# Sample Size for a Nonparametric Prediction Interval for a Continuous Distribution Compute the sample size necessary for a nonparametric prediction interval to contain at least $k$ out of the next $m$ future observations with probability $(1-\alpha)100\%$ for a continuous distribution. ```r predIntNparN(k = m, m = 1, lpl.rank = ifelse(pi.type == "upper", 0, 1), n.plus.one.minus.upl.rank = ifelse(pi.type == "lower", 0, 1), pi.type = "two.sided", conf.level = 0.95, n.max = 5000, maxiter = 1000) ``` ## Arguments - `k`: vector of positive integers specifying the minimum number of future observations out of `m` that should be contained in the prediction interval. The default value is `k=m`. - `m`: vector of positive integers specifying the number of future observations. The default value is `m=1`. - `lpl.rank`: vector of positive integers indicating the rank of the order statistic to use for the lower bound of the prediction interval. If `pi.type="two-sided"` or `pi.type="lower"`, the default value is `lpl.rank=1` (implying the minimum value is used as the lower bound of the prediction interval). If `pi.type="upper"`, this argument 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"`. - `conf.level`: numeric vector of values between 0 and 1 indicating the confidence level associated with the prediction interval. The default value is `conf.level=0.95`. - `n.max`: positive integer greater than 1 indicating the maximum possible sample size. The default value is `n.max=5000`. - `maxiter`: positive integer indicating the maximum number of iterations to use in the `uniroot` search algorithm. The default value is `maxiter=1000`. ## Details If the arguments `k`, `m`, `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 `predIntNparN` initially computes the required sample size $n$ by solving Equation (11) or (12) in the help file for `predIntNpar` for $n$, depending on the value of the argument `pi.type`. If `k < m`, `lpl.rank > 1` (two-sided and lower prediction intervals only), or `n.plus.one.minus.upl.rank > 1` (two-sided and upper prediction intervals only), then this initial value of $n$ is used as the upper bound in a binary search based on Equation (8) in the help file for `predIntNpar` and is implemented via the function `uniroot` with the argument `tolerance` set to `1`. ## Returns vector of positive integers indicating the required sample size(s) for the specified nonparametric prediction interval(s). ## References See the help file for `predIntNpar`. ## Author(s) Steven P. Millard (EnvStats@ProbStatInfo.com ) ## Note See the help file for `predIntNpar`. ## See Also `predIntNpar`, `predIntNparConfLevel`, `plotPredIntNparDesign`. ## Examples ```r # Look at how the required sample size for a nonparametric prediction 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 predIntNparN(conf.level = seq(0.5, 0.9, by = 0.1)) #[1] 3 4 6 9 19 #---------- # Look at how the required sample size for a nonparametric prediction interval # increases with number of future observations (m): 1:5 #[1] 1 2 3 4 5 predIntNparN(m = 1:5) #[1] 39 78 116 155 193 #---------- # Look at how the required sample size for a nonparametric prediction interval # increases with minimum number of observations that must be contained within # the interval (k): predIntNparN(k = 1:5, m = 5) #[1] 4 7 13 30 193 #---------- # Look at how the required sample size for a nonparametric prediction interval # increases with the rank of the lower prediction limit: predIntNparN(lpl.rank = 1:5) #[1] 39 59 79 100 119 #========== # Example 18-3 of USEPA (2009, p.18-19) shows how to construct # a one-sided upper nonparametric prediction interval for the next # 4 future observations of trichloroethylene (TCE) at a downgradient well. # The data for this example are stored in EPA.09.Ex.18.3.TCE.df. # There are 6 monthly observations of TCE (ppb) at 3 background wells, # and 4 monthly observations of TCE at a compliance well. # Look at the data #----------------- EPA.09.Ex.18.3.TCE.df # Month Well Well.type TCE.ppb.orig TCE.ppb Censored #1 1 BW-1 Background <5 5.0 TRUE #2 2 BW-1 Background <5 5.0 TRUE #3 3 BW-1 Background 8 8.0 FALSE #... #22 4 CW-4 Compliance <5 5.0 TRUE #23 5 CW-4 Compliance 8 8.0 FALSE #24 6 CW-4 Compliance 14 14.0 FALSE longToWide(EPA.09.Ex.18.3.TCE.df, "TCE.ppb.orig", "Month", "Well", paste.row.name = TRUE) # BW-1 BW-2 BW-3 CW-4 #Month.1 <5 7 <5 #Month.2 <5 6.5 <5 #Month.3 8 <5 10.5 7.5 #Month.4 <5 6 <5 <5 #Month.5 9 12 <5 8 #Month.6 10 <5 9 14 # If we construct the prediction limit based on the background well # data using the maximum value as the upper prediction limit, # the associated confidence level is only 82%. #----------------------------------------------------------------- predIntNparConfLevel(n = 18, m = 4, pi.type = "upper") #[1] 0.8181818 # We would have to collect an additional 18 observations to achieve a # confidence level of at least 90%: predIntNparN(m = 4, pi.type = "upper", conf.level = 0.9) #[1] 36 predIntNparConfLevel(n = 36, m = 4, pi.type = "upper") #[1] 0.9 ```

predIntNparN function