predIntNparSimultaneousConfLevel {EnvStats}

Confidence Level of Simultaneous Nonparametric Prediction Interval for Continuous Distribution

Description

Compute the confidence level associated with a nonparametric simultaneous prediction interval based on one of three possible rules: k-of-m, California, or Modified California. Observations are assumed to come from from a continuous distribution.

Usage

  predIntNparSimultaneousConfLevel(n, n.median = 1, k = 1, m = 2, r = 1, 
    rule = "k.of.m", lpl.rank = ifelse(pi.type == "upper", 0, 1), 
    n.plus.one.minus.upl.rank = ifelse(pi.type == "lower", 0, 1), 
    pi.type = "upper", integrate.args.list = NULL)

Arguments

n

vector of positive integers specifying the sample sizes. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are not allowed.

n.median

vector of positive odd integers specifying the sample size associated with the future medians. The default value is n.median=1 (i.e., individual observations). Note that all future medians must be based on the same sample size.

k

for the k-of-m rule (rule="k.of.m"), a vector of positive integers specifying the minimum number of observations (or medians) out of m observations (or medians) (all obtained on one future sampling “occassion”) the prediction interval should contain. The default value is k=1. This argument is ignored when the argument rule is not equal to "k.of.m".

m

vector of positive integers specifying the maximum number of future observations (or medians) on one future sampling “occasion”. The default value is m=2, except when rule="Modified.CA", in which case this argument is ignored and m is automatically set equal to 4.

r

vector of positive integers specifying the number of future sampling “occasions”. The default value is r=1.

rule

character string specifying which rule to use. The possible values are "k.of.m" (k-of-m rule; the default), "CA" (California rule), and "Modified.CA" (modified California rule).

lpl.rank

vector of positive integers indicating the rank of the order statistic to use for the lower bound of the prediction interval. When pi.type="lower", the default value is lpl.rank=1 (implying the minimum value of x is used as the lower bound of the prediction interval). When pi.type="upper", the argument lpl.rank is set equal to 0.

n.plus.one.minus.upl.rank

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

pi.type

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

integrate.args.list

list of arguments to supply to the integrate function. The default value is NULL.

Details

If the arguments n, k, m, r, lpl.rank, and n.plus.one.minus.upl.rank are not all the same length, they are replicated to be the same length as the length of the longest argument.

The function predIntNparSimultaneousConfLevel computes the confidence level based on Equation (8), (9), or (10) in the help file for predIntNparSimultaneous, depending on the value of the argument rule.

Note that when rule="k.of.m" and r=1, this is equivalent to a standard nonparametric prediction interval and you can use the function predIntNparConfLevel instead.

Value

vector of values between 0 and 1 indicating the confidence level associated with the specified simultaneous nonparametric prediction interval.

Note

See the help file for predIntNparSimultaneous.

Author(s)

Steven P. Millard ([e-mail])

References

See the help file for predIntNparSimultaneous.

See Also

predIntNparSimultaneous, predIntNparSimultaneousN, plotPredIntNparSimultaneousDesign, predIntNparSimultaneousTestPower, predIntNpar, tolIntNpar.

Examples

  # For the 1-of-3 rule with r=20 future sampling occasions, look at how the 
  # confidence level of a simultaneous nonparametric prediction interval 
  # increases with increasing sample size:

  seq(5, 25, by = 5) 
  #[1] 5 10 15 20 25 

  conf <- predIntNparSimultaneousConfLevel(n = seq(5, 25, by = 5), 
    k = 1, m = 3, r = 20)
  round(conf, 2) 
  #[1] 0.82 0.95 0.98 0.99 0.99

  #----------

  # For the 1-of-m rule with r=20 future sampling occasions, look at how the 
  # confidence level of a simultaneous nonparametric prediction interval 
  # increases as the number of future observations increases:

  1:5
  #[1] 1 2 3 4 5

  conf <- predIntNparSimultaneousConfLevel(n = 10, k = 1, m = 1:5, r = 20)
  round(conf, 2) 
  #[1] 0.33 0.81 0.95 0.98 0.99

