tolIntNparN {EnvStats}

Sample Size for Nonparametric Tolerance Interval for Continuous Distribution

Description

Compute the sample size necessary for a nonparametric tolerance interval (for a continuous distribution) with a specified coverage and, in the case of a \beta-content tolerance interval, a specified confidence level, given the ranks of the order statistics used for the interval.

Usage

  tolIntNparN(coverage = 0.95, conf.level = 0.95, cov.type = "content", 
    ltl.rank = ifelse(ti.type == "upper", 0, 1), 
    n.plus.one.minus.utl.rank = ifelse(ti.type == "lower", 0, 1),  
    ti.type = "two.sided")

Arguments

coverage

numeric vector of values between 0 and 1 indicating the desired coverage of the tolerance interval.

conf.level

numeric vector of values between 0 and 1 indicating the confidence level of the tolerance interval.

cov.type

character string specifying the coverage type for the tolerance interval. The possible values are "content" (\beta-content; the default), and "expectation" (\beta-expectation).

ltl.rank

vector of positive integers indicating the rank of the order statistic to use for the lower bound of the tolerance interval. If ti.type="two-sided" or
ti.type="lower", the default value is ltl.rank=1 (implying the minimum value of x is used as the lower bound of the tolerance interval). If
ti.type="upper", this argument is set equal to 0.

n.plus.one.minus.utl.rank

vector of positive integers related to the rank of the order statistic to use for the upper bound of the tolerance interval. A value of n.plus.one.minus.utl.rank=1 (the default) means use the first largest value, and in general a value of
n.plus.one.minus.utl.rank=i means use the i'th largest value. If
ti.type="lower", this argument is set equal to 0.

ti.type

character string indicating what kind of tolerance interval to compute. The possible values are "two-sided" (the default), "lower", and "upper".

Details

If the arguments coverage, conf.level, ltl.rank, and n.plus.one.minus.utl.rank are not all the same length, they are replicated to be the same length as the length of the longest argument.

The help file for tolIntNpar explains how nonparametric tolerance intervals are constructed.

Computing Required Sample Size for a \beta-Content Tolerance Interval (cov.type="content")
For a \beta-content tolerance interval, if the coverage C=\beta is specified, then the associated confidence level (1-\alpha)100\% is computed as:

1 - \alpha = 1 - F(\beta, v-u, w+u) \;\;\;\;\;\; (1)

where F(y, \delta, \gamma) denotes the cumulative distribution function of a beta random variable with parameters shape1=\delta and shape2=\gamma evaluated at y. The value of 1-\alpha is determined by the argument conf.level. The value of \beta is determined by the argument coverage. The value of u is determined by the argument ltl.rank. The value of w is determined by the argument
n.plus.one.minus.utl.rank. Once these values have been determined, the above equation can be solved implicitly for n, since

v = n + 1 - w \;\;\;\;\;\; (2)

Computing Required Sample Size for a \beta-Expectation Tolerance Interval (cov.type="expectation")
For a \beta-expectation tolerance interval, the expected coverage is simply the mean of a beta random variable with parameters shape1=v-u and shape2=w+u, which is given by:

E(C) = \frac{v-u}{n+1} \;\;\;\;\;\; (3)

or, using Equation (2) above, we can re-write the formula for the expected coverage as:

E(C) = \frac{n+1-w-u}{n+1} = 1 - \frac{u+w}{n+1} \;\;\;\;\;\; (4)

Thus, for user-specified values of u (ltl.rank), w (n.plus.one.minus.utl.rank), and expected coverage, the required sample size is computed as:

n = Ceiling\{ [ \frac{u+w}{1-E(C)} ] - 1 \} \;\;\;\;\;\; (5)

where Ceiling(x) denotes the smallest integer greater than or equal to x. (See the R help file for ceiling).

Value

A vector of positive integers indicating the required sample size(s) for the specified nonparametric tolerance interval(s).

Note

See the help file for tolIntNpar.

In the course of designing a sampling program, an environmental scientist may wish to determine the relationship between sample size, coverage, and confidence level if one of the objectives of the sampling program is to produce tolerance intervals. The functions tolIntNparN, tolIntNparCoverage, tolIntNparConfLevel, and plotTolIntNparDesign can be used to investigate these relationships for constructing nonparametric tolerance intervals.

Author(s)

Steven P. Millard ([e-mail])

References

See the help file for tolIntNpar.

See Also

tolIntNpar, tolIntNparConfLevel, tolIntNparCoverage, plotTolIntNparDesign.

Examples

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

  seq(0.5, 0.9, by = 0.1) 
  #[1] 0.5 0.6 0.7 0.8 0.9 

  tolIntNparN(conf.level = seq(0.5, 0.9, by = 0.1)) 
  #[1] 34 40 49 59 77

  #----------

  # Look at how the required sample size for a nonparametric tolerance interval increases 
  # with increasing coverage:

  tolIntNparN(coverage = seq(0.5, 0.9, by = 0.1)) 
  #[1]  8 10 14 22 46

  #----------

  # Look at how the required sample size for a nonparametric tolerance interval increases 
  # with the rank of the lower tolerance limit:

  tolIntNparN(ltl.rank = 1:5) 
  #[1]  93 124 153 181 208

  #==========

  # Example 17-4 on page 17-21 of USEPA (2009) uses copper concentrations (ppb) from 3 
  # background wells to set an upper limit for 2 compliance wells.  The maximum value from 
  # the 3 wells is set to the 95% confidence upper tolerance limit, and we need to 
  # determine the coverage of this tolerance interval.  

  tolIntNparCoverage(n = 24, conf.level = 0.95, ti.type = "upper")
  #[1] 0.8826538

  # Here we will modify the example and determine the sample size required to produce 
  # a tolerance interval with 95% confidence level AND 95% coverage. 

  tolIntNparN(coverage = 0.95, conf.level = 0.95, ti.type = "upper")
  #[1] 59