# tolIntNparCoverage: Coverage for Nonparametric Tolerance Interval for Continuous... In EnvStats: Package for Environmental Statistics, Including US EPA Guidance

## Description

Compute the coverage associated with a nonparametric tolerance interval for a continuous distribution given the sample size, confidence level, coverage type (β-content versus β-expectation), and ranks of the order statistics used for the interval.

## Usage

 ```1 2 3 4``` ``` tolIntNparCoverage(n, 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

 `n` vector of positive integers specifying the sample sizes. Missing (`NA`), undefined (`NaN`), and infinite (`Inf`, `-Inf`) values are not allowed. `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"` (β-content; the default), and `"expectation"` (β-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 `n`, `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 β-content tolerance intervals are constructed and how the coverage associated with the tolerance interval is computed based on specified values for the sample size, the confidence level, and the ranks of the order statistics used for the bounds of the tolerance interval.

## Value

vector of values between 0 and 1 indicating the coverage associated with the specified nonparametric tolerance interval.

## 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`, `tolIntNparConfLevel`, `tolIntNparCoverage`, and `plotTolIntNparDesign` can be used to investigate these relationships for constructing nonparametric tolerance intervals.

## Author(s)

Steven P. Millard ([email protected])

## References

See the help file for `tolIntNpar`.

`tolIntNpar`, `tolIntNparN`, `tolIntNparConfLevel`, `plotTolIntNparDesign`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37``` ``` # Look at how the coverage of a nonparametric tolerance interval increases with # increasing sample size: seq(10, 60, by=10) #[1] 10 20 30 40 50 60 round(tolIntNparCoverage(n = seq(10, 60, by = 10)), 2) #[1] 0.61 0.78 0.85 0.89 0.91 0.92 #--------- # Look at how the coverage of a nonparametric tolerance interval decreases with # increasing confidence level: seq(0.5, 0.9, by=0.1) #[1] 0.5 0.6 0.7 0.8 0.9 round(tolIntNparCoverage(n = 10, conf.level = seq(0.5, 0.9, by = 0.1)), 2) #[1] 0.84 0.81 0.77 0.73 0.66 #---------- # Look at how the coverage of a nonparametric tolerance interval decreases with # the rank of the lower tolerance limit: round(tolIntNparCoverage(n = 60, ltl.rank = 1:5), 2) #[1] 0.92 0.90 0.88 0.85 0.83 #========== # 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 ```