# Confidence Level of Simultaneous Nonparametric Prediction Interval for Continuous Distribution 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. ```r 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. ## Returns vector of values between 0 and 1 indicating the confidence level associated with the specified simultaneous nonparametric prediction interval. ## References See the help file for `predIntNparSimultaneous`. ## Author(s) Steven P. Millard (EnvStats@ProbStatInfo.com ) ## Note See the help file for `predIntNparSimultaneous`. ## See Also `predIntNparSimultaneous`, `predIntNparSimultaneousN`, `plotPredIntNparSimultaneousDesign`, `predIntNparSimultaneousTestPower`, `predIntNpar`, `tolIntNpar`. ## Examples ```r # 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) ```

predIntNparSimultaneousConfLevel function