  #----------

  # For the 1-of-3 rule, look at how the confidence level of a simultaneous 
  # nonparametric prediction interval decreases with number of future sampling 
  # occasions (r):

  seq(5, 20, by = 5)
  #[1]  5 10 15 20

  conf <- predIntNparSimultaneousConfLevel(n = 10, k = 1, m = 3, 
    r = seq(5, 20, by = 5))

  round(conf, 2) 
  #[1] 0.98 0.97 0.96 0.95

  #----------

  # For the 1-of-3 rule with r=20 future sampling occasions, look at how the 
  # confidence level of a simultaneous nonparametric prediction interval 
  # decreases as the rank of the upper prediction limit decreases:

  conf <- predIntNparSimultaneousConfLevel(n = 10, k = 1, m = 3, r = 20, 
    n.plus.one.minus.upl.rank = 1:5)

  round(conf, 2) 
  #[1] 0.95 0.82 0.63 0.43 0.25

  #----------

  # Clean up
  #---------
  rm(conf)

  #==========

  # Example 19-5 of USEPA (2009, p. 19-33) shows how to compute nonparametric upper 
  # simultaneous prediction limits for various rules based on trace mercury data (ppb) 
  # collected in the past year from a site with four background wells and 10 compliance 
  # wells (data for two of the compliance wells  are shown in the guidance document).  
  # The facility must monitor the 10 compliance wells for five constituents 
  # (including mercury) annually.
  
  # Here we will compute the confidence level associated with two different sampling plans: 
  # 1) the 1-of-2 retesting plan for a median of order 3 using the background maximum and 
  # 2) the 1-of-4 plan on individual observations using the 3rd highest background value.
  # The data for this example are stored in EPA.09.Ex.19.5.mercury.df.

  # We will pool data from 4 background wells that were sampled on 
  # a number of different occasions, giving us a sample size of 
  # n = 20 to use to construct the prediction limit.

  # There are 10 compliance wells and we will monitor 5 different 
  # constituents at each well annually.  For this example, USEPA (2009) 
  # recommends setting r to the product of the number of compliance wells and 
  # the number of evaluations per year.  

  # To determine the minimum confidence level we require for 
  # the simultaneous prediction interval, USEPA (2009) recommends 
  # setting the maximum allowed individual Type I Error level per constituent to:
 
  # 1 - (1 - SWFPR)^(1 / Number of Constituents)
  
  # which translates to setting the confidence limit to 

  # (1 - SWFPR)^(1 / Number of Constituents)

  # where SWFPR = site-wide false positive rate.  For this example, we 
  # will set SWFPR = 0.1.  Thus, the required individual Type I Error level 
  # and confidence level per constituent are given as follows:

  # n  = 20 based on 4 Background Wells
  # nw = 10 Compliance Wells
  # nc =  5 Constituents
  # ne =  1 Evaluation per year

  n  <- 20
  nw <- 10
  nc <-  5
  ne <-  1

  # Set number of future sampling occasions r to 
  # Number Compliance Wells x Number Evaluations per Year
  r  <-  nw * ne

  conf.level <- (1 - 0.1)^(1 / nc)
  conf.level
  #[1] 0.9791484

  # So the required confidence level is 0.98, or 98%.
  # Now determine the confidence level associated with each plan.
  # Note that both plans achieve the required confidence level.
 
  # 1) the 1-of-2 retesting plan for a median of order 3 using the 
  #    background maximum

  predIntNparSimultaneousConfLevel(n = 20, n.median = 3, k = 1, m = 2, r = r)
  #[1] 0.9940354


  # 2) the 1-of-4 plan on individual observations using the 3rd highest 
  #    background value.

  predIntNparSimultaneousConfLevel(n = 20, k = 1, m = 4, r = r, 
    n.plus.one.minus.upl.rank = 3)
  #[1] 0.9864909
 
  #==========

  # Cleanup
  #--------
  rm(n, nw, nc, ne, r, conf.level)