in compliance. If a test of hypothesis is conducted periodically over time to test compliance and/or several tests are performed during each time period, and the facility or location is always in compliance, and each single test has a Type I error rate of α, and the result of each test is independent of the result of any other test (usually not a reasonable assumption), then the number of times the facility is declared out of compliance when in fact it is in compliance is a binomial random variable with probability of “success” p=α being the probability of being declared out of compliance (see USEPA, 2009).

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.

Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.

Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York, NY, Chapter 11.

Johnson, N. L., S. Kotz, and A.W. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, Chapter 3.

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.

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

ebinom, Binomial, estimate.object.

Examples

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  # Generate 20 observations from a binomial distribution with 
  # parameters size=1 and prob=0.2, then estimate the 'prob' 
  # parameter and the 90'th percentile. 
  # (Note: the call to set.seed simply allows you to reproduce this example. 

  set.seed(251) 
  dat <- rbinom(20, size = 1, prob = 0.2) 
  eqbinom(dat, p = 0.9) 

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Binomial
  #
  #Estimated Parameter(s):          size = 20.0
  #                                 prob =  0.1
  #
  #Estimation Method:               mle/mme/mvue for 'prob'
  #
  #Estimated Quantile(s):           90'th %ile = 4
  #
  #Quantile Estimation Method:      Quantile(s) Based on
  #                                 mle/mme/mvue for 'prob' Estimators
  #
  #Data:                            dat
  #
  #Sample Size:                     20
  #
  #
  #

  #----------
  # Clean up

  rm(dat)

